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

theorem
  (X (/\) Y) (\) Z = (X (\) Z) (/\) (Y (\) Z)
proof
A1: X (\) Z c= X by Th56;
  thus (X (/\) Y) (\) Z = (X (\) Z) (/\) Y by Th62
    .= (X (\) Z) (\) ((X (\) Z) (\) Y) by Th68
    .= (X (\) Z) (\) (X (\) (Z (\/) Y)) by Th73
    .= ((X (\) Z) (\) X) (\/) ((X (\) Z) (/\) (Z (\/) Y)) by Th64
    .= EmptyMS I (\/) ((X (\) Z) (/\) (Z (\/) Y)) by A1,Th52
    .= (X (\) Z) (/\) (Y (\/) Z) by Th22,Th43
    .= (X (\) Z) (/\) ((Y (\) Z) (\/) Z) by Th67
    .= (X (\) Z) (/\) (Y (\) Z) (\/) (X (\) Z) (/\) Z by Th32
    .= (X (\) Z) (/\) (Y (\) Z) (\/) EmptyMS I by Th63
    .= (X (\) Z) (/\) (Y (\) Z) by Th22,Th43;
end;
