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

theorem Th32:
  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 Def5
      .= X.i /\ (Y.i \/ Z.i) by A1,Def4
      .= X.i /\ Y.i \/ X.i /\ Z.i by XBOOLE_1:23
      .= (X (/\) Y).i \/ X.i /\ Z.i by A1,Def5
      .= (X (/\) Y).i \/ (X (/\) Z).i by A1,Def5
      .= (X (/\) Y (\/) X (/\) Z).i by A1,Def4;
end;
