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

theorem
  SFX is_finer_than SFY implies union SFX c= union SFY
proof
  assume
A1: for X st X in SFX ex Y st Y in SFY & X c= Y;
  thus union SFX c= union SFY
  proof
    let x be object;
    assume x in union SFX;
    then consider Y such that
A2: x in Y and
A3: Y in SFX by TARSKI:def 4;
    ex Z st Z in SFY & Y c= Z by A1,A3;
    hence thesis by A2,TARSKI:def 4;
  end;
end;
