reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;

theorem
  DIFFERENCE(SFX,SFX) is_finer_than SFX
proof
  let X;
  assume X in DIFFERENCE(SFX,SFX);
  then consider Z1,Z2 such that
A1: Z1 in SFX and
  Z2 in SFX and
A2: X = Z1 \ Z2 by Def6;
  take Z1;
  thus thesis by A1,A2;
end;
