reserve x, y for set;

theorem Th11:
  for P being upper-bounded non empty Poset st the InternalRel
  of P is well-ordering holds P is algebraic
proof
  let P be upper-bounded non empty Poset;
  assume
A1: the InternalRel of P is well-ordering;
  then reconsider L = P as connected complete continuous non empty Poset by Th9
;
  now
    let x,y be Element of L;
    assume x << y;
    then x is compact & x <= x & x <= y or x < y by WAYBEL_3:1;
    then consider z being Element of L such that
A2: z is compact and
A3: x <= z & z <= y by A1,Th10;
    take z;
    thus z in the carrier of CompactSublatt L by A2,WAYBEL_8:def 1;
    thus x <= z & z <= y by A3;
  end;
  hence thesis by WAYBEL_8:7;
end;
