
theorem lemonto:
for X being non empty set,
    Z being set
for r being Renaming of X,Z holds r" is onto
proof
let X be non empty set, Z be set;
let r be Renaming of X,Z;
A: now let o be object;
   assume o in X; then
   H: o in dom r by defr; then
   r.o in rng r by FUNCT_1:3; then
   r.o in dom(r") by FUNCT_2:def 1; then
   r".(r.o) in rng(r") by FUNCT_1:3;
   hence o in rng(r") by H,FUNCT_1:34;
   end;
now let o be object;
  assume o in rng(r");
  then o in dom r by FUNCT_1:33;
  hence o in X;
  end;
hence thesis by A,TARSKI:2;
end;
