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

theorem Th72:
  (X (\/) Y) (\) Z = (X (\) Z) (\/) (Y (\) Z)
proof
    let i be object;
    assume
A1: i in I;
    hence ((X (\/) Y) (\) Z).i = (X (\/) Y).i \ Z.i by Def6
      .= X.i \/ Y.i \ Z.i by A1,Def4
      .= X.i \ Z.i \/ (Y.i \ Z.i) by XBOOLE_1:42
      .= X.i \ Z.i \/ (Y (\) Z).i by A1,Def6
      .= (X (\) Z).i \/ (Y (\) Z).i by A1,Def6
      .= ((X (\) Z) (\/) (Y (\) Z)).i by A1,Def4;
end;
