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

theorem
  X c= Z implies X (\/) Y (/\) Z = (X (\/) Y) (/\) Z
proof
  assume
A1: X c= Z;
    let i be object;
    assume
A2: i in I;
    then
A3: X.i c= Z.i by A1;
    thus (X (\/) Y (/\) Z).i = X.i \/ (Y (/\) Z).i by A2,Def4
      .= X.i \/ Y.i /\ Z.i by A2,Def5
      .= (X.i \/ Y.i) /\ Z.i by A3,XBOOLE_1:30
      .= (X (\/) Y).i /\ Z.i by A2,Def4
      .= ((X (\/) Y) (/\) Z).i by A2,Def5;
end;
