reserve x, y for set;

theorem Th9:
  for P being upper-bounded non empty Poset st the InternalRel of
  P is well-ordering holds P is connected complete continuous
proof
  let P be upper-bounded non empty Poset such that
A1: the InternalRel of P is well-ordering;
A2: field the InternalRel of P = the carrier of P by ORDERS_1:15;
  thus
A3: P is connected
  proof
    let x,y being Element of P;
A4: x = y or x <> y;
    the InternalRel of P is_connected_in the carrier of P & the
InternalRel of P is_reflexive_in the carrier of P by A1,A2,RELAT_2:def 9,def 14
    ;
    then [x,y] in the InternalRel of P or [y,x] in the InternalRel of P by A4;
    hence x <= y or y <= x;
  end;
  thus P is complete
  proof
    set y = the Element of P;
    let X be set;
    defpred P[object] means
 for y being Element of P st y in X holds [y,$1] in
    the InternalRel of P;
    consider Y being set such that
A5: for x being object holds  x in Y iff x in the carrier of P & P[x]
     from XBOOLE_0:sch 1;
A6: Y c= the carrier of P
    by A5;
    the InternalRel of P is upper-bounded by Th8;
    then consider x such that
A7: for y st y in field the InternalRel of P holds [y,x] in the
    InternalRel of P;
    [y,x] in the InternalRel of P by A2,A7;
    then reconsider x as Element of P by A2,RELAT_1:15;
    for y being Element of P st y in X holds [y,x] in the InternalRel of
    P by A2,A7;
    then x in Y by A5;
    then consider a being object such that
A8: a in Y and
A9: for b being object st b in Y holds [a,b] in the InternalRel of P
    by A1,A2,A6,WELLORD1:6;
    reconsider a as Element of P by A6,A8;
    take a;
    thus for y be Element of P st y in X holds y <= a by A5,A8;
    let b be Element of P;
    assume
A10: for c being Element of P st c in X holds c <= b;
    for c being Element of P st c in X
     holds [c,b] in the InternalRel of P by ORDERS_2:def 5,A10;
    then b in Y by A5;
    then [a,b] in the InternalRel of P by A9;
    hence a <= b;
  end;
  hence thesis by A3;
end;
