let C be non empty set ; :: thesis: for V being non empty Abelian addLoopStr
for f1, f2 being PartFunc of C,V holds f1 + f2 = f2 + f1
let V be non empty Abelian addLoopStr ; :: thesis: for f1, f2 being PartFunc of C,V holds f1 + f2 = f2 + f1
let f1, f2 be PartFunc of C,V; :: thesis: f1 + f2 = f2 + f1
A1: dom (f1 + f2) = (dom f2) /\ (dom f1) by Def1
.= dom (f2 + f1) by Def1 ;

now :: thesis: for c being Element of C st c in dom (f1 + f2) holds
(f1 + f2) /. c = (f2 + f1) /. c

let c be Element of C; :: thesis: ( c in dom (f1 + f2) implies (f1 + f2) /. c = (f2 + f1) /. c )
assume A2: c in dom (f1 + f2) ; :: thesis: (f1 + f2) /. c = (f2 + f1) /. c
hence (f1 + f2) /. c = (f2 /. c) + (f1 /. c) by Def1
.= (f2 + f1) /. c by A1, A2, Def1 ;
:: thesis: verum

end;

hence f1 + f2 = f2 + f1 by A1, PARTFUN2:1; :: thesis: verum