
theorem Th3: :: 1.2(iii), p. 39
  for L being non empty Poset st L is with_suprema or L is /\-complete
  for x,y,z being Element of L
  st x << z & y << z holds ex_sup_of {x,y}, L & x "\/" y << z
proof
  let L be non empty Poset such that
A1: L is with_suprema or L is /\-complete;
  let x,y,z be Element of L;
  assume
A2: z >> x;
  then
A3: z >= x by Th1;
  assume
A4: z >> y;
  then
A5: z >= y by Th1;
  thus
A6: now per cases by A1;
    suppose L is with_suprema;
      hence ex_sup_of {x,y},L by YELLOW_0:20;
    end;
    suppose
A7:   L is /\-complete;
      set X = {a where a is Element of L: a >= x & a >= y};
      X c= the carrier of L
      proof
        let a be object;
        assume a in X;
        then ex z being Element of L st a = z & z >= x & z >= y;
        hence thesis;
      end;
      then reconsider X as Subset of L;
      z in X by A3,A5;
      then ex_inf_of X,L by A7,WAYBEL_0:76;
      then consider c being Element of L such that
A8:   c is_<=_than X and
A9:   for d being Element of L st d is_<=_than X holds d <= c;
A10:  c is_>=_than {x,y}
      proof
        let a be Element of L;
        assume
A11:    a in {x,y};
        a is_<=_than X
        proof
          let b be Element of L;
          assume b in X;
          then ex z being Element of L st b = z & z >= x & z >= y;
          hence thesis by A11,TARSKI:def 2;
        end;
        hence thesis by A9;
      end;
      now
        let a be Element of L;
        assume
A12:    a is_>=_than {x,y};
        then
A13:    a >= x by YELLOW_0:8;
        a >= y by A12,YELLOW_0:8;
        then a in X by A13;
        hence c <= a by A8;
      end;
      hence ex_sup_of {x,y},L by A10,YELLOW_0:15;
    end;
  end;
  let D be non empty directed Subset of L;
  assume
A14: z <= sup D;
  then consider d1 being Element of L such that
A15: d1 in D and
A16: x <= d1 by A2;
  consider d2 being Element of L such that
A17: d2 in D and
A18: y <= d2 by A4,A14;
  consider d being Element of L such that
A19: d in D and
A20: d1 <= d and
A21: d2 <= d by A15,A17,WAYBEL_0:def 1;
A22: x <= d by A16,A20,ORDERS_2:3;
A23: y <= d by A18,A21,ORDERS_2:3;
  take d;
  thus thesis by A6,A19,A22,A23,YELLOW_0:18;
end;
