
theorem Th27:
  for I being non empty set
  for J being RelStr-yielding non-Empty ManySortedSet of I
  for x being Function holds x is Element of product J iff dom x = I &
  for i being Element of I holds x.i is Element of J.i
proof
  let I be non empty set;
  let J be RelStr-yielding non-Empty ManySortedSet of I;
  let x be Function;
A1: the carrier of product J = product Carrier J by YELLOW_1:def 4;
A2: dom Carrier J = I by PARTFUN1:def 2;
  hereby
    assume
A3: x is Element of product J;
    hence dom x = I by A1,A2,CARD_3:9;
    let i be Element of I;
    reconsider y = x as Element of product J by A3;
    y.i is Element of J.i;
    hence x.i is Element of J.i;
  end;
  assume that
A4: dom x = I and
A5: for i being Element of I holds x.i is Element of J.i;
  now
    let i be object;
    assume i in dom Carrier J;
    then reconsider j = i as Element of I by PARTFUN1:def 2;
A6: x.j is Element of J.j by A5;
    ex R being 1-sorted st R = J.j & (Carrier J).j = the carrier of R by
PRALG_1:def 15;
    hence x.i in (Carrier J).i by A6;
  end;
  hence thesis by A1,A2,A4,CARD_3:9;
end;
