
theorem Th25:
  for q be set,
  F be multMagma-Family of {q},
  G be non empty multMagma st
  F = q .--> G holds
  for y be (the carrier of G)-valued total {q} -defined Function holds
  y in the carrier of product F & y.q in the carrier of G &
  y= q .--> y.q
  proof
    let q be set,
    F be multMagma-Family of {q},
    G be non empty multMagma;
    assume A1: F = q .--> G;
    A2: q in {q} by TARSKI:def 1;
    A3: the carrier of product F = product Carrier F by GROUP_7:def 2;
    ex R being 1-sorted st
    R = F . q & (Carrier F) . q = the carrier of R
    by PRALG_1:def 15,A2; then
    A4: (Carrier F) . q = the carrier of G by A2,A1,FUNCOP_1:7;
    A5: dom (Carrier F) = {q} by PARTFUN1:def 2;
    let y be (the carrier of G)-valued total {q} -defined Function;
    A6: dom y = {q} by PARTFUN1:def 2; then
    y.q in rng y by FUNCT_1:3,A2; then
    reconsider z=y.q as Element of G;
    A7: for x be object st x in dom y holds y.x = z by TARSKI:def 1;
    now let i be object;
      assume A8:i in dom y; then
      A9: i = q by TARSKI:def 1;
      y.i = z by TARSKI:def 1,A8;
      hence y.i in (Carrier F) . i by A4,A9;
    end;
    hence thesis by A7,FUNCOP_1:11,A5,A6,CARD_3:9,A3;
  end;
