
theorem Th1:
  for P1, P2 being non empty Poset, K being non empty Chain of P1,
      h being monotone Function of P1, P2 holds
    h.:K is non empty Chain of P2
  proof
  let P1, P2 be non empty Poset,
      K be non empty Chain of P1,
      h be monotone Function of P1, P2;
  set R = the InternalRel of P2;
  set K2 = h.:K;
  for x, y be object
st x in K2 & y in K2 & x <>y holds [x,y] in R or [y,x] in R
  proof
    let x, y be object;
    assume A1:x in K2 & y in K2 & x <>y;
    then reconsider x,y as Element of P2;
    consider a be object such that
A2: a in dom h & a in K & x = h.a
                                            by A1,FUNCT_1:def 6;
    consider b be object such that
A3: b in dom h & b in K & y = h.b
                                            by A1,FUNCT_1:def 6;
    reconsider a,b as Element of P1 by A2,A3;
    a<=b or b<=a by A2,A3,ORDERS_2:11;
    then x<=y or y<=x by A2,A3,ORDERS_3:def 5;
    hence thesis by ORDERS_2:def 5;
  end;
  then A4:R is_connected_in K2 by RELAT_2:def 6;
  for x be object st x in K2 holds [x,x] in R
  proof
    let x be object;
    assume x in K2;
    then reconsider x as Element of P2;
    x<=x;
    hence thesis by ORDERS_2:def 5;
  end;
  then R is_reflexive_in K2 by RELAT_2:def 1;
  then R is_strongly_connected_in K2 by A4,ORDERS_1:7;
  then reconsider K2 as Chain of P2 by ORDERS_2:def 7;
  consider a be object such that A5:a in K by XBOOLE_0:7;
  a in the carrier of P1 by A5;
  then a in dom h by FUNCT_2:def 1;
  then h.a in K2 by A5,FUNCT_1:def 6;
  hence thesis;
  end;
