let X, X1, X2 be set ; :: thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
let Y, Y1, Y2 be complex-functions-membered set ; :: thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
let f be PartFunc of X,Y; :: thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
let f1 be PartFunc of X1,Y1; :: thesis: for f2 being PartFunc of X2,Y2 holds f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
let f2 be PartFunc of X2,Y2; :: thesis: f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
set f3 = f <##> f1;
set f4 = f <##> f2;
set f5 = f1 <++> f2;
A4:
dom (f <##> f1) = (dom f) /\ (dom f1)
by Def46;
A5:
dom (f <##> f2) = (dom f) /\ (dom f2)
by Def46;
A6:
dom (f <##> (f1 <++> f2)) = (dom f) /\ (dom (f1 <++> f2))
by Def46;
A7:
dom ((f <##> f1) <++> (f <##> f2)) = (dom (f <##> f1)) /\ (dom (f <##> f2))
by Def44;
dom (f1 <++> f2) = (dom f1) /\ (dom f2)
by Def44;
hence A8:
dom (f <##> (f1 <++> f2)) = dom ((f <##> f1) <++> (f <##> f2))
by A7, A4, A5, A6, Th1; :: according to FUNCT_1:def 17 :: thesis: for b1 being set holds
( not b1 in dom (f <##> (f1 <++> f2)) or (f <##> (f1 <++> f2)) . b1 = ((f <##> f1) <++> (f <##> f2)) . b1 )
let x be set ; :: thesis: ( not x in dom (f <##> (f1 <++> f2)) or (f <##> (f1 <++> f2)) . x = ((f <##> f1) <++> (f <##> f2)) . x )
assume A9:
x in dom (f <##> (f1 <++> f2))
; :: thesis: (f <##> (f1 <++> f2)) . x = ((f <##> f1) <++> (f <##> f2)) . x
then A10:
x in dom (f <##> f1)
by A7, A8, XBOOLE_0:def 4;
A11:
x in dom (f <##> f2)
by A7, A9, A8, XBOOLE_0:def 4;
A12:
x in dom (f1 <++> f2)
by A6, A9, XBOOLE_0:def 4;
thus (f <##> (f1 <++> f2)) . x =
(f . x) (#) ((f1 <++> f2) . x)
by A9, Def46
.=
(f . x) (#) ((f1 . x) + (f2 . x))
by A12, Def44
.=
((f . x) (#) (f1 . x)) + ((f . x) (#) (f2 . x))
by RFUNCT_1:22
.=
((f <##> f1) . x) + ((f . x) (#) (f2 . x))
by A10, Def46
.=
((f <##> f1) . x) + ((f <##> f2) . x)
by A11, Def46
.=
((f <##> f1) <++> (f <##> f2)) . x
by A9, A8, Def44
; :: thesis: verum