Theorem List for Intuitionistic Logic Explorer - 9501-9600 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | rpdivcld 9501 |
Closure law for division of positive reals. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | ltrecd 9502 |
The reciprocal of both sides of 'less than'. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | lerecd 9503 |
The reciprocal of both sides of 'less than or equal to'. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | ltrec1d 9504 |
Reciprocal swap in a 'less than' relation. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | lerec2d 9505 |
Reciprocal swap in a 'less than or equal to' relation. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | lediv2ad 9506 |
Division of both sides of 'less than or equal to' into a nonnegative
number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltdiv2d 9507 |
Division of a positive number by both sides of 'less than'.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lediv2d 9508 |
Division of a positive number by both sides of 'less than or equal to'.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ledivdivd 9509 |
Invert ratios of positive numbers and swap their ordering.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | divge1 9510 |
The ratio of a number over a smaller positive number is larger than 1.
(Contributed by Glauco Siliprandi, 5-Apr-2020.)
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Theorem | divlt1lt 9511 |
A real number divided by a positive real number is less than 1 iff the
real number is less than the positive real number. (Contributed by AV,
25-May-2020.)
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Theorem | divle1le 9512 |
A real number divided by a positive real number is less than or equal to 1
iff the real number is less than or equal to the positive real number.
(Contributed by AV, 29-Jun-2021.)
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Theorem | ledivge1le 9513 |
If a number is less than or equal to another number, the number divided by
a positive number greater than or equal to one is less than or equal to
the other number. (Contributed by AV, 29-Jun-2021.)
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Theorem | ge0p1rpd 9514 |
A nonnegative number plus one is a positive number. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | rerpdivcld 9515 |
Closure law for division of a real by a positive real. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | ltsubrpd 9516 |
Subtracting a positive real from another number decreases it.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltaddrpd 9517 |
Adding a positive number to another number increases it. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | ltaddrp2d 9518 |
Adding a positive number to another number increases it. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | ltmulgt11d 9519 |
Multiplication by a number greater than 1. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | ltmulgt12d 9520 |
Multiplication by a number greater than 1. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | gt0divd 9521 |
Division of a positive number by a positive number. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | ge0divd 9522 |
Division of a nonnegative number by a positive number. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | rpgecld 9523 |
A number greater or equal to a positive real is positive real.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | divge0d 9524 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmul1d 9525 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmul2d 9526 |
Multiplication of both sides of 'less than' by a positive number.
Theorem I.19 of [Apostol] p. 20.
(Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | lemul1d 9527 |
Multiplication of both sides of 'less than or equal to' by a positive
number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lemul2d 9528 |
Multiplication of both sides of 'less than or equal to' by a positive
number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltdiv1d 9529 |
Division of both sides of 'less than' by a positive number.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lediv1d 9530 |
Division of both sides of a less than or equal to relation by a positive
number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmuldivd 9531 |
'Less than' relationship between division and multiplication.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmuldiv2d 9532 |
'Less than' relationship between division and multiplication.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lemuldivd 9533 |
'Less than or equal to' relationship between division and
multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | lemuldiv2d 9534 |
'Less than or equal to' relationship between division and
multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | ltdivmuld 9535 |
'Less than' relationship between division and multiplication.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltdivmul2d 9536 |
'Less than' relationship between division and multiplication.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ledivmuld 9537 |
'Less than or equal to' relationship between division and
multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ledivmul2d 9538 |
'Less than or equal to' relationship between division and
multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmul1dd 9539 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | ltmul2dd 9540 |
Multiplication of both sides of 'less than' by a positive number.
Theorem I.19 of [Apostol] p. 20.
(Contributed by Mario Carneiro,
30-May-2016.)
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Theorem | ltdiv1dd 9541 |
Division of both sides of 'less than' by a positive number.
(Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | lediv1dd 9542 |
Division of both sides of a less than or equal to relation by a
positive number. (Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | lediv12ad 9543 |
Comparison of ratio of two nonnegative numbers. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | ltdiv23d 9544 |
Swap denominator with other side of 'less than'. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | lediv23d 9545 |
Swap denominator with other side of 'less than or equal to'.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | mul2lt0rlt0 9546 |
If the result of a multiplication is strictly negative, then
multiplicands are of different signs. (Contributed by Thierry Arnoux,
19-Sep-2018.)
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Theorem | mul2lt0rgt0 9547 |
If the result of a multiplication is strictly negative, then
multiplicands are of different signs. (Contributed by Thierry Arnoux,
19-Sep-2018.)
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Theorem | mul2lt0llt0 9548 |
If the result of a multiplication is strictly negative, then
multiplicands are of different signs. (Contributed by Thierry Arnoux,
19-Sep-2018.)
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Theorem | mul2lt0lgt0 9549 |
If the result of a multiplication is strictly negative, then
multiplicands are of different signs. (Contributed by Thierry Arnoux,
2-Oct-2018.)
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Theorem | mul2lt0np 9550 |
The product of multiplicands of different signs is negative.
(Contributed by Jim Kingdon, 25-Feb-2024.)
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Theorem | mul2lt0pn 9551 |
The product of multiplicands of different signs is negative.
(Contributed by Jim Kingdon, 25-Feb-2024.)
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Theorem | lt2mul2divd 9552 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | nnledivrp 9553 |
Division of a positive integer by a positive number is less than or equal
to the integer iff the number is greater than or equal to 1. (Contributed
by AV, 19-Jun-2021.)
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Theorem | nn0ledivnn 9554 |
Division of a nonnegative integer by a positive integer is less than or
equal to the integer. (Contributed by AV, 19-Jun-2021.)
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Theorem | addlelt 9555 |
If the sum of a real number and a positive real number is less than or
equal to a third real number, the first real number is less than the third
real number. (Contributed by AV, 1-Jul-2021.)
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4.5.2 Infinity and the extended real number
system (cont.)
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Syntax | cxne 9556 |
Extend class notation to include the negative of an extended real.
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Syntax | cxad 9557 |
Extend class notation to include addition of extended reals.
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Syntax | cxmu 9558 |
Extend class notation to include multiplication of extended reals.
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Definition | df-xneg 9559 |
Define the negative of an extended real number. (Contributed by FL,
26-Dec-2011.)
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Definition | df-xadd 9560* |
Define addition over extended real numbers. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Definition | df-xmul 9561* |
Define multiplication over extended real numbers. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Theorem | ltxr 9562 |
The 'less than' binary relation on the set of extended reals.
Definition 12-3.1 of [Gleason] p. 173.
(Contributed by NM,
14-Oct-2005.)
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Theorem | elxr 9563 |
Membership in the set of extended reals. (Contributed by NM,
14-Oct-2005.)
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Theorem | xrnemnf 9564 |
An extended real other than minus infinity is real or positive infinite.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xrnepnf 9565 |
An extended real other than plus infinity is real or negative infinite.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xrltnr 9566 |
The extended real 'less than' is irreflexive. (Contributed by NM,
14-Oct-2005.)
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Theorem | ltpnf 9567 |
Any (finite) real is less than plus infinity. (Contributed by NM,
14-Oct-2005.)
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Theorem | 0ltpnf 9568 |
Zero is less than plus infinity (common case). (Contributed by David A.
Wheeler, 8-Dec-2018.)
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Theorem | mnflt 9569 |
Minus infinity is less than any (finite) real. (Contributed by NM,
14-Oct-2005.)
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Theorem | mnflt0 9570 |
Minus infinity is less than 0 (common case). (Contributed by David A.
Wheeler, 8-Dec-2018.)
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Theorem | mnfltpnf 9571 |
Minus infinity is less than plus infinity. (Contributed by NM,
14-Oct-2005.)
