
theorem Th36:
  for I being non empty set for J,K being RelStr-yielding
  non-Empty ManySortedSet of I st for i being Element of I holds K.i is full
  SubRelStr of J.i holds product K is full SubRelStr of product J
proof
  let I be non empty set;
  let J,K be RelStr-yielding non-Empty ManySortedSet of I;
  assume
A1: for i being Element of I holds K.i is full SubRelStr of J.i;
  then for i being Element of I holds K.i is SubRelStr of J.i;
  then reconsider S = product K as non empty SubRelStr of product J by Th35;
A2: (the InternalRel of product J)|_2 the carrier of S = (the InternalRel of
  product J)/\[:the carrier of S,the carrier of S:] by WELLORD1:def 6;
  S is full
  proof
    the InternalRel of S c= the InternalRel of product J by YELLOW_0:def 13;
    hence
    the InternalRel of S c= (the InternalRel of product J)|_2 the carrier
    of S by A2,XBOOLE_1:19;
    let x,y be object;
    assume
A3: [x,y] in (the InternalRel of product J)|_2 the carrier of S;
    then
A4: [x,y] in the InternalRel of product J by A2,XBOOLE_0:def 4;
    [x,y] in [:the carrier of S, the carrier of S:] by A2,A3,XBOOLE_0:def 4;
    then reconsider x, y as Element of S by ZFMISC_1:87;
    reconsider a = x, b = y as Element of product J by YELLOW_0:58;
    reconsider x, y as Element of product K;
A5: a <= b by A4,ORDERS_2:def 5;
    now
      let i be Element of I;
A6:   K.i is full SubRelStr of J.i by A1;
      a.i <= b.i by A5,WAYBEL_3:28;
      hence x.i <= y.i by A6,YELLOW_0:60;
    end;
    then x <= y by WAYBEL_3:28;
    hence thesis by ORDERS_2:def 5;
  end;
  hence thesis;
end;
