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

theorem Th53:
  X c= Y implies X (\) Z c= Y (\) Z
proof
  assume
A1: X c= Y;
  now
    let i be object;
    assume
A2: i in I;
    then
A3: (X (\) Z).i = X.i \ Z.i & (Y (\) Z).i = Y.i \ Z.i by Def6;
    X.i c= Y.i by A1,A2;
    hence (X (\) Z).i c= (Y (\) Z).i by A3,XBOOLE_1:33;
  end;
  hence thesis;
end;