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Theorem | mnfltxr 9572 |
Minus infinity is less than an extended real that is either real or plus
infinity. (Contributed by NM, 2-Feb-2006.)
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Theorem | pnfnlt 9573 |
No extended real is greater than plus infinity. (Contributed by NM,
15-Oct-2005.)
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Theorem | nltmnf 9574 |
No extended real is less than minus infinity. (Contributed by NM,
15-Oct-2005.)
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Theorem | pnfge 9575 |
Plus infinity is an upper bound for extended reals. (Contributed by NM,
30-Jan-2006.)
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Theorem | 0lepnf 9576 |
0 less than or equal to positive infinity. (Contributed by David A.
Wheeler, 8-Dec-2018.)
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Theorem | nn0pnfge0 9577 |
If a number is a nonnegative integer or positive infinity, it is greater
than or equal to 0. (Contributed by Alexander van der Vekens,
6-Jan-2018.)
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Theorem | mnfle 9578 |
Minus infinity is less than or equal to any extended real. (Contributed
by NM, 19-Jan-2006.)
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Theorem | xrltnsym 9579 |
Ordering on the extended reals is not symmetric. (Contributed by NM,
15-Oct-2005.)
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Theorem | xrltnsym2 9580 |
'Less than' is antisymmetric and irreflexive for extended reals.
(Contributed by NM, 6-Feb-2007.)
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Theorem | xrlttr 9581 |
Ordering on the extended reals is transitive. (Contributed by NM,
15-Oct-2005.)
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Theorem | xrltso 9582 |
'Less than' is a weakly linear ordering on the extended reals.
(Contributed by NM, 15-Oct-2005.)
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Theorem | xrlttri3 9583 |
Extended real version of lttri3 7844. (Contributed by NM, 9-Feb-2006.)
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Theorem | xrltle 9584 |
'Less than' implies 'less than or equal' for extended reals. (Contributed
by NM, 19-Jan-2006.)
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Theorem | xrltled 9585 |
'Less than' implies 'less than or equal to' for extended reals.
Deduction form of xrltle 9584. (Contributed by Glauco Siliprandi,
11-Dec-2019.)
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Theorem | xrleid 9586 |
'Less than or equal to' is reflexive for extended reals. (Contributed by
NM, 7-Feb-2007.)
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Theorem | xrleidd 9587 |
'Less than or equal to' is reflexive for extended reals. Deduction form
of xrleid 9586. (Contributed by Glauco Siliprandi,
26-Jun-2021.)
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Theorem | xrletri3 9588 |
Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
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Theorem | xrlelttr 9589 |
Transitive law for ordering on extended reals. (Contributed by NM,
19-Jan-2006.)
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Theorem | xrltletr 9590 |
Transitive law for ordering on extended reals. (Contributed by NM,
19-Jan-2006.)
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Theorem | xrletr 9591 |
Transitive law for ordering on extended reals. (Contributed by NM,
9-Feb-2006.)
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Theorem | xrlttrd 9592 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
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Theorem | xrlelttrd 9593 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
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Theorem | xrltletrd 9594 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
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Theorem | xrletrd 9595 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
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Theorem | xrltne 9596 |
'Less than' implies not equal for extended reals. (Contributed by NM,
20-Jan-2006.)
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Theorem | nltpnft 9597 |
An extended real is not less than plus infinity iff they are equal.
(Contributed by NM, 30-Jan-2006.)
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Theorem | npnflt 9598 |
An extended real is less than plus infinity iff they are not equal.
(Contributed by Jim Kingdon, 17-Apr-2023.)
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Theorem | xgepnf 9599 |
An extended real which is greater than plus infinity is plus infinity.
(Contributed by Thierry Arnoux, 18-Dec-2016.)
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Theorem | ngtmnft 9600 |
An extended real is not greater than minus infinity iff they are equal.
(Contributed by NM, 2-Feb-2006.)
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