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) = (<-> f1) <++> f2
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) = (<-> f1) <++> f2
let f1 be PartFunc of X1,Y1; :: thesis: for f2 being PartFunc of X2,Y2 holds <-> (f1 <--> f2) = (<-> f1) <++> f2
let f2 be PartFunc of X2,Y2; :: thesis: <-> (f1 <--> f2) = (<-> f1) <++> f2
set f3 = f1 <--> f2;
set f4 = <-> f1;
A1: dom (<-> (f1 <--> f2)) = dom (f1 <--> f2) by Def33;
A2: ( dom (f1 <--> f2) = (dom f1) /\ (dom f2) & dom (<-> f1) = dom f1 ) by Def33, Def46;
hence A3: dom (<-> (f1 <--> f2)) = dom ((<-> f1) <++> f2) by A1, Def45; :: according to FUNCT_1:def 11 :: thesis: for b₁ being object holds
( not b₁ in dom (<-> (f1 <--> f2)) or (<-> (f1 <--> f2)) . b₁ = ((<-> f1) <++> f2) . b₁ )
let x be object ; :: thesis: ( not x in dom (<-> (f1 <--> f2)) or (<-> (f1 <--> f2)) . x = ((<-> f1) <++> f2) . x )
assume A4: x in dom (<-> (f1 <--> f2)) ; :: thesis: (<-> (f1 <--> f2)) . x = ((<-> f1) <++> f2) . x
then A5: x in dom (<-> f1) by A2, A1, XBOOLE_0:def 4;
thus (<-> (f1 <--> f2)) . x = - ((f1 <--> f2) . x) by A4, Def33
.= - ((f1 . x) - (f2 . x)) by A1, A4, Def46
.= (- (f1 . x)) - (- (f2 . x)) by Th17
.= ((<-> f1) . x) + (f2 . x) by A5, Def33
.= ((<-> f1) <++> f2) . x by A3, A4, Def45 ; :: thesis: verum