
theorem LM281:
  for F be FinSequence of REAL
  st for i be Element of dom F holds 0 <= F.i & F.i <= 1
  holds 0 <= Product F & Product F <= 1
  proof
    let F be FinSequence of REAL;
    assume
    A1: for i be Element of dom F holds 0 <= F.i & F.i <= 1;
    per cases;
    suppose
      ex i be Nat st i in dom F & F.i = 0;
      hence 0 <= Product F & Product F <= 1 by RVSUM_1:103;
    end;
    suppose
      A2: not ex i be Nat st i in dom F & F.i = 0;
      A3: for k being Element of NAT st k in dom F holds F . k > 0
      proof
        let k be Element of NAT;
        assume
        A4: k in dom F;
        F.k <> 0 by A2,A4;
        hence F.k > 0 by A1,A4;
      end;
      hence 0 <= Product F by NAT_4:42;
      1 is Element of REAL by XREAL_0:def 1; then
      reconsider G = (len F) |-> 1 as FinSequence of REAL by FINSEQ_2:63;
      A6: len G = len F by CARD_1:def 7;
      for k being Element of NAT st k in dom F holds F.k <= G.k & F.k > 0
      proof
        let k being Element of NAT;
        assume
        A7: k in dom F;
        A9: k in Seg len F by A7,FINSEQ_1:def 3;
        F.k <= 1 by A1,A7;
        hence F.k <= G.k by A9,FINSEQ_2:57;
        thus thesis by A3,A7;
      end; then
      Product F <= Product G by A6,NAT_4:54;
      hence Product F <= 1 by RVSUM_1:102;
    end;
  end;
