
theorem
  for V, V9, C, C9 being set st V c= V9 & C c= C9 holds SubstPoset (V, C
  ) is full SubRelStr of SubstPoset (V9, C9)
proof
  let V, V9, C, C9 be set;
  set K = SubstPoset (V, C), L = SubstPoset (V9, C9);
A1: the carrier of K = SubstitutionSet (V, C) & the carrier of L =
  SubstitutionSet (V9, C9) by SUBSTLAT:def 4;
  assume V c= V9 & C c= C9;
  then
A2: the carrier of K c= the carrier of L by A1,Th9;
  now
    let a, b be object;
    assume
A3: [a,b] in the InternalRel of K;
    then reconsider a9 = a, b9 = b as Element of K by ZFMISC_1:87;
    a in the carrier of K & b in the carrier of K by A3,ZFMISC_1:87;
    then reconsider a1 = a, b1 = b as Element of L by A2;
    a9 <= b9 by A3,ORDERS_2:def 5;
    then for x being set st x in a9 ex y being set st y in b9 & y c= x by Th12;
    then a1 <= b1 by Th12;
    hence [a,b] in the InternalRel of L by ORDERS_2:def 5;
  end;
  then the InternalRel of K c= the InternalRel of L by RELAT_1:def 3;
  then reconsider K as SubRelStr of L by A2,YELLOW_0:def 13;
  now
    let x, y be object;
    assume
A4: [x,y] in (the InternalRel of L) |_2 the carrier of K;
    then
A5: [x,y] in (the InternalRel of L) by XBOOLE_0:def 4;
    then reconsider p = x, q = y as Element of L by ZFMISC_1:87;
    [x,y] in [:the carrier of K, the carrier of K:] by A4,XBOOLE_0:def 4;
    then reconsider p9 = x, q9 = y as Element of K by ZFMISC_1:87;
    p <= q by A5,ORDERS_2:def 5;
    then for a being set st a in p ex b being set st b in q & b c= a by Th12;
    then p9 <= q9 by Th12;
    hence [x,y] in the InternalRel of K by ORDERS_2:def 5;
  end;
  then
A6: (the InternalRel of L) |_2 the carrier of K c= the InternalRel of K by
RELAT_1:def 3;
  now
    let x, y be object;
    assume
A7: [x,y] in the InternalRel of K;
    then reconsider p = x, q = y as Element of K by ZFMISC_1:87;
    reconsider p9 = p, q9 = q as Element of L by A2;
    p <= q by A7,ORDERS_2:def 5;
    then for a being set st a in p ex b being set st b in q & b c= a by Th12;
    then p9 <= q9 by Th12;
    then [p9,q9] in the InternalRel of L by ORDERS_2:def 5;
    hence [x,y] in (the InternalRel of L) |_2 the carrier of K by A7,
XBOOLE_0:def 4;
  end;
  then the InternalRel of K c= (the InternalRel of L) |_2 the carrier of K by
RELAT_1:def 3;
  then the InternalRel of K = (the InternalRel of L) |_2 the carrier of K by A6
;
  hence thesis by YELLOW_0:def 14;
end;
