
theorem Th29:
  for I be non empty finite set,
  F be associative Group-like multMagma-Family of I,
  x be total I -defined Function
  st for p be Element of I holds x.p in F.p
  holds x in the carrier of product F
  proof
    let I be non empty finite set,
    F be associative Group-like multMagma-Family of I,
    x be total I -defined Function;
    assume A1: for p be Element of I holds x.p in F.p;
    A2: dom (Carrier F) = I by PARTFUN1:def 2;
    A3: the carrier of product F = product (Carrier F) by GROUP_7:def 2;
    A4: dom x = I by PARTFUN1:def 2;
    now let i be object;
      assume A5:i in dom (Carrier F);
      consider R being 1-sorted such that
      A6:
      R = F . i & (Carrier F) . i = the carrier of R by PRALG_1:def 15,A5;
      reconsider i0=i as Element of I by A5;
      x.i0 in the carrier of R by A6,STRUCT_0:def 5,A1;
      hence x.i in (Carrier F) . i by A6;
    end;
    hence thesis by A3,A2,A4,CARD_3:def 5;
  end;
