reserve x,A,B,X,X9,Y,Y9,Z,V for set;

theorem
  (X /\ Y) \/ (Y /\ Z) \/ (Z /\ X) = (X \/ Y) /\ (Y \/ Z) /\ (Z \/ X)
proof
  thus X /\ Y \/ Y /\ Z \/ Z /\ X = (X /\ Y \/ Y /\ Z \/ Z) /\ (X /\ Y \/ Y /\
  Z \/ X) by Th24
    .= (X /\ Y \/ (Y /\ Z \/ Z)) /\ (X /\ Y \/ Y /\ Z \/ X) by Th4
    .= (X /\ Y \/ Z) /\ (X /\ Y \/ Y /\ Z \/ X) by Th22
    .= (X /\ Y \/ Z) /\ (X /\ Y \/ X \/ Y /\ Z) by Th4
    .= (X /\ Y \/ Z) /\ (X \/ Y /\ Z) by Th22
    .= (X \/ Z) /\ (Y \/ Z) /\ (X \/ Y /\ Z) by Th24
    .= (X \/ Z) /\ (Y \/ Z) /\ ((X \/ Y) /\ (X \/ Z)) by Th24
    .= (X \/ Y) /\ ((Y \/ Z) /\ (X \/ Z) /\ (X \/ Z)) by Th16
    .= (X \/ Y) /\ ((Y \/ Z) /\ ((X \/ Z) /\ (X \/ Z))) by Th16
    .= (X \/ Y) /\ (Y \/ Z) /\ (Z \/ X) by Th16;
end;
