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 <--> f2) <--> f1
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 <--> f2) <--> f1
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 <--> f2) <--> f1
let f1 be PartFunc of X1,Y1; :: thesis: for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = (f <--> f2) <--> f1
let f2 be PartFunc of X2,Y2; :: thesis: (f <--> f1) <--> f2 = (f <--> f2) <--> f1
set f3 = f <--> f1;
set f4 = f <--> f2;
A3:
dom (f <--> f1) = (dom f) /\ (dom f1)
by Def45;
A4:
dom (f <--> f2) = (dom f) /\ (dom f2)
by Def45;
A5:
dom ((f <--> f1) <--> f2) = (dom (f <--> f1)) /\ (dom f2)
by Def45;
A6:
dom ((f <--> f2) <--> f1) = (dom (f <--> f2)) /\ (dom f1)
by Def45;
hence A7:
dom ((f <--> f1) <--> f2) = dom ((f <--> f2) <--> f1)
by A3, A5, A4, XBOOLE_1:16; :: 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 <--> f2) <--> f1) . b1 )
let x be set ; :: thesis: ( not x in dom ((f <--> f1) <--> f2) or ((f <--> f1) <--> f2) . x = ((f <--> f2) <--> f1) . x )
assume A8:
x in dom ((f <--> f1) <--> f2)
; :: thesis: ((f <--> f1) <--> f2) . x = ((f <--> f2) <--> f1) . x
then A9:
x in dom (f <--> f1)
by A5, XBOOLE_0:def 4;
A10:
x in dom (f <--> f2)
by A6, A8, A7, XBOOLE_0:def 4;
thus ((f <--> f1) <--> f2) . x =
((f <--> f1) . x) - (f2 . x)
by A8, Def45
.=
((f . x) - (f1 . x)) - (f2 . x)
by A9, Def45
.=
((f . x) - (f2 . x)) - (f1 . x)
by RFUNCT_1:35
.=
((f <--> f2) . x) - (f1 . x)
by A10, Def45
.=
((f <--> f2) <--> f1) . x
by A8, A7, Def45
; :: thesis: verum