
theorem Th22:
  for I0,I be non empty finite set,
  F0 be associative Group-like multMagma-Family of I0,
  F be associative Group-like multMagma-Family of I,
  H,K be Group,
  q be Element of I,
  k be Element of K,
  g be Function st
  g in the carrier of product F0 &
  not q in I0 & I = I0 \/ {q} & F = F0 +* (q .--> K) holds
  g +* (q .--> k) in the carrier of product F
  proof
    let I0,I be non empty finite set,
    F0 be associative Group-like multMagma-Family of I0,
    F be associative Group-like multMagma-Family of I,
    H,K be Group,
    q be Element of I,
    k be Element of K,
    g be Function;
    assume A1: g in the carrier of product F0 & not q in I0 & I = I0 \/ {q} &
    F = F0 +* (q .--> K);
    set HK = <*H,K*>;
    A2: dom Carrier F0 = I0 by PARTFUN1:def 2;
    A3: dom (q .--> k) = {q};
    A4: dom (q .--> K) = {q};
    A5: dom F0 = I0 by PARTFUN1:def 2;
    set w = g +* (q .--> k);
    A6:g in product (Carrier F0) by A1,GROUP_7:def 2;
    then A7:
    ex g0 be Function st g = g0 & dom g0 = dom (Carrier F0) &
    for y be object st y in dom (Carrier F0) holds g0.y in (Carrier F0).y
    by CARD_3:def 5;
    dom w = (dom g) \/ (dom (q .--> k)) by FUNCT_4:def 1
    .= I0 \/ (dom (q .--> k)) by PARTFUN1:def 2,A6
    .= I by A1;
    then
    A8: dom w = dom (Carrier F) by PARTFUN1:def 2;
    for x be object st x in dom (Carrier F)
    holds w.x in (Carrier F).x
    proof
      let x be object;
      assume A9: x in dom (Carrier F);
      per cases by A1,XBOOLE_0:def 3,A9;
      suppose A10: x in I0;
        A11: not x in {q} by A1,A10,TARSKI:def 1;
        then
        A12: F.x = F0.x by A1,FUNCT_4:def 1,A5,A4,A9;
        consider R1 being 1-sorted such that
        A13: R1 = F0.x & (Carrier F0).x = the carrier of R1
        by PRALG_1:def 15,A10;
        consider R2 being 1-sorted such that
        A14: R2 = F.x
        & (Carrier F).x = the carrier of R2 by PRALG_1:def 15,A9;
        w.x = g.x by FUNCT_4:def 1,A7,A2,A3,A9,A1,A11;
        hence w.x in (Carrier F).x by A13,A14,A12,A10,A2,A7;
      end;
      suppose A15: x in {q}; then
        F.x = (q .--> K).x by A1,FUNCT_4:def 1,A5,A4; then
        A16: F.x = K by A15,FUNCOP_1:7;
        A17: w.x = (q .--> k).x by A7,A2,A3,A1,FUNCT_4:def 1,A15
        .= k by A15,FUNCOP_1:7;
        ex R2 being 1-sorted st R2 = F.x
        & (Carrier F).x = the carrier of R2 by PRALG_1:def 15,A15;
        hence w.x in (Carrier F).x by A17,A16;
      end;
    end;
    then w in product (Carrier F) by A8,CARD_3:def 5;
    hence thesis by GROUP_7:def 2;
  end;
