
theorem ProductA:
  for f be positive real-valued FinSequence,
      i be Nat,
      a be Real st i in dom f holds
    Product (f+*(i,a)) = (Product f) * a / (f.i)
  proof
    let f be positive real-valued FinSequence,
        i be Nat,
        a be Real;
    reconsider w = f as FinSequence of REAL by RVSUM_1:145;
    reconsider aa = a as Element of REAL by XREAL_0:def 1;
    assume
A1: i in dom f; then
Z1: Product (w+*(i,a)) = Product ((w | (i-'1)) ^ <*aa*> ^ (w/^i))
       by CopyForValued
    .= Product ((w | (i-'1)) ^ <*aa*>) * Product (w/^i) by RVSUM_1:97
    .= Product (w | (i-'1)) * Product <*aa*> * Product (w/^i) by RVSUM_1:97
    .= aa * (Product (w | (i-'1)) * Product (w/^i))
    .= aa * Product ((w | (i-'1)) ^ (w/^i)) by RVSUM_1:97;
    reconsider fi = f.i as Real;
ZZ: fi <> 0 by A1,PosDef;
    1 <= i & i <= len w by A1,FINSEQ_3:25; then
zz: Product w = Product ((w | (i-'1)) ^ <*f.i*> ^ (w/^i)) by FINSEQ_5:84
    .= Product ((w | (i-'1)) ^ <*f.i*>) * Product (w/^i) by RVSUM_1:97
    .= Product (w | (i-'1)) * Product <*f.i*> * Product (w/^i) by RVSUM_1:97
    .= fi * (Product (w | (i-'1)) * Product (w/^i))
    .= fi * Product ((w | (i-'1)) ^ (w/^i)) by RVSUM_1:97;
    Product (w+*(i,a)) = aa * (Product w / fi) by ZZ,XCMPLX_1:89,zz,Z1
      .= aa * (Product w * (1 / fi)) by XCMPLX_1:99
      .= Product w * aa * (1 / fi)
      .= (Product f) * a / (f.i) by XCMPLX_1:99;
    hence thesis;
  end;
