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Theorem dalem-cly 33205
Description: Lemma for dalem9 33206. Center of perspectivity  C is not in plane  Y (when  Y and  Z are different planes). (Contributed by NM, 13-Aug-2012.)
Hypotheses
Ref Expression
dalema.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalem-cly.o  |-  O  =  ( LPlanes `  K )
dalem-cly.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem-cly.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem-cly  |-  ( (
ph  /\  Y  =/=  Z )  ->  -.  C  .<_  Y )

Proof of Theorem dalem-cly
StepHypRef Expression
1 dalema.ph . . . . . . 7  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
21dalemkelat 33158 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
41, 3dalemceb 33172 . . . . . 6  |-  ( ph  ->  C  e.  ( Base `  K ) )
5 dalem-cly.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
61, 5dalemyeb 33183 . . . . . 6  |-  ( ph  ->  Y  e.  ( Base `  K ) )
7 eqid 2422 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
9 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
107, 8, 9latleeqj1 16308 . . . . . 6  |-  ( ( K  e.  Lat  /\  C  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( C  .<_  Y  <->  ( C  .\/  Y )  =  Y ) )
112, 4, 6, 10syl3anc 1264 . . . . 5  |-  ( ph  ->  ( C  .<_  Y  <->  ( C  .\/  Y )  =  Y ) )
121dalemclpjs 33168 . . . . . . . . . . . . 13  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
131dalemkehl 33157 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  HL )
14 dalem-cly.y . . . . . . . . . . . . . . 15  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
151, 8, 9, 3, 5, 14dalemcea 33194 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  A )
161dalemsea 33163 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  A )
171dalempea 33160 . . . . . . . . . . . . . 14  |-  ( ph  ->  P  e.  A )
181dalemqea 33161 . . . . . . . . . . . . . . 15  |-  ( ph  ->  Q  e.  A )
191dalem-clpjq 33171 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
208, 9, 3atnlej1 32913 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  C  .<_  ( P  .\/  Q
) )  ->  C  =/=  P )
2113, 15, 17, 18, 19, 20syl131anc 1277 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  P )
228, 9, 3hlatexch1 32929 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  S  e.  A  /\  P  e.  A
)  /\  C  =/=  P )  ->  ( C  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  C ) ) )
2313, 15, 16, 17, 21, 22syl131anc 1277 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  .<_  ( P 
.\/  S )  ->  S  .<_  ( P  .\/  C ) ) )
2412, 23mpd 15 . . . . . . . . . . . 12  |-  ( ph  ->  S  .<_  ( P  .\/  C ) )
259, 3hlatjcom 32902 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  P  e.  A )  ->  ( C  .\/  P
)  =  ( P 
.\/  C ) )
2613, 15, 17, 25syl3anc 1264 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  P
)  =  ( P 
.\/  C ) )
2724, 26breqtrrd 4450 . . . . . . . . . . 11  |-  ( ph  ->  S  .<_  ( C  .\/  P ) )
281dalemclqjt 33169 . . . . . . . . . . . . 13  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
291dalemtea 33164 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  e.  A )
301dalemrea 33162 . . . . . . . . . . . . . . 15  |-  ( ph  ->  R  e.  A )
31 simp312 1153 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) )  ->  -.  C  .<_  ( Q  .\/  R ) )
321, 31sylbi 198 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  C  .<_  ( Q 
.\/  R ) )
338, 9, 3atnlej1 32913 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  C  .<_  ( Q  .\/  R
) )  ->  C  =/=  Q )
3413, 15, 18, 30, 32, 33syl131anc 1277 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  Q )
358, 9, 3hlatexch1 32929 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  T  e.  A  /\  Q  e.  A
)  /\  C  =/=  Q )  ->  ( C  .<_  ( Q  .\/  T
