reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve p for Prime;

theorem
  for f being with_values_greater_or_equal_one real-valued FinSequence
  holds Product f >= 1
  proof
    let f be with_values_greater_or_equal_one real-valued FinSequence;
    defpred P[FinSequence of REAL] means
    for g being with_values_greater_or_equal_one real-valued FinSequence st
    g = $1 holds Product $1 >= 1;
A1: P[<*>REAL] by RVSUM_1:94;
A2: for p being FinSequence of REAL for x being Element of REAL
    st P[p] holds P[p^<*x*>]
    proof
      let p be FinSequence of REAL;
      let x be Element of REAL;
      assume
A3:   P[p];
      let g be with_values_greater_or_equal_one real-valued FinSequence
      such that
A4:   g = p^<*x*>;
      <*x*> is with_values_greater_or_equal_one by A4,Th19;
      then
A5:   x >= 1 by Th20;
      p is with_values_greater_or_equal_one by A4,Th19;
      then
A6:   Product p >= 1 by A3;
      Product(p^<*x*>) = x * Product p by RVSUM_1:96;
      then Product(p^<*x*>) >= 1*1 by A5,A6,XREAL_1:66;
      hence thesis;
    end;
A7: for p being FinSequence of REAL holds P[p] from FINSEQ_2:sch 2(A1,A2);
    f is FinSequence of REAL by FINSEQ_1:106;
    hence thesis by A7;
  end;
