let X, X1 be set ; :: thesis: for Y, Y1 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for g, h being complex-valued Function st f1 = f <#> g holds
f1 </> h = f <#> (g /" h)
let Y, Y1 be complex-functions-membered set ; :: thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for g, h being complex-valued Function st f1 = f <#> g holds
f1 </> h = f <#> (g /" h)
let f be PartFunc of X,Y; :: thesis: for f1 being PartFunc of X1,Y1
for g, h being complex-valued Function st f1 = f <#> g holds
f1 </> h = f <#> (g /" h)
let f1 be PartFunc of X1,Y1; :: thesis: for g, h being complex-valued Function st f1 = f <#> g holds
f1 </> h = f <#> (g /" h)
let g, h be complex-valued Function; :: thesis: ( f1 = f <#> g implies f1 </> h = f <#> (g /" h) )
assume A1: f1 = f <#> g ; :: thesis: f1 </> h = f <#> (g /" h)
then A2: dom f1 = (dom f) /\ (dom g) by Def42;
A3: dom (f1 </> h) = (dom f1) /\ (dom h) by Th100;
A4: dom (f <#> (g /" h)) = (dom f) /\ (dom (g /" h)) by Def42;
dom (g /" h) = (dom g) /\ (dom h) by VALUED_1:16;
hence A5: dom (f1 </> h) = dom (f <#> (g /" h)) by A2, A3, A4, XBOOLE_1:16; :: according to FUNCT_1:def 17 :: thesis: for b₁ being set holds
( not b₁ in dom (f1 </> h) or (f1 </> h) . b₁ = (f <#> (g /" h)) . b₁ )
let x be set ; :: thesis: ( not x in dom (f1 </> h) or (f1 </> h) . x = (f <#> (g /" h)) . x )
assume A6: x in dom (f1 </> h) ; :: thesis: (f1 </> h) . x = (f <#> (g /" h)) . x
then A7: x in dom f1 by A3, XBOOLE_0:def 4;
thus (f1 </> h) . x = (f1 . x) (/) (h . x) by A6, Th101
.= ((f . x) (#) (g . x)) (/) (h . x) by A1, A7, Def42
.= (f . x) (#) ((g . x) / (h . x)) by Th12
.= (f . x) (#) ((g /" h) . x) by VALUED_1:17
.= (f <#> (g /" h)) . x by A6, A5, Def42 ; :: thesis: verum