
theorem LemmaOne:
  for R being Function st R~ is Function holds
    R is one-to-one
  proof
    let R be Function;
    assume
A0: R~ is Function;
    assume not R is one-to-one; then
    consider x1,x2 being object such that
A1: x1 in dom R & x2 in dom R & R.x1 = R.x2 & x1 <> x2 by FUNCT_1:def 4;
    [x1,R.x1] in R by A1,FUNCT_1:def 2; then
A2: [R.x1,x1] in R~ by RELAT_1:def 7;
    [x2,R.x2] in R by A1,FUNCT_1:def 2; then
    [R.x2,x2] in R~ by RELAT_1:def 7;
    hence thesis by A1,A2,A0,FUNCT_1:def 1;
  end;
