
theorem Th29:
for X be set, F be Function, x,y be object
 st not x in X & not y in X holds F|X = Swap(F,x,y)|X
proof
    let X be set, F be Function, x,y be object;
    assume A1: not x in X & not y in X;
    dom F = dom Swap(F,x,y) by FUNCT_7:99; then
    dom(F|X) = dom Swap(F,x,y) /\ X by RELAT_1:61; then
A2: dom(F|X) = dom(Swap(F,x,y)|X) by RELAT_1:61;

    now let z be object;
     assume z in dom(F|X); then
A3:  z in X by RELAT_1:57; then
     Swap(F,x,y).z = F.z by A1,EXCHSORT:33; then
     (F|X).z = Swap(F,x,y).z by A3,FUNCT_1:49;
     hence (F|X).z = (Swap(F,x,y)|X).z by A3,FUNCT_1:49;
    end;
    hence thesis by A2,FUNCT_1:2;
end;
