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 Th34:
  f is one-to-one & y in rng f implies y = f.((f").y) & y = (f*f") .y
proof
  assume
A1: f is one-to-one;
  assume
A2: y in rng f;
  hence
A3: y = f.((f").y) by A1,Th31;
  rng f = dom(f") by A1,Th32;
  hence thesis by A2,A3,Th13;
end;
