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Theorem sharhght 38620
Description: Let  A B C be a triangle, and let  D lie on the line  A B. Then (doubled) areas of triangles  A D C and  C D B relate as lengths of corresponding bases  A D and  D B. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
Hypotheses
Ref Expression
sharhght.sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
sharhght.a  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
sharhght.b  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
Assertion
Ref Expression
sharhght  |-  ( ph  ->  ( ( ( C  -  A ) G ( D  -  A
) )  x.  ( B  -  D )
)  =  ( ( ( C  -  B
) G ( D  -  B ) )  x.  ( A  -  D ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y
Allowed substitution hints:    ph( x, y)    G( x, y)

Proof of Theorem sharhght
StepHypRef Expression
1 sharhght.a . . . . . . . . 9  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
21simp3d 1044 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
31simp1d 1042 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
42, 3subcld 10005 . . . . . . 7  |-  ( ph  ->  ( C  -  A
)  e.  CC )
54adantr 472 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  A )  e.  CC )
6 sharhght.b . . . . . . . . 9  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
76simpld 466 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
87, 3subcld 10005 . . . . . . 7  |-  ( ph  ->  ( D  -  A
)  e.  CC )
98adantr 472 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  A )  e.  CC )
10 sharhght.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1110sigarim 38606 . . . . . 6  |-  ( ( ( C  -  A
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  A ) G ( D  -  A
) )  e.  RR )
125, 9, 11syl2anc 673 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  RR )
1312recnd 9687 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  CC )
1413mul01d 9850 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  0 )  =  0 )
151simp2d 1043 . . . . . 6  |-  ( ph  ->  B  e.  CC )
1615adantr 472 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  e.  CC )
17 simpr 468 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  =  D )
1816, 17subeq0bd 10066 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  ( B  -  D )  =  0 )
1918oveq2d 6324 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  A ) G ( D  -  A ) )  x.  0 ) )
202, 15subcld 10005 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
2120adantr 472 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  B )  e.  CC )
227, 15subcld 10005 . . . . . . . 8  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2322adantr 472 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  e.  CC )
2410sigarval 38605 . . . . . . 7  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  =  ( Im `  ( ( * `  ( C  -  B ) )  x.  ( D  -  B ) ) ) )
2521, 23, 24syl2anc 673 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  ( Im `  ( ( * `  ( C  -  B
) )  x.  ( D  -  B )
) ) )
267adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  e.  CC )
2717eqcomd 2477 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  =  B )
2826, 27subeq0bd 10066 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  =  0 )
2928oveq2d 6324 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  ( ( * `
 ( C  -  B ) )  x.  0 ) )
3021cjcld 13336 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  (
* `  ( C  -  B ) )  e.  CC )
3130mul01d 9850 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  0 )  =  0 )
3229, 31eqtrd 2505 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  0 )
3332fveq2d 5883 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  ( (
* `  ( C  -  B ) )  x.  ( D  -  B
) ) )  =  ( Im `  0
) )
34 0red 9662 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  0  e.  RR )
3534reim0d 13365 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  0 )  =  0 )
3625, 33, 353eqtrd 2509 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  0 )
3736oveq1d 6323 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  ( 0  x.  ( A  -  D
) ) )
383adantr 472 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  A  e.  CC )
3938, 26subcld 10005 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  ( A  -  D )  e.  CC )
4039mul02d 9849 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
0  x.  ( A  -  D ) )  =  0 )
4137, 40eqtrd 2505 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  0 )
4214, 19, 413eqtr4d 2515 . 2  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
432adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  C  e.  CC )
4415adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  B  e.  CC )
453adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  A  e.  CC )
4643, 44, 45npncand 10029 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
)  +  ( B  -  A ) )  =  ( C  -  A ) )
4746oveq1d 6323 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( C  -  A ) G ( D  -  A
) ) )
4843, 44subcld 10005 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( C  -  B )  e.  CC )
498adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  e.  CC )
5044, 45subcld 10005 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  A )  e.  CC )
5110sigaraf 38608 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  A
)  e.  CC  /\  ( B  -  A
)  e.  CC )  ->  ( ( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A
) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
