
theorem Th8:
  for f being Function st f is one-to-one holds ~f is one-to-one
proof
  let f be Function such that
A1: f is one-to-one;
  let x1,x2 be object;
  consider X,Y being set such that
A2: dom~f c= [:X,Y:] by FUNCT_4:44;
  assume
A3: x1 in dom~f;
  then consider x11,x12 being object such that x11 in X and x12 in Y and
A4: x1 = [x11,x12] by A2,ZFMISC_1:84;
  assume
A5: x2 in dom~f;
  then consider x21,x22 being object such that x21 in X and x22 in Y and
A6: x2 = [x21,x22] by A2,ZFMISC_1:84;
  assume
A7: (~f).x1 = (~f).x2;
A8: [x12,x11] in dom f by A3,A4,FUNCT_4:42;
A9: [x22,x21] in dom f by A5,A6,FUNCT_4:42;
  f.(x12,x11) = ~f.(x11,x12) by A3,A4,FUNCT_4:43
    .= (~f).(x21,x22) by A4,A6,A7
    .= f.(x22,x21) by A5,A6,FUNCT_4:43;
  then
A10: [x12,x11] = [x22,x21] by A1,A8,A9;
  then x12 = x22 by XTUPLE_0:1;
  hence thesis by A4,A6,A10,XTUPLE_0:1;
end;
