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

theorem
  SFY <> {} & SFY is_coarser_than SFX implies meet SFX c= meet SFY
proof
  assume that
A1: SFY <> {} and
A2: for Z2 st Z2 in SFY ex Z1 st Z1 in SFX & Z1 c= Z2;
  meet SFX c= meet SFY
  proof
    let x be object such that
A3: x in meet SFX;
    for Z st Z in SFY holds x in Z
    proof
      let Z;
      assume Z in SFY;
      then consider Z1 such that
A4:   Z1 in SFX and
A5:   Z1 c= Z by A2;
      x in Z1 by A3,A4,Def1;
      hence thesis by A5;
    end;
    hence thesis by A1,Def1;
  end;
  hence thesis;
end;
