reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;
reserve f for BinOp of D;
reserve a,a1,a2,b,b1,b2,A,B,C,X,Y,Z,x,x1,x2,y,y1,y2,z for set,
U,U1,U2,U3 for non empty set, u,u1,u2 for Element of U,
P,Q,R for Relation, f,f1,f2,g,g1,g2 for Function,
k,m,n for Nat, kk,mm,nn for Element of NAT, m1, n1 for non zero Nat,
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;

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;
