reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

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;
