let X1, X2 be set ; :: thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <--> f2 = <-> (f2 <--> f1)
let Y1, Y2 be complex-functions-membered set ; :: thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <--> f2 = <-> (f2 <--> f1)
let f1 be PartFunc of X1,Y1; :: thesis: for f2 being PartFunc of X2,Y2 holds f1 <--> f2 = <-> (f2 <--> f1)
let f2 be PartFunc of X2,Y2; :: thesis: f1 <--> f2 = <-> (f2 <--> f1)
set f = f2 <--> f1;
A1: ( dom (f1 <--> f2) = (dom f1) /\ (dom f2) & dom (f2 <--> f1) = (dom f2) /\ (dom f1) ) by Def46;
hence A2: dom (f1 <--> f2) = dom (<-> (f2 <--> f1)) by Def33; :: according to FUNCT_1:def 11 :: thesis: for b₁ being object holds
( not b₁ in dom (f1 <--> f2) or (f1 <--> f2) . b₁ = (<-> (f2 <--> f1)) . b₁ )
let x be object ; :: thesis: ( not x in dom (f1 <--> f2) or (f1 <--> f2) . x = (<-> (f2 <--> f1)) . x )
assume A3: x in dom (f1 <--> f2) ; :: thesis: (f1 <--> f2) . x = (<-> (f2 <--> f1)) . x
hence (f1 <--> f2) . x = (f1 . x) - (f2 . x) by Def46
.= - ((f2 . x) - (f1 . x)) by Th18
.= - ((f2 <--> f1) . x) by A1, A3, Def46
.= (<-> (f2 <--> f1)) . x by A2, A3, Def33 ;
:: thesis: verum