reserve i,j,e,u for object;
reserve I for set; 
reserve x,X,Y,Z,V for ManySortedSet of I;

theorem Th90:
  (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 Th72
    .= ( S1 (\/) ((Y (\) X) (\) Z)) (\/) (Z (\) ((X (\) Y) (\/) (Y (\) X)))
                by Th73
    .= ( S1 (\/) S2) (\/) (Z (\) ((X (\) Y) (\/) (Y (\) X))) by Th73
    .= ( S1 (\/) S2) (\/) (Z (\) ((X (\/) Y) (\) (X (/\) Y))) by Th77
    .= (S1 (\/) S2) (\/) (S4 (\/) S3) by Th64
    .= (S1 (\/) S2 (\/) S4) (\/) S3 by Th28
    .= (S1 (\/) S4 (\/) S2) (\/) S3 by Th28
    .= (S1 (\/) S4) (\/) (S2 (\/) S3) by Th28
    .= (S1 (\/) X (/\) (Y (/\) Z)) (\/) (S2 (\/) S3) by Th29
    .= X (\) ((Y (\/) Z) (\) (Y (/\) Z)) (\/) (S2 (\/) S3) by Th64
    .= X (\) ((Y (\) Z) (\/) (Z (\) Y)) (\/) (S2 (\/) (Z (\) (Y (\/) X)))
               by Th77
    .= X (\) ((Y (\) Z) (\/) (Z (\) Y)) (\/) ((Y (\) (Z (\/) X))
              (\/) (Z (\) Y (\) X)) by Th73
    .= X (\) ((Y (\) Z) (\/) (Z (\) Y)) (\/) ((Y (\) Z (\) X)
         (\/) (Z (\) Y (\) X)) by Th73
    .= X (\+\) (Y (\+\) Z) by Th72;
end;
