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
  f is one-to-one & rng g = dom f & f*g = id rng f implies g = f"
proof
  assume that
A1: f is one-to-one and
A2: rng g = dom f & f*g = id rng f;
  f"*f = id dom f & dom(f") = rng f by A1,Th32,Th38;
  hence thesis by A2,Lm1;
end;
