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Type | Label | Description |
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Statement | ||
Theorem | 2ndconst 6001 |
The mapping of a restriction of the ![]() |
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Theorem | dfmpt2 6002* |
Alternate definition for the maps-to notation df-mpt2 5671 (although it
requires that ![]() |
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Theorem | cnvf1olem 6003 | Lemma for cnvf1o 6004. (Contributed by Mario Carneiro, 27-Apr-2014.) |
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Theorem | cnvf1o 6004* | Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
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Theorem | f2ndf 6005 |
The ![]() ![]() ![]() ![]() |
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Theorem | fo2ndf 6006 |
The ![]() ![]() ![]() ![]() |
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Theorem | f1o2ndf1 6007 |
The ![]() ![]() ![]() ![]() |
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Theorem | algrflem 6008 | Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
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Theorem | algrflemg 6009 | Lemma for algrf 11366 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.) |
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Theorem | xporderlem 6010* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
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Theorem | poxp 6011* | A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
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Theorem | spc2ed 6012* | Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
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Theorem | cnvoprab 6013* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
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Theorem | f1od2 6014* | Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
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Theorem | disjxp1 6015* | The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | disjsnxp 6016* | The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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The following theorems are about maps-to operations (see df-mpt2 5671) where the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 5740, ovmpt2x 5787 and fmpt2x 5984). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short. | ||
Theorem | opeliunxp2f 6017* |
Membership in a union of Cartesian products, using bound-variable
hypothesis for ![]() |
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Theorem | mpt2xopn0yelv 6018* | If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
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Theorem | mpt2xopoveq 6019* | Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.) |
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Theorem | mpt2xopovel 6020* | Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.) |
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Theorem | sprmpt2 6021* | The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
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Theorem | isprmpt2 6022* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
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Syntax | ctpos 6023 | The transposition of a function. |
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Definition | df-tpos 6024* |
Define the transposition of a function, which is a function
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Theorem | tposss 6025 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposeq 6026 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposeqd 6027 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
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Theorem | tposssxp 6028 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
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Theorem | reltpos 6029 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | brtpos2 6030 |
Value of the transposition at a pair ![]() ![]() ![]() ![]() ![]() |
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Theorem | brtpos0 6031 | The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | reldmtpos 6032 |
Necessary and sufficient condition for ![]() ![]() |
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Theorem | brtposg 6033 | The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) |
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Theorem | ottposg 6034 | The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) |
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Theorem | dmtpos 6035 |
The domain of tpos ![]() ![]() ![]() |
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Theorem | rntpos 6036 |
The range of tpos ![]() ![]() ![]() |
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Theorem | tposexg 6037 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | ovtposg 6038 |
The transposition swaps the arguments in a two-argument function. When
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Theorem | tposfun 6039 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | dftpos2 6040* |
Alternate definition of tpos when ![]() |
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Theorem | dftpos3 6041* |
Alternate definition of tpos when ![]() |
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Theorem | dftpos4 6042* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpostpos 6043 |
Value of the double transposition for a general class ![]() |
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Theorem | tpostpos2 6044 | Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
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Theorem | tposfn2 6045 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfo2 6046 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf2 6047 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf12 6048 | Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf1o2 6049 | Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfo 6050 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf 6051 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfn 6052 | Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpos0 6053 | Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposco 6054 | Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpossym 6055* | Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tposeqi 6056 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposex 6057 | A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | nftpos 6058 | Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposoprab 6059* | Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposmpt2 6060* | Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | pwuninel2 6061 | The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | 2pwuninelg 6062 | The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.) |
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Theorem | iunon 6063* |
The indexed union of a set of ordinal numbers ![]() ![]() ![]() ![]() |
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Syntax | wsmo 6064 | Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals. |
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Definition | df-smo 6065* | Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.) |
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Theorem | dfsmo2 6066* | Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.) |
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Theorem | issmo 6067* |
Conditions for which ![]() |
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Theorem | issmo2 6068* | Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.) |
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Theorem | smoeq 6069 | Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.) |
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Theorem | smodm 6070 | The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.) |
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Theorem | smores 6071 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
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Theorem | smores3 6072 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.) |
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Theorem | smores2 6073 | A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
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Theorem | smodm2 6074 | The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.) |
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Theorem | smofvon2dm 6075 | The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.) |
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Theorem | iordsmo 6076 | The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.) |
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Theorem | smo0 6077 | The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.) |
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Theorem | smofvon 6078 |
If ![]() ![]() ![]() ![]() |
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Theorem | smoel 6079 |
If ![]() ![]() ![]() ![]() |
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Theorem | smoiun 6080* | The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.) |
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Theorem | smoiso 6081 |
If ![]() ![]() ![]() ![]() |
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Theorem | smoel2 6082 | A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.) |
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Syntax | crecs 6083 | Notation for a function defined by strong transfinite recursion. |
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Definition | df-recs 6084* |
Define a function recs![]() ![]() ![]() ![]() ![]() (Contributed by Stefan O'Rear, 18-Jan-2015.) |
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Theorem | recseq 6085 | Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
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Theorem | nfrecs 6086 | Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
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Theorem | tfrlem1 6087* | A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
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Theorem | tfrlem3ag 6088* |
Lemma for transfinite recursion. This lemma just changes some bound
variables in ![]() |
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Theorem | tfrlem3a 6089* |
Lemma for transfinite recursion. Let ![]() ![]() |
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Theorem | tfrlem3 6090* |
Lemma for transfinite recursion. Let ![]() ![]() |
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Theorem | tfrlem3-2d 6091* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
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Theorem | tfrlem4 6092* |
Lemma for transfinite recursion. ![]() ![]() |
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Theorem | tfrlem5 6093* | Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
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Theorem | recsfval 6094* | Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.) |
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Theorem | tfrlem6 6095* | Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
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Theorem | tfrlem7 6096* | Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
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Theorem | tfrlem8 6097* | Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.) |
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Theorem | tfrlem9 6098* | Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.) |
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Theorem | tfrfun 6099 | Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.) |
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Theorem | tfr2a 6100 | A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.) |
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