theorem (x1=x2 implies f+*(x1.-->y1)+*(x2.-->y2)=f+*(x2.-->y2)) &
(x1<>x2 implies f+*(x1.-->y1)+*(x2.-->y2)=f+*(x2.-->y2)+*(x1.-->y1)) ::#Th43
proof
set f1=x1.-->y1, f2=x2.-->y2, LH=f+*f1+*f2;
hereby
assume x1=x2; then {x1} = dom f2;
then dom f1 = dom f2; then
f1+*f2 = f2 by FUNCT_4:19; hence LH=f+*f2 by FUNCT_4:14;
end;
assume x1<>x2; then {x1} misses {x2} by ZFMISC_1:11; then
f1 tolerates f2 by FUNCOP_1:87; then f+*(f1+*f2)=f+*(f2+*f1) by FUNCT_4:34 .=
f+*f2+*f1 by FUNCT_4:14; hence LH=f+*f2+*f1 by FUNCT_4:14;
end;
