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

theorem
  X <> {} & (for Z1 st Z1 in X holds Z c= Z1) implies Z c= meet X
proof
  assume that
A1: X <> {} and
A2: for Z1 st Z1 in X holds Z c= Z1;
  thus Z c= meet X
  proof
    let x be object such that
A3: x in Z;
    for Y st Y in X holds x in Y
    proof
      let Y;
      assume Y in X;
      then Z c= Y by A2;
      hence thesis by A3;
    end;
    hence thesis by A1,Def1;
  end;
end;
