theorem Th39:
  f is one-to-one implies f" is one-to-one
proof
  assume
A1: f is one-to-one;
  let y1,y2;
  assume that
A2: y1 in dom(f") and
A3: y2 in dom(f");
  y1 in rng f by A1,A2,Th31;
  then
A4: y1 = f.((f").y1) by A1,Th34;
  y2 in rng f by A1,A3,Th31;
  hence thesis by A1,A4,Th34;
end;
