
theorem Th7:
  for f being Function st [:f,f:] is one-to-one holds f is one-to-one
proof
  let f be Function such that
A1: [:f,f:] is one-to-one;
  let x1,x2 be object such that
A2: x1 in dom f and
A3: x2 in dom f and
A4: f.x1 = f.x2;
A5: dom[:f,f:] = [:dom f,dom f:] by FUNCT_3:def 8;
  then
A6: [x1,x1] in dom[:f,f:] by A2,ZFMISC_1:87;
A7: [x2,x2] in dom[:f,f:] by A3,A5,ZFMISC_1:87;
  [:f,f:].(x1,x1) = [f.x2,f.x2] by A4,A5,A6,FUNCT_3:65
    .= [:f,f:].(x2,x2) by A5,A7,FUNCT_3:65;
  then [x1,x1] = [x2,x2] by A1,A6,A7;
  hence thesis by XTUPLE_0:1;
end;
