reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th4:
  for a being Complex, p,q being complex-valued FinSequence st
  len p = len q &
  ex i being Nat st i in dom p & q.i = a * p.i &
  for j being Nat st j in dom p & i <> j holds q.j = p.j
  holds Product q = a * Product p
  proof
    let a be Complex;
    let p,q be complex-valued FinSequence such that
A1: len p = len q;
    given i being Nat such that
A2: i in dom p and
A3: q.i = a * p.i and
A4: for j being Nat st j in dom p & i <> j holds q.j = p.j;
    reconsider i as Element of NAT by ORDINAL1:def 12;
A5: the carrier of F_Complex = COMPLEX by COMPLFLD:def 1;
    then reconsider a1 = a as Element of F_Complex by XCMPLX_0:def 2;
    reconsider p1 = p, q1 = q as FinSequence of F_Complex by A5,FINSEQ_1:107;
A6: dom p1 = dom q1 by A1,FINSEQ_3:29;
    then q1/.i = q.i & p1/.i = p.i by A2,PARTFUN1:def 6;
    then
A7: q1/.i = a1 * p1/.i by A3;
    for j being Element of NAT st j in dom p1 & i <> j holds q1/.j = p1/.j
    proof
      let j be Element of NAT;
      assume
A8:   j in dom p1;
      then q1/.j = q.j & p1/.j = p.j by A6,PARTFUN1:def 6;
      hence thesis by A4,A8;
    end;
    then
A9: Product q1 = a1 * Product p1 by A1,A2,A7,POLYNOM2:6;
    Product p = Product p1 & Product q = Product q1 by Th3;
    hence Product q = a * Product p by A9;
  end;
