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Theorem sharhght 38187
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 1019 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
31simp1d 1017 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
42, 3subcld 9987 . . . . . . 7  |-  ( ph  ->  ( C  -  A
)  e.  CC )
54adantr 466 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  A )  e.  CC )
6 sharhght.b . . . . . . . . 9  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
76simpld 460 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
87, 3subcld 9987 . . . . . . 7  |-  ( ph  ->  ( D  -  A
)  e.  CC )
98adantr 466 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  A )  e.  CC )
10 sharhght.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1110sigarim 38173 . . . . . 6  |-  ( ( ( C  -  A
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  A ) G ( D  -  A
) )  e.  RR )
125, 9, 11syl2anc 665 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  RR )
1312recnd 9670 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  CC )
1413mul01d 9833 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  0 )  =  0 )
151simp2d 1018 . . . . . 6  |-  ( ph  ->  B  e.  CC )
1615adantr 466 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  e.  CC )
17 simpr 462 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  =  D )
1816, 17subeq0bd 10046 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  ( B  -  D )  =  0 )
1918oveq2d 6318 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  A ) G ( D  -  A ) )  x.  0 ) )
202, 15subcld 9987 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
2120adantr 466 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  B )  e.  CC )
227, 15subcld 9987 . . . . . . . 8  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2322adantr 466 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  e.  CC )
2410sigarval 38172 . . . . . . 7  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  =  ( Im `  ( ( * `  ( C  -  B ) )  x.  ( D  -  B ) ) ) )
2521, 23, 24syl2anc 665 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  ( Im `  ( ( * `  ( C  -  B
) )  x.  ( D  -  B )
) ) )
267adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  e.  CC )
2717eqcomd 2430 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  =  B )
2826, 27subeq0bd 10046 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  =  0 )
2928oveq2d 6318 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  ( ( * `
 ( C  -  B ) )  x.  0 ) )
3021cjcld 13248 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  (
* `  ( C  -  B ) )  e.  CC )
3130mul01d 9833 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  0 )  =  0 )
3229, 31eqtrd 2463 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  0 )
3332fveq2d 5882 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  ( (
* `  ( C  -  B ) )  x.  ( D  -  B
) ) )  =  ( Im `  0
) )
34 0red 9645 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  0  e.  RR )
3534reim0d 13277 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  0 )  =  0 )
3625, 33, 353eqtrd 2467 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  0 )
3736oveq1d 6317 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  ( 0  x.  ( A  -  D
) ) )
383adantr 466 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  A  e.  CC )
3938, 26subcld 9987 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  ( A  -  D )  e.  CC )
4039mul02d 9832 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
0  x.  ( A  -  D ) )  =  0 )
4137, 40eqtrd 2463 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  0 )
4214, 19, 413eqtr4d 2473 . 2  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
432adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  C  e.  CC )
4415adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  B  e.  CC )
453adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  A  e.  CC )
4643, 44, 45npncand 10011 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
)  +  ( B  -  A ) )  =  ( C  -  A ) )
4746oveq1d 6317 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( C  -  A ) G ( D  -  A
) ) )
4843, 44subcld 9987 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( C  -  B )  e.  CC )
498adantr 466 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  e.  CC )
5044, 45subcld 9987 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  A )  e.  CC )
5110sigaraf 38175 . . . . . . . 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 1264 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
5347, 52eqtr3d 2465 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
546simprd 464 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  0 )
5554adantr 466 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  0 )
567adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  D  e.  CC )
5710sigarperm 38182 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
5845, 44, 56, 57syl3anc 1264 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
5955, 58eqtr3d 2465 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  0  =  ( ( B  -  A ) G ( D  -  A
) ) )
6059oveq2d 6318 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
6110sigarim 38173 . . . . . . . . 9  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  A
) )  e.  RR )
6248, 49, 61syl2anc 665 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  RR )
6362recnd 9670 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  CC )
6463addid1d 9834 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6553, 60, 643eqtr2d 2469 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6644, 56negsubdi2d 10003 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( B  -  D )  =  ( D  -  B ) )
6766eqcomd 2430 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  =  -u ( B  -  D ) )
6867oveq1d 6317 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
6944, 56subcld 9987 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  e.  CC )
70 simpr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  B  =  D )  ->  -.  B  =  D )
7170neqned 2627 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  B  =/=  D )
7244, 56, 71subne0d 9996 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  =/=  0 )
7369, 69, 72divnegd 10397 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
7469, 72dividd 10382 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( B  -  D
)  /  ( B  -  D ) )  =  1 )
7574negeqd 9870 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  -u 1 )
7668, 73, 753eqtr2d 2469 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  -u 1 )
7776oveq1d 6317 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( -u 1  x.  ( A  -  D
) ) )
7845, 56subcld 9987 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  ( A  -  D )  e.  CC )
7978mulm1d 10071 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( -u 1  x.  ( A  -  D ) )  =  -u ( A  -  D ) )
8045, 56negsubdi2d 10003 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( A  -  D )  =  ( D  -  A ) )
8177, 79, 803eqtrd 2467 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( D  -  A ) )
8256, 44subcld 9987 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  e.  CC )
8382, 69, 78, 72div32d 10407 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8481, 83eqtr3d 2465 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8584oveq2d 6318 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( ( D  -  B )  x.  (
( A  -  D
)  /  ( B  -  D ) ) ) ) )
8656, 45, 443jca 1185 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  e.  CC  /\  A  e.  CC  /\  B  e.  CC ) )
8710, 86, 70, 55sigardiv 38183 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  RR )
8810sigarls 38179 . . . . . 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 1264 . . . . 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 2467 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
9190oveq1d 6317 . . 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 38173 . . . . . 6  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  RR )
9392recnd 9670 . . . . 5  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  CC )
9448, 82, 93syl2anc 665 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  e.  CC )
9578, 69, 72divcld 10384 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  CC )
9694, 95, 69mulassd 9667 . . 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 10385 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( A  -  D )  /  ( B  -  D )
)  x.  ( B  -  D ) )  =  ( A  -  D ) )
9897oveq2d 6318 . . 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 2467 . 2  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
10042, 99pm2.61dan 798 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 370    /\ w3a 982    = wceq 1437    e. wcel 1868   ` cfv 5598  (class class class)co 6302    |-> cmpt2 6304   CCcc 9538   RRcr 9539   0cc0 9540   1c1 9541    + caddc 9543    x. cmul 9545    - cmin 9861   -ucneg 9862    / cdiv 10270   *ccj 13148   Imcim 13150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-2 10669  df-cj 13151  df-re 13152  df-im 13153
This theorem is referenced by:  cevathlem2  38190
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