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Theorem cdleme22f 34272
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115.  F,  N represent f(t), ft(s) respectively. If s  <_ t  \/ v, then ft(s)  <_ f(t)  \/ v. (Contributed by NM, 6-Dec-2012.)
Hypotheses
Ref Expression
cdleme22.l  |-  .<_  =  ( le `  K )
cdleme22.j  |-  .\/  =  ( join `  K )
cdleme22.m  |-  ./\  =  ( meet `  K )
cdleme22.a  |-  A  =  ( Atoms `  K )
cdleme22.h  |-  H  =  ( LHyp `  K
)
cdleme22f.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme22f.f  |-  F  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
cdleme22f.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )
Assertion
Ref Expression
cdleme22f  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  N  .<_  ( F  .\/  V
) )

Proof of Theorem cdleme22f
StepHypRef Expression
1 cdleme22f.n . 2  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )
2 simp11l 1099 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  K  e.  HL )
3 hllat 33290 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  K  e.  Lat )
5 simp12l 1101 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  P  e.  A )
6 simp13l 1103 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  Q  e.  A )
7 eqid 2450 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme22.j . . . . . 6  |-  .\/  =  ( join `  K )
9 cdleme22.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 33293 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1219 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
12 simp11r 1100 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  W  e.  H )
13 simp22 1022 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  T  e.  A )
14 cdleme22.l . . . . . . 7  |-  .<_  =  ( le `  K )
15 cdleme22.m . . . . . . 7  |-  ./\  =  ( meet `  K )
16 cdleme22.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
17 cdleme22f.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdleme22f.f . . . . . . 7  |-  F  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
1914, 8, 15, 9, 16, 17, 18, 7cdleme1b 34152 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  T  e.  A ) )  ->  F  e.  ( Base `  K ) )
202, 12, 5, 6, 13, 19syl23anc 1226 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  F  e.  ( Base `  K
) )
21 simp21l 1105 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  S  e.  A )
227, 8, 9hlatjcl 33293 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
232, 21, 13, 22syl3anc 1219 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( S  .\/  T )  e.  ( Base `  K
) )
247, 16lhpbase 33924 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2512, 24syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  W  e.  ( Base `  K
) )
267, 15latmcl 15310 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( S  .\/  T )  ./\  W )  e.  ( Base `  K ) )
274, 23, 25, 26syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  (
( S  .\/  T
)  ./\  W )  e.  ( Base `  K
) )
287, 8latjcl 15309 . . . . 5  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  (
( S  .\/  T
)  ./\  W )  e.  ( Base `  K
) )  ->  ( F  .\/  ( ( S 
.\/  T )  ./\  W ) )  e.  (
Base `  K )
)
294, 20, 27, 28syl3anc 1219 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( F  .\/  ( ( S 
.\/  T )  ./\  W ) )  e.  (
Base `  K )
)
307, 14, 15latmle2 15335 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
)  e.  ( Base `  K ) )  -> 
( ( P  .\/  Q )  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )  .<_  ( F 
.\/  ( ( S 
.\/  T )  ./\  W ) ) )
314, 11, 29, 30syl3anc 1219 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  (
( P  .\/  Q
)  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )  .<_  ( F 
.\/  ( ( S 
.\/  T )  ./\  W ) ) )
32 simp21 1021 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
33 simp3l 1016 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  S  =/=  T )
34 simp23l 1109 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  V  e.  A )
35 simp23r 1110 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  V  .<_  W )
36 simp3r 1017 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  S  .<_  ( T  .\/  V
) )
378, 9hlatjcom 33294 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  ( T  .\/  V
)  =  ( V 
.\/  T ) )
382, 13, 34, 37syl3anc 1219 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( T  .\/  V )  =  ( V  .\/  T
) )
3936, 38breqtrd 4400 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  S  .<_  ( V  .\/  T
) )
40 hlcvl 33286 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CvLat )
412, 40syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  K  e.  CvLat )
4214, 8, 9cvlatexch2 33264 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( S  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .<_  ( V  .\/  T
)  ->  V  .<_  ( S  .\/  T ) ) )
4341, 21, 34, 13, 33, 42syl131anc 1232 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( S  .<_  ( V  .\/  T )  ->  V  .<_  ( S  .\/  T ) ) )
4439, 43mpd 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  V  .<_  ( S  .\/  T
) )
45 eqid 2450 . . . . . 6  |-  ( ( S  .\/  T ) 
./\  W )  =  ( ( S  .\/  T )  ./\  W )
4614, 8, 15, 9, 16, 45cdleme22aa 34265 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  S  =/=  T )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( S  .\/  T ) ) )  ->  V  =  ( ( S  .\/  T )  ./\  W ) )
472, 12, 32, 13, 33, 34, 35, 44, 46syl233anc 1248 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  V  =  ( ( S 
.\/  T )  ./\  W ) )
4847oveq2d 6192 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( F  .\/  V )  =  ( F  .\/  (
( S  .\/  T
)  ./\  W )
) )
4931, 48breqtrrd 4402 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  (
( P  .\/  Q
)  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )  .<_  ( F 
.\/  V ) )
501, 49syl5eqbr 4409 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  N  .<_  ( F  .\/  V
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   Basecbs 14262   lecple 14333   joincjn 15202   meetcmee 15203   Latclat 15303   Atomscatm 33190   CvLatclc 33192   HLchlt 33277   LHypclh 33910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-p1 15298  df-lat 15304  df-clat 15366  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-lhyp 33914
This theorem is referenced by:  cdleme22f2  34273  cdleme26fALTN  34288  cdleme26f  34289
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