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

theorem Th64:
  X (\) (Y (\) Z) = (X (\) Y) (\/) X (/\) Z
proof
    let i be object;
    assume
A1: i in I;
    hence (X (\) (Y (\) Z)).i = X.i \ (Y (\) Z).i by Def6
      .= X.i \ (Y.i \ Z.i) by A1,Def6
      .= X.i \ Y.i \/ X.i /\ Z.i by XBOOLE_1:52
      .= X.i \ Y.i \/ (X (/\) Z).i by A1,Def5
      .= (X (\) Y).i \/ (X (/\) Z).i by A1,Def6
      .= ((X (\) Y) (\/) X (/\) Z).i by A1,Def4;
end;
