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Theorem ps-2b 32499
Description: Variation of projective geometry axiom ps-2 32495. (Contributed by NM, 3-Jul-2012.)
Hypotheses
Ref Expression
ps-2b.l  |-  .<_  =  ( le `  K )
ps-2b.j  |-  .\/  =  ( join `  K )
ps-2b.m  |-  ./\  =  ( meet `  K )
ps-2b.z  |-  .0.  =  ( 0. `  K )
ps-2b.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
ps-2b  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
.0.  )

Proof of Theorem ps-2b
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 simp11 1027 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  K  e.  HL )
2 simp12 1028 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  P  e.  A )
3 simp13 1029 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  Q  e.  A )
4 simp21 1030 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  R  e.  A )
52, 3, 43jca 1177 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
6 simp22 1031 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  S  e.  A )
7 simp23 1032 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  T  e.  A )
86, 7jca 530 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( S  e.  A  /\  T  e.  A
) )
9 simp31 1033 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  -.  P  .<_  ( Q 
.\/  R ) )
10 simp32 1034 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  S  =/=  T )
119, 10jca 530 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T
) )
12 simp33 1035 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) )
13 ps-2b.l . . . 4  |-  .<_  =  ( le `  K )
14 ps-2b.j . . . 4  |-  .\/  =  ( join `  K )
15 ps-2b.a . . . 4  |-  A  =  ( Atoms `  K )
1613, 14, 15ps-2 32495 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( -.  P  .<_  ( Q  .\/  R
)  /\  S  =/=  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  E. u  e.  A  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
171, 5, 8, 11, 12, 16syl32anc 1238 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  E. u  e.  A  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
18 simp111 1126 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  K  e.  HL )
19 hlatl 32378 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
2018, 19syl 17 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  K  e.  AtLat )
21 hllat 32381 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2218, 21syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  K  e.  Lat )
23 simp112 1127 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  P  e.  A )
24 simp121 1129 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  R  e.  A )
25 eqid 2402 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2625, 14, 15hlatjcl 32384 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
2718, 23, 24, 26syl3anc 1230 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( P  .\/  R
)  e.  ( Base `  K ) )
28 simp122 1130 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  S  e.  A )
29 simp123 1131 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  T  e.  A )
3025, 14, 15hlatjcl 32384 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
3118, 28, 29, 30syl3anc 1230 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
32 ps-2b.m . . . . . 6  |-  ./\  =  ( meet `  K )
3325, 32latmcl 16006 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  ( Base `  K ) )
3422, 27, 31, 33syl3anc 1230 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  ( Base `  K
) )
35 simp2 998 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  u  e.  A )
36 simp3 999 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
3725, 15atbase 32307 . . . . . . 7  |-  ( u  e.  A  ->  u  e.  ( Base `  K
) )
3835, 37syl 17 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  u  e.  ( Base `  K ) )
3925, 13, 32latlem12 16032 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( u  e.  ( Base `  K )  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
) )  ->  (
( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) )  <->  u  .<_  ( ( P  .\/  R
)  ./\  ( S  .\/  T ) ) ) )
4022, 38, 27, 31, 39syl13anc 1232 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( ( u  .<_  ( P  .\/  R )  /\  u  .<_  ( S 
.\/  T ) )  <-> 
u  .<_  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) ) ) )
4136, 40mpbid 210 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  u  .<_  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) ) )
42 ps-2b.z . . . . 5  |-  .0.  =  ( 0. `  K )
4325, 13, 42, 15atlen0 32328 . . . 4  |-  ( ( ( K  e.  AtLat  /\  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  ( Base `  K
)  /\  u  e.  A )  /\  u  .<_  ( ( P  .\/  R )  ./\  ( S  .\/  T ) ) )  ->  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) )  =/=  .0.  )
4420, 34, 35, 41, 43syl31anc 1233 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
.0.  )
4544rexlimdv3a 2898 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( E. u  e.  A  ( u  .<_  ( P  .\/  R )  /\  u  .<_  ( S 
.\/  T ) )  ->  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) )  =/=  .0.  )
)
4617, 45mpd 15 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
.0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   0.cp0 15991   Latclat 15999   Atomscatm 32281   AtLatcal 32282   HLchlt 32368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369
This theorem is referenced by:  ps-2c  32545
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