let X, X1, X2 be set ; 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 <--> f2)
let Y, Y1, Y2 be 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 <--> f2)
let f be PartFunc of X,Y; for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <++> f1) <--> f2 = f <++> (f1 <--> f2)
let f1 be PartFunc of X1,Y1; for f2 being PartFunc of X2,Y2 holds (f <++> f1) <--> f2 = f <++> (f1 <--> f2)
let f2 be PartFunc of X2,Y2; (f <++> f1) <--> f2 = f <++> (f1 <--> f2)
set f3 = f <++> f1;
set f4 = f1 <--> f2;
A1:
dom ((f <++> f1) <--> f2) = (dom (f <++> f1)) /\ (dom f2)
by Def46;
A2:
dom (f <++> (f1 <--> f2)) = (dom f) /\ (dom (f1 <--> f2))
by Def45;
( dom (f <++> f1) = (dom f) /\ (dom f1) & dom (f1 <--> f2) = (dom f1) /\ (dom f2) )
by Def45, Def46;
hence A3:
dom ((f <++> f1) <--> f2) = dom (f <++> (f1 <--> f2))
by A1, A2, XBOOLE_1:16; FUNCT_1:def 11 for b1 being object holds
( not b1 in dom ((f <++> f1) <--> f2) or ((f <++> f1) <--> f2) . b1 = (f <++> (f1 <--> f2)) . b1 )
let x be object ; ( not x in dom ((f <++> f1) <--> f2) or ((f <++> f1) <--> f2) . x = (f <++> (f1 <--> f2)) . x )
assume A4:
x in dom ((f <++> f1) <--> f2)
; ((f <++> f1) <--> f2) . x = (f <++> (f1 <--> f2)) . x
then A5:
x in dom (f1 <--> f2)
by A2, A3, XBOOLE_0:def 4;
A6:
x in dom (f <++> f1)
by A1, A4, XBOOLE_0:def 4;
thus ((f <++> f1) <--> f2) . x =
((f <++> f1) . x) - (f2 . x)
by A4, Def46
.=
((f . x) + (f1 . x)) - (f2 . x)
by A6, Def45
.=
(f . x) + ((f1 . x) - (f2 . x))
by RFUNCT_1:8
.=
(f . x) + ((f1 <--> f2) . x)
by A5, Def46
.=
(f <++> (f1 <--> f2)) . x
by A3, A4, Def45
; verum