reserve X1,X2,X3,X4 for set;

theorem Thm03:
  (X1 \ (X2 \/ X3)) \/ ((X1 /\ X4) \ X2) =
  ((X1 /\ X4) \ (X2 \/ X3)) \/
  (X1 \ (X2 \/ X3 \/ X4)) \/ ((X1 /\ X3 /\ X4) \ X2)
  proof
    set M1=X1 \ (X2 \/ X3);
    set M2=(X1 /\ X4) \ X2;
    set M3=((X1 /\ X4) \ (X2 \/ X3));
    set M4=(X1 \ (X2 \/ X3 \/ X4));
    set M5=((X1 /\ X3 /\ X4) \ X2);
    set Z=M3 \/ M4 \/ M5;
    M1 c= Z & M2 c= Z by Lm1,Lm2;
    then M1 \/ M2 c= Z \/ Z by XBOOLE_1:13;
    hence M1 \/ M2 c= Z;
    M3 c= M2 & M4 c= M1 & M5 c= M2 by XBOOLE_1:7,XBOOLE_1:34,Lm3;
    then M3 \/ M4 c= (M1 \/ M2) \/ (M1 \/ M2) & M5 c= M1 \/ M2
    by XBOOLE_1:10,XBOOLE_1:13;
    hence M3 \/ M4 \/ M5 c= M1 \/ M2 by XBOOLE_1:13;
  end;
