
theorem Th33:
  for G being commutative Group,
  I0,I be non empty finite set,
  q be Element of I,
  x be (the carrier of G)-valued total I -defined Function,
  x0 be (the carrier of G)-valued total I0 -defined Function,
  k be Element of G st
  not q in I0 & I = I0 \/ {q} & x = x0 +* (q .--> k)
  holds
  Product x = (Product x0)*k
  proof
    let G be commutative Group,
    I0,I be non empty finite set,
    q be Element of I,
    x be (the carrier of G)-valued total I -defined Function,
    x0 be (the carrier of G)-valued total I0 -defined Function,
    k be Element of G;
    assume A1: not q in I0 & I = I0 \/ {q} & x = x0 +* (q .--> k);
    reconsider y = (q .--> k) as (the carrier of G)-valued
    total {q} -defined Function;
    A2: I0 misses {q}
    proof
      assume I0 meets {q}; then
      consider x be object such that
      A3: x in I0 & x in {q} by XBOOLE_0:3;
      thus contradiction by A3,A1,TARSKI:def 1;
    end;
    Product x = (Product x0) * (Product y) by A2,A1,Th8;
    hence thesis by Th9;
  end;
