
theorem Th1:
  for n,m be Nat holds n <= m implies InclPoset(n) is full
  SubRelStr of InclPoset(m)
proof
  let n,m be Nat;
A1: the InternalRel of InclPoset m = RelIncl m by YELLOW_1:1;
  assume n <= m;
  then
A2: Segm n c= Segm m by NAT_1:39;
A3: RelIncl n c= RelIncl m
  proof
    let x be object;
    assume x in RelIncl n;
    then x in (RelIncl m) |_2 n by A2,WELLORD2:7;
    hence thesis by XBOOLE_0:def 4;
  end;
  the carrier of InclPoset(m)=m by YELLOW_1:1;
  then
A4: the carrier of InclPoset(n) c= the carrier of InclPoset(m) by A2,YELLOW_1:1
;
A5: the InternalRel of InclPoset n = RelIncl n by YELLOW_1:1;
  then (RelIncl m) |_2 n = the InternalRel of InclPoset n by A2,WELLORD2:7;
  then the InternalRel of InclPoset(n) = (the InternalRel of InclPoset(m))|_2
  the carrier of InclPoset(n) by A1,YELLOW_1:1;
  hence thesis by A4,A5,A1,A3,YELLOW_0:def 13,def 14;
end;
