reserve i,j,n,n1,n2,m,k,l,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat,
        F for XFinSequence,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;
reserve x,y,x1,u,w for Nat;

theorem
  for x,y,x1 being Nat st x1 = 0 holds
    y = Product (1+(x1 * idseq x)) iff y = 1
proof
  let x,y,x1 be Nat such that A1: x1 = 0;
A2: len (idseq x) =x;
rng (idseq x) c= REAL;
then idseq x is FinSequence of REAL by FINSEQ_1:def 4;
then idseq x is Element of x-tuples_on REAL by A2,FINSEQ_2:92;
then x1 * idseq x = x|->0 by A1,RVSUM_1:53;
then 1+(x1 * idseq x) = x|->(1+0) by Th21;
hence thesis by RVSUM_1:102;
end;
