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Theorem cdleme42ke 38236
Description: Part of proof of Lemma E in [Crawley] p. 113. Remove 𝑅𝑆 condition. TODO: FIX COMMENT. (Contributed by NM, 2-Apr-2013.)
Hypotheses
Ref Expression
cdleme41.b 𝐵 = (Base‘𝐾)
cdleme41.l = (le‘𝐾)
cdleme41.j = (join‘𝐾)
cdleme41.m = (meet‘𝐾)
cdleme41.a 𝐴 = (Atoms‘𝐾)
cdleme41.h 𝐻 = (LHyp‘𝐾)
cdleme41.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme41.d 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme41.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme41.g 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
cdleme41.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
cdleme41.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme41.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
cdleme41.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
cdleme34e.v 𝑉 = ((𝑅 𝑆) 𝑊)
Assertion
Ref Expression
cdleme42ke ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝐹𝑅) (𝐹𝑆)) = ((𝐹𝑅) 𝑉))
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑆,𝑠   𝑈,𝑠   𝑊,𝑠   𝑦,𝑡,𝐴,𝑠   𝐵,𝑠,𝑡,𝑦   𝑦,𝐷   𝑦,𝐺   𝐸,𝑠,𝑦   𝐻,𝑠,𝑡,𝑦   𝑡, ,𝑦   𝐾,𝑠,𝑡,𝑦   𝑡, ,𝑦   𝑡, ,𝑦   𝑡,𝑃,𝑦   𝑡,𝑄,𝑦   𝑡,𝑅,𝑦   𝑡,𝑆,𝑦   𝑡,𝑈,𝑦   𝑡,𝑊,𝑦   𝑥,𝑧,𝐴   𝑥,𝐵,𝑧   𝑧,𝐸,𝑠   𝑧,𝐻   𝑥, ,𝑧   𝑧,𝐾   𝑥, ,𝑧   𝑥, ,𝑧   𝑥,𝑁,𝑧   𝑥,𝑃,𝑧   𝑥,𝑄,𝑧   𝑥,𝑅,𝑧   𝑥,𝑆,𝑧   𝑥,𝑈,𝑧   𝑥,𝑊,𝑧,𝑠,𝑡,𝑦   𝑉,𝑠,𝑡,𝑥,𝑧
Allowed substitution hints:   𝐷(𝑥,𝑧,𝑡,𝑠)   𝐸(𝑥,𝑡)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑥,𝑧,𝑡,𝑠)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐾(𝑥)   𝑁(𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑉(𝑦)

Proof of Theorem cdleme42ke
StepHypRef Expression
1 simpl1l 1226 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → 𝐾 ∈ HL)
2 simpr2 1197 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
3 cdleme41.b . . . . . . . 8 𝐵 = (Base‘𝐾)
4 cdleme41.l . . . . . . . 8 = (le‘𝐾)
5 cdleme41.j . . . . . . . 8 = (join‘𝐾)
6 cdleme41.m . . . . . . . 8 = (meet‘𝐾)
7 cdleme41.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
8 cdleme41.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
9 cdleme41.u . . . . . . . 8 𝑈 = ((𝑃 𝑄) 𝑊)
10 cdleme41.d . . . . . . . 8 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
11 cdleme41.e . . . . . . . 8 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
12 cdleme41.g . . . . . . . 8 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
13 cdleme41.i . . . . . . . 8 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
14 cdleme41.n . . . . . . . 8 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
15 cdleme41.o . . . . . . . 8 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
16 cdleme41.f . . . . . . . 8 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
173, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16cdleme32fvaw 38190 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝐹𝑅) ∈ 𝐴 ∧ ¬ (𝐹𝑅) 𝑊))
182, 17syldan 594 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝐹𝑅) ∈ 𝐴 ∧ ¬ (𝐹𝑅) 𝑊))
1918simpld 498 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → (𝐹𝑅) ∈ 𝐴)
205, 7hlatjidm 37120 . . . . 5 ((𝐾 ∈ HL ∧ (𝐹𝑅) ∈ 𝐴) → ((𝐹𝑅) (𝐹𝑅)) = (𝐹𝑅))
211, 19, 20syl2anc 587 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝐹𝑅) (𝐹𝑅)) = (𝐹𝑅))
22 fveq2 6717 . . . . 5 (𝑅 = 𝑆 → (𝐹𝑅) = (𝐹𝑆))
