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

theorem Th54:
  X c= Y implies Z (\) Y c= Z (\) X
proof
  assume
A1: X c= Y;
  now
    let i be object;
    assume
A2: i in I;
    then
A3: (Z (\) X).i = Z.i \ X.i & (Z (\) Y).i = Z.i \ Y.i by Def6;
    X.i c= Y.i by A1,A2;
    hence (Z (\) Y).i c= (Z (\) X).i by A3,XBOOLE_1:34;
  end;
  hence thesis;
end;