)  ->  T  .<_  ( Q  .\/  C ) ) )
3613, 15, 29, 18, 34, 35syl131anc 1277 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  .<_  ( Q 
.\/  T )  ->  T  .<_  ( Q  .\/  C ) ) )
3728, 36mpd 15 . . . . . . . . . . . 12  |-  ( ph  ->  T  .<_  ( Q  .\/  C ) )
389, 3hlatjcom 32902 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  Q  e.  A )  ->  ( C  .\/  Q
)  =  ( Q 
.\/  C ) )
3913, 15, 18, 38syl3anc 1264 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  Q
)  =  ( Q 
.\/  C ) )
4037, 39breqtrrd 4450 . . . . . . . . . . 11  |-  ( ph  ->  T  .<_  ( C  .\/  Q ) )
411, 3dalemseb 33176 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  ( Base `  K ) )
427, 9, 3hlatjcl 32901 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  P  e.  A )  ->  ( C  .\/  P
)  e.  ( Base `  K ) )
4313, 15, 17, 42syl3anc 1264 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  P
)  e.  ( Base `  K ) )
441, 3dalemteb 33177 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  ( Base `  K ) )
457, 9, 3hlatjcl 32901 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  Q  e.  A )  ->  ( C  .\/  Q
)  e.  ( Base `  K ) )
4613, 15, 18, 45syl3anc 1264 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  Q
)  e.  ( Base `  K ) )
477, 8, 9latjlej12 16312 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( C  .\/  P )  e.  ( Base `  K
) )  /\  ( T  e.  ( Base `  K )  /\  ( C  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  ( C  .\/  P )  /\  T  .<_  ( C 
.\/  Q ) )  ->  ( S  .\/  T )  .<_  ( ( C  .\/  P )  .\/  ( C  .\/  Q ) ) ) )
482, 41, 43, 44, 46, 47syl122anc 1273 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  .<_  ( C  .\/  P )  /\  T  .<_  ( C 
.\/  Q ) )  ->  ( S  .\/  T )  .<_  ( ( C  .\/  P )  .\/  ( C  .\/  Q ) ) ) )
4927, 40, 48mp2and 683 . . . . . . . . . 10  |-  ( ph  ->  ( S  .\/  T
)  .<_  ( ( C 
.\/  P )  .\/  ( C  .\/  Q ) ) )
501, 3dalempeb 33173 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  ( Base `  K ) )
511, 3dalemqeb 33174 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  ( Base `  K ) )
527, 9latjjdi 16348 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) ) )  -> 
( C  .\/  ( P  .\/  Q ) )  =  ( ( C 
.\/  P )  .\/  ( C  .\/  Q ) ) )
532, 4, 50, 51, 52syl13anc 1266 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  ( P  .\/  Q ) )  =  ( ( C 
.\/  P )  .\/  ( C  .\/  Q ) ) )
5449, 53breqtrrd 4450 . . . . . . . . 9  |-  ( ph  ->  ( S  .\/  T
)  .<_  ( C  .\/  ( P  .\/  Q ) ) )
551dalemclrju 33170 . . . . . . . . . . 11  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
561dalemuea 33165 . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  A )
57 simp313 1154 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) )  ->  -.  C  .<_  ( R  .\/  P ) )
581, 57sylbi 198 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  C  .<_  ( R 
.\/  P ) )
598, 9, 3atnlej1 32913 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  -.  C  .<_  ( R  .\/  P
) )  ->  C  =/=  R )
6013, 15, 30, 17, 58, 59syl131anc 1277 . . . . . . . . . . . 12  |-  ( ph  ->  C  =/=  R )
618, 9, 3hlatexch1 32929 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  U  e.  A  /\  R  e.  A
)  /\  C  =/=  R )  ->  ( C  .<_  ( R  .\/  U
)  ->  U  .<_  ( R  .\/  C ) ) )
6213, 15, 56, 30, 60, 61syl131anc 1277 . . . . . . . . . . 11  |-  ( ph  ->  ( C  .<_  ( R 
.\/  U )  ->  U  .<_  ( R  .\/  C ) ) )
6355, 62mpd 15 . . . . . . . . . 10  |-  ( ph  ->  U  .<_  ( R  .\/  C ) )
649, 3hlatjcom 32902 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  C  e.  A  /\  R  e.  A )  ->  ( C  .\/  R
)  =  ( R 
.\/  C ) )
6513, 15, 30, 64syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  R
)  =  ( R 
.\/  C ) )
6663, 65breqtrrd 4450 . . . . . . . . 9  |-  ( ph  ->  U  .<_  ( C  .\/  R ) )