2322oveq2d 7229 . . . 4 (𝑅 = 𝑆 → ((𝐹𝑅) (𝐹𝑅)) = ((𝐹𝑅) (𝐹𝑆)))
2421, 23sylan9req 2799 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) ∧ 𝑅 = 𝑆) → (𝐹𝑅) = ((𝐹𝑅) (𝐹𝑆)))
25 simpr2l 1234 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → 𝑅𝐴)
265, 7hlatjidm 37120 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑅𝐴) → (𝑅 𝑅) = 𝑅)
271, 25, 26syl2anc 587 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → (𝑅 𝑅) = 𝑅)
2827oveq1d 7228 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝑅 𝑅) 𝑊) = (𝑅 𝑊))
29 simpl1 1193 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
30 eqid 2737 . . . . . . . . 9 (0.‘𝐾) = (0.‘𝐾)
314, 6, 30, 7, 8lhpmat 37781 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
3229, 2, 31syl2anc 587 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → (𝑅 𝑊) = (0.‘𝐾))
3328, 32eqtrd 2777 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝑅 𝑅) 𝑊) = (0.‘𝐾))
3433oveq2d 7229 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝐹𝑅) ((𝑅 𝑅) 𝑊)) = ((𝐹𝑅) (0.‘𝐾)))
35 hlol 37112 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OL)
361, 35syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → 𝐾 ∈ OL)
373, 7atbase 37040 . . . . . . 7 ((𝐹𝑅) ∈ 𝐴 → (𝐹𝑅) ∈ 𝐵)
3819, 37syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → (𝐹𝑅) ∈ 𝐵)
393, 5, 30olj01 36976 . . . . . 6 ((𝐾 ∈ OL ∧ (𝐹𝑅) ∈ 𝐵) → ((𝐹𝑅) (0.‘𝐾)) = (𝐹𝑅))
4036, 38, 39syl2anc 587 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝐹𝑅) (0.‘𝐾)) = (𝐹𝑅))
4134, 40eqtrd 2777 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝐹𝑅) ((𝑅 𝑅) 𝑊)) = (𝐹𝑅))
42 oveq2 7221 . . . . . . 7 (𝑅 = 𝑆 → (𝑅 𝑅) = (𝑅 𝑆))
4342oveq1d 7228 . . . . . 6 (𝑅 = 𝑆 → ((𝑅 𝑅) 𝑊) = ((𝑅 𝑆) 𝑊))
44 cdleme34e.v . . . . . 6 𝑉 = ((𝑅 𝑆) 𝑊)
4543, 44eqtr4di 2796 . . . . 5 (𝑅 = 𝑆 → ((𝑅 𝑅) 𝑊) = 𝑉)
4645oveq2d 7229 . . . 4 (𝑅 = 𝑆 → ((𝐹𝑅) ((𝑅 𝑅) 𝑊)) = ((𝐹𝑅) 𝑉))
4741, 46sylan9req 2799 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) ∧ 𝑅 = 𝑆) → (𝐹𝑅) = ((𝐹𝑅) 𝑉))
4824, 47eqtr3d 2779 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) ∧ 𝑅 = 𝑆) → ((𝐹𝑅) (𝐹𝑆)) = ((𝐹𝑅) 𝑉))
493, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44cdleme42k 38235 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → ((𝐹𝑅) (𝐹𝑆)) = ((𝐹𝑅) 𝑉))
50493expa 1120 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) ∧ 𝑅𝑆) → ((𝐹𝑅) (𝐹𝑆)) = ((𝐹𝑅) 𝑉))
5148, 50pm2.61dane 3029 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝐹𝑅) (𝐹𝑆)) = ((𝐹𝑅) 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wral 3061  ifcif 4439   class class class wbr 5053  cmpt 5135  cfv 6380  crio 7169  (class class class)co 7213  Basecbs 16760  lecple 16809  joincjn 17818  meetcmee 17819  0.cp0 17929  OLcol 36925  Atomscatm 37014  HLchlt 37101  LHypclh 37735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-riotaBAD 36704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-undef 8015  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-p1 17932  df-lat 17938  df-clat 18005  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-llines 37249  df-lplanes 37250  df-lvols 37251  df-lines 37252  df-psubsp 37254  df-pmap 37255  df-padd 37547  df-lhyp 37739
This theorem is referenced by:  cdleme42keg  38237  cdleme42mN  38238  cdlemeg46fjv  38274
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