
theorem Th1:
  for L be LATTICE for x,y be Element of L holds x <= y implies {x,
  y} is Chain of x,y
proof
  let L be LATTICE;
  let x,y be Element of L;
A1: x in {x,y} & y in {x,y} by TARSKI:def 2;
  assume
A2: x <= y;
A3: for z be Element of L st z in {x,y} holds x <= z & z <= y
  proof
    let z be Element of L;
    assume
A4: z in {x,y};
    per cases by A4,TARSKI:def 2;
    suppose
      z=x;
      hence thesis by A2;
    end;
    suppose
      z=y;
      hence thesis by A2;
    end;
  end;
  {x,y} is Chain of L by A2,ORDERS_2:9;
  hence thesis by A2,A1,A3,Def2;
end;
