
theorem Th4:
  for I,J be non empty set, a be Function of I,J, x,y be Function
  st dom x = I & dom y = J & a is bijective
  holds x = y * a iff y = x * a"
  proof
    let I,J be non empty set,
        a be Function of I,J,
        x,y be Function;
    assume that
    A1: dom x = I and
    A2: dom y = J and
    A3: a is bijective;
    A4: dom a = I & rng a = J by A3,GROUP_6:61;
    hereby
      assume
      A5: x = y * a;
      thus y = y * (id J) by A2,RELAT_1:51
            .= y * (a * a") by A3,A4,FUNCT_2:29
            .= x * a" by A5,RELAT_1:36;
    end;
    assume
    A6: y = x * a";
    thus x = x * (id I) by A1,RELAT_1:51
          .= x * (a" * a) by A3,A4,FUNCT_2:29
          .= y * a by A6,RELAT_1:36;
  end;