671, 9, 3dalemsjteb 33180 . . . . . . . . . 10  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
681, 9, 3dalempjqeb 33179 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
697, 9latjcl 16296 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  C  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( C  .\/  ( P  .\/  Q ) )  e.  (
Base `  K )
)
702, 4, 68, 69syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  ( P  .\/  Q ) )  e.  ( Base `  K
) )
711, 3dalemueb 33178 . . . . . . . . . 10  |-  ( ph  ->  U  e.  ( Base `  K ) )
727, 9, 3hlatjcl 32901 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  C  e.  A  /\  R  e.  A )  ->  ( C  .\/  R
)  e.  ( Base `  K ) )
7313, 15, 30, 72syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  R
)  e.  ( Base `  K ) )
747, 8, 9latjlej12 16312 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( S  .\/  T )  e.  ( Base `  K )  /\  ( C  .\/  ( P  .\/  Q ) )  e.  (
Base `  K )
)  /\  ( U  e.  ( Base `  K
)  /\  ( C  .\/  R )  e.  (
Base `  K )
) )  ->  (
( ( S  .\/  T )  .<_  ( C  .\/  ( P  .\/  Q
) )  /\  U  .<_  ( C  .\/  R
) )  ->  (
( S  .\/  T
)  .\/  U )  .<_  ( ( C  .\/  ( P  .\/  Q ) )  .\/  ( C 
.\/  R ) ) ) )
752, 67, 70, 71, 73, 74syl122anc 1273 . . . . . . . . 9  |-  ( ph  ->  ( ( ( S 
.\/  T )  .<_  ( C  .\/  ( P 
.\/  Q ) )  /\  U  .<_  ( C 
.\/  R ) )  ->  ( ( S 
.\/  T )  .\/  U )  .<_  ( ( C  .\/  ( P  .\/  Q ) )  .\/  ( C  .\/  R ) ) ) )
7654, 66, 75mp2and 683 . . . . . . . 8  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  ( ( C  .\/  ( P  .\/  Q ) )  .\/  ( C 
.\/  R ) ) )
771, 3dalemreb 33175 . . . . . . . . 9  |-  ( ph  ->  R  e.  ( Base `  K ) )
787, 9latjjdi 16348 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
) )  ->  ( C  .\/  ( ( P 
.\/  Q )  .\/  R ) )  =  ( ( C  .\/  ( P  .\/  Q ) ) 
.\/  ( C  .\/  R ) ) )
792, 4, 68, 77, 78syl13anc 1266 . . . . . . . 8  |-  ( ph  ->  ( C  .\/  (
( P  .\/  Q
)  .\/  R )
)  =  ( ( C  .\/  ( P 
.\/  Q ) ) 
.\/  ( C  .\/  R ) ) )
8076, 79breqtrrd 4450 . . . . . . 7  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  ( C  .\/  (
( P  .\/  Q
)  .\/  R )
) )
81 dalem-cly.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
8214oveq2i 6316 . . . . . . 7  |-  ( C 
.\/  Y )  =  ( C  .\/  (
( P  .\/  Q
)  .\/  R )
)
8380, 81, 823brtr4g 4456 . . . . . 6  |-  ( ph  ->  Z  .<_  ( C  .\/  Y ) )
84 breq2 4427 . . . . . 6  |-  ( ( C  .\/  Y )  =  Y  ->  ( Z  .<_  ( C  .\/  Y )  <->  Z  .<_  Y ) )
8583, 84syl5ibcom 223 . . . . 5  |-  ( ph  ->  ( ( C  .\/  Y )  =  Y  ->  Z  .<_  Y ) )
8611, 85sylbid 218 . . . 4  |-  ( ph  ->  ( C  .<_  Y  ->  Z  .<_  Y ) )
871dalemzeo 33167 . . . . . 6  |-  ( ph  ->  Z  e.  O )
881dalemyeo 33166 . . . . . 6  |-  ( ph  ->  Y  e.  O )
898, 5lplncmp 33096 . . . . . 6  |-  ( ( K  e.  HL  /\  Z  e.  O  /\  Y  e.  O )  ->  ( Z  .<_  Y  <->  Z  =  Y ) )
9013, 87, 88, 89syl3anc 1264 . . . . 5  |-  ( ph  ->  ( Z  .<_  Y  <->  Z  =  Y ) )
91 eqcom 2431 . . . . 5  |-  ( Z  =  Y  <->  Y  =  Z )
9290, 91syl6bb 264 . . . 4  |-  ( ph  ->  ( Z  .<_  Y  <->  Y  =  Z ) )
9386, 92sylibd 217 . . 3  |-  ( ph  ->  ( C  .<_  Y  ->  Y  =  Z )
)
9493necon3ad 2630 . 2  |-  ( ph  ->  ( Y  =/=  Z  ->  -.  C  .<_  Y ) )
9594imp 430 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  -.  C  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   Basecbs 15120   lecple 15196   joincjn 16188   Latclat 16290   Atomscatm 32798   HLchlt 32885   LPlanesclpl 33026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-lat 16291  df-clat 16353  df-oposet 32711  df-ol 32713  df-oml 32714  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857  df-hlat 32886  df-llines 33032  df-lplanes 33033
This theorem is referenced by:  dalem9  33206
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