
theorem Th30:
  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,
  K be Group,
  q be Element of I,
  x be Element of product F st
  not q in I0 & I = I0 \/ {q} & F = F0 +* (q .--> K) holds
  ex x0 be total I0 -defined Function,
  k be Element of K st x0 in product F0
  & x = x0 +* (q .--> k) & for p be Element of I0 holds x0.p in F0.p
  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,
    K be Group,
    q be Element of I,
    x be Element of product F;
    assume A1: not q in I0 & I = I0 \/ {q} & F = F0 +* (q .--> K);
    reconsider y=x as total I-defined Function by Lm6;
    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 F0 = I0 by PARTFUN1:def 2;
    A6: q in {q} by TARSKI:def 1;
    A7: q in dom (q .--> K) by TARSKI:def 1;
    A8: q in dom F0 \/ dom (q .--> K) by A6,XBOOLE_0:def 3;
    A9: F.q = (q .--> K).q by A7,A8,A1,FUNCT_4:def 1
    .= K by FUNCOP_1:7,A6;
    ex R be non empty multMagma st
    R = F.q & y.q in the carrier of R by Lm7; then
    reconsider k=y.q as Element of K by A9;
    reconsider y0 = y|I0 as I0-defined Function;
    A10: the carrier of product F0 = product (Carrier F0) by GROUP_7:def 2;
    I = dom y by PARTFUN1:def 2; then
    A11: dom y0 = I0 by RELAT_1:62,A1,XBOOLE_1:7; then
    reconsider y0 as total I0-defined Function by PARTFUN1:def 2;
    A12: dom (Carrier F0) = I0 by PARTFUN1:def 2;
    A13: for i be Element of I0
    holds y0.i in (Carrier F0).i & y0.i in F0.i
    proof
      let i be Element of I0;
      A14: i in dom F0 \/ dom (q .--> K) by A4,XBOOLE_0:def 3;
      i <> q by A1; then
      A15:not i in dom (q .--> K) by FUNCOP_1:75;
      A16: i in I by TARSKI:def 3,XBOOLE_1:7,A1;
      consider R being 1-sorted such that
A17:  R = F0 . i & (Carrier F0) . i = the carrier of R by PRALG_1:def 15;
      ex R being 1-sorted st
      R = F . i & (Carrier F) . i = the carrier of R by A16,PRALG_1:def 15;
      then
      A18: (Carrier F0) . i = (Carrier F) . i
        by A15,A14,FUNCT_4:def 1,A1,A17;
      ex g be Function st y = g & dom g = dom (Carrier F) &
      for i be object st i in dom Carrier F
      holds g.i in (Carrier F).i by CARD_3:def 5,A3;
      then y.i in (Carrier F).i by A16,A2;
      hence thesis by A17,FUNCT_1:49,A18;
    end;
    for i be object st i in dom (Carrier F0)
    holds y0.i in (Carrier F0).i by A13; then
    A19: y0 in the carrier of (product F0) by A10,A11,A12,CARD_3:def 5;
    A20: dom y = I by PARTFUN1:def 2; then
    y|{q} = q .--> k by FUNCT_7:6;
    then y = y|I0 +* (q .--> k) by A1,A20,FUNCT_4:70;
    hence thesis by A19,STRUCT_0:def 5,A13;
  end;
