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

theorem
  (X \+\ Y) \+\ Z = X \+\ (Y \+\ Z)
proof
  set S1 = X \ (Y \/ Z), S2 = Y \ (X \/ Z), S3 = Z \ (X \/ Y), S4 = X /\ Y /\
  Z;
  thus (X \+\ Y) \+\ Z = (((X \ Y) \ Z) \/ ((Y \ X) \ Z)) \/ (Z \ ((X \ Y) \/
  (Y \ X))) by Th42
    .= (S1 \/ ((Y \ X) \ Z)) \/ (Z \ ((X \ Y) \/ (Y \ X))) by Th41
    .= (S1 \/ S2) \/ (Z \ ((X \ Y) \/ (Y \ X))) by Th41
    .= (S1 \/ S2) \/ (Z \ ((X \/ Y) \ (X /\ Y))) by Th55
    .= (S1 \/ S2) \/ (S4 \/ S3) by Th52
    .= (S1 \/ S2 \/ S4) \/ S3 by Th4
    .= (S1 \/ S4 \/ S2) \/ S3 by Th4
    .= (S1 \/ S4) \/ (S2 \/ S3) by Th4
    .= (S1 \/ X /\ (Y /\ Z)) \/ (S2 \/ S3) by Th16
    .= X \ ((Y \/ Z) \ (Y /\ Z)) \/ (S2 \/ S3) by Th52
    .= X \ ((Y \ Z) \/ (Z \ Y)) \/ (S2 \/ (Z \ (Y \/ X))) by Th55
    .= X \ ((Y \ Z) \/ (Z \ Y)) \/ ((Y \ (Z \/ X)) \/ (Z \ Y \ X)) by Th41
    .= X \ ((Y \ Z) \/ (Z \ Y)) \/ ((Y \ Z \ X) \/ (Z \ Y \ X)) by Th41
    .= X \+\ (Y \+\ Z) by Th42;
end;
