
theorem Th35:
  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 SubRelStr
  of J.i holds product K is SubRelStr of product J
proof
  let I be non empty set;
  let J,K be RelStr-yielding non-Empty ManySortedSet of I such that
A1: for i being Element of I holds K.i is SubRelStr of J.i;
A2: now
    let i be object;
    assume i in I;
    then reconsider j = i as Element of I;
A3: ex R being 1-sorted st R = J.j & (Carrier J).j = the carrier of R by
PRALG_1:def 15;
A4: ex R being 1-sorted st R = K.j & (Carrier K).j = the carrier of R by
PRALG_1:def 15;
    K.j is SubRelStr of J.j by A1;
    hence (Carrier K).i c= (Carrier J).i by A3,A4,YELLOW_0:def 13;
  end;
A5: dom Carrier K = I by PARTFUN1:def 2;
A6: dom Carrier J = I by PARTFUN1:def 2;
A7: the carrier of product J = product Carrier J by YELLOW_1:def 4;
  the carrier of product K = product Carrier K by YELLOW_1:def 4;
  hence
A8: the carrier of product K c= the carrier of product J by A7,A6,A5,A2,
CARD_3:27;
  let x,y be object;
  assume
A9: [x,y] in the InternalRel of product K;
  reconsider x,y as Element of product K by A9,ZFMISC_1:87;
  reconsider f = x, g = y as Element of product J by A8;
A10: x <= y by A9,ORDERS_2:def 5;
  now
    let i be Element of I;
A11: x.i <= y.i by A10,WAYBEL_3:28;
    K.i is SubRelStr of J.i by A1;
    hence f.i <= g.i by A11,YELLOW_0:59;
  end;
  then f <= g by WAYBEL_3:28;
  hence thesis by ORDERS_2:def 5;
end;