5248, 49, 50, 51syl3anc 1292 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
5347, 52eqtr3d 2507 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
546simprd 470 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  0 )
5554adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  0 )
567adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  D  e.  CC )
5710sigarperm 38615 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
5845, 44, 56, 57syl3anc 1292 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
5955, 58eqtr3d 2507 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  0  =  ( ( B  -  A ) G ( D  -  A
) ) )
6059oveq2d 6324 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
6110sigarim 38606 . . . . . . . . 9  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  A
) )  e.  RR )
6248, 49, 61syl2anc 673 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  RR )
6362recnd 9687 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  CC )
6463addid1d 9851 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6553, 60, 643eqtr2d 2511 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6644, 56negsubdi2d 10021 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( B  -  D )  =  ( D  -  B ) )
6766eqcomd 2477 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  =  -u ( B  -  D ) )
6867oveq1d 6323 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
6944, 56subcld 10005 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  e.  CC )
70 simpr 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  B  =  D )  ->  -.  B  =  D )
7170neqned 2650 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  B  =/=  D )
7244, 56, 71subne0d 10014 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  =/=  0 )
7369, 69, 72divnegd 10418 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
7469, 72dividd 10403 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( B  -  D
)  /  ( B  -  D ) )  =  1 )
7574negeqd 9889 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  -u 1 )
7668, 73, 753eqtr2d 2511 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  -u 1 )
7776oveq1d 6323 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( -u 1  x.  ( A  -  D
) ) )
7845, 56subcld 10005 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  ( A  -  D )  e.  CC )
7978mulm1d 10091 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( -u 1  x.  ( A  -  D ) )  =  -u ( A  -  D ) )
8045, 56negsubdi2d 10021 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( A  -  D )  =  ( D  -  A ) )
8177, 79, 803eqtrd 2509 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( D  -  A ) )
8256, 44subcld 10005 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  e.  CC )
8382, 69, 78, 72div32d 10428 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8481, 83eqtr3d 2507 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8584oveq2d 6324 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( ( D  -  B )  x.  (
( A  -  D
)  /  ( B  -  D ) ) ) ) )
8656, 45, 443jca 1210 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  e.  CC  /\  A  e.  CC  /\  B  e.  CC ) )
8710, 86, 70, 55sigardiv 38616 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  RR )
8810sigarls 38612 . . . . . 6  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC  /\  ( ( A  -  D )  /  ( B  -  D )
)  e.  RR )  ->  ( ( C  -  B ) G ( ( D  -  B )  x.  (
( A  -  D
)  /  ( B  -  D ) ) ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( A  -  D )  /  ( B  -  D ) ) ) )
8948, 82, 87, 88syl3anc 1292 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( ( D  -  B )  x.  ( ( A  -  D )  / 
( B  -  D
) ) ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
9065, 85, 893eqtrd 2509 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
9190oveq1d 6323 . . 3  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( ( C  -  B
) G ( D  -  B ) )  x.  ( ( A  -  D )  / 
( B  -  D
) ) )  x.  ( B  -  D
) ) )
9210sigarim 38606 . . . . . 6  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  RR )
9392recnd 9687 . . . . 5  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  CC )
9448, 82, 93syl2anc 673 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  e.  CC )
9578, 69, 72divcld 10405 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  CC )
9694, 95, 69mulassd 9684 . . 3  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( ( C  -  B ) G ( D  -  B
) )  x.  (
( A  -  D
)  /  ( B  -  D ) ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( ( A  -  D )  / 
( B  -  D
) )  x.  ( B  -  D )
) ) )
9778, 69, 72divcan1d 10406 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( A  -  D )  /  ( B  -  D )
)  x.  ( B  -  D ) )  =  ( A  -  D ) )
9897oveq2d 6324 . . 3  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( ( ( A  -  D
)  /  ( B  -  D ) )  x.  ( B  -  D ) ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
9991, 96, 983eqtrd 2509 . 2  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
10042, 99pm2.61dan 808 1  |-  ( ph  ->  ( ( ( C  -  A ) G ( D  -  A
) )  x.  ( B  -  D )
)  =  ( ( ( C  -  B
) G ( D  -  B ) )  x.  ( A  -  D ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    - cmin 9880   -ucneg 9881    / cdiv 10291   *ccj 13236   Imcim 13238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-2 10690  df-cj 13239  df-re 13240  df-im 13241
This theorem is referenced by:  cevathlem2  38623
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