reserve
  I for non empty set,
  F for associative Group-like multMagma-Family of I,
  i, j for Element of I;

theorem Th5:
  for g1,g2 be Element of product F
  st g1 in ProjSet(F,i) & g2 in ProjSet(F,i)
  holds g1 * g2 in ProjSet(F,i)
  proof
    let g1,g2 be Element of product F;
    assume A1:
    g1 in ProjSet(F,i) & g2 in ProjSet(F,i);
    consider z1 be Element of (F.i) such that
    A2: g1 = 1_product F +* (i,z1) by Def1,A1;
    consider z2 be Element of (F.i) such that
    A3: g2 = 1_product F +* (i,z2) by Def1,A1;
    g1*g2 = ((1_(product F))+* (i,z1*z2)) by Th3,A2,A3;
    hence g1*g2 in ProjSet(F,i) by Def1;
  end;
