let X be set ; :: thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <#> g) </> h = f <#> (g /" h)
let Y be complex-functions-membered set ; :: thesis: for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <#> g) </> h = f <#> (g /" h)
let f be PartFunc of X,Y; :: thesis: for g, h being complex-valued Function holds (f <#> g) </> h = f <#> (g /" h)
let g, h be complex-valued Function; :: thesis: (f <#> g) </> h = f <#> (g /" h)
set f1 = f <#> g;
A1: dom (g /" h) = (dom g) /\ (dom h) by VALUED_1:16;
A2: dom ((f <#> g) </> h) = (dom (f <#> g)) /\ (dom h) by Th71;
( dom (f <#> g) = (dom f) /\ (dom g) & dom (f <#> (g /" h)) = (dom f) /\ (dom (g /" h)) ) by Def43;
hence A3: dom ((f <#> g) </> h) = dom (f <#> (g /" h)) by A2, A1, XBOOLE_1:16; :: according to FUNCT_1:def 11 :: thesis: for b₁ being object holds
( not b₁ in dom ((f <#> g) </> h) or ((f <#> g) </> h) . b₁ = (f <#> (g /" h)) . b₁ )
let x be object ; :: thesis: ( not x in dom ((f <#> g) </> h) or ((f <#> g) </> h) . x = (f <#> (g /" h)) . x )
assume A4: x in dom ((f <#> g) </> h) ; :: thesis: ((f <#> g) </> h) . x = (f <#> (g /" h)) . x
then A5: x in dom (f <#> g) by A2, XBOOLE_0:def 4;
thus ((f <#> g) </> h) . x = ((f <#> g) . x) (/) (h . x) by A4, Th72
.= ((f . x) (#) (g . x)) (/) (h . x) by A5, Def43
.= (f . x) (#) ((g . x) / (h . x)) by Th16
.= (f . x) (#) ((g /" h) . x) by VALUED_1:17
.= (f <#> (g /" h)) . x by A3, A4, Def43 ; :: thesis: verum