
theorem LM090:
  for x being real-valued FinSequence, i being Nat
  st x is positive & 1 <= i <= len x holds
  x | i is positive & x | i <> {}
  proof
    let x be real-valued FinSequence;
    let i be Nat;
    assume that
    A2: x is positive and
    A3: 1 <= i and
    A4: i <= len x;
    A5: dom (x | i) c= dom x by RELAT_1:60;
    A6: len (x | i) = i by A4,FINSEQ_1:59;
    for j being Nat st j in dom (x | i)
    holds 0 < (x | i) . j
    proof
      let j be Nat;
      assume A7: j in dom (x | i); then
      A8: 0 < x . j by A2,A5;
      Seg i c= Seg len x by A4,FINSEQ_1:5; then
      Seg i c= dom x by FINSEQ_1:def 3; then
      dom (x | i) = Seg i by RELAT_1:62; then
      j <= i by A7,FINSEQ_1:1;
      hence thesis by A8,FINSEQ_3:112;
    end;
    hence x | i is positive & x | i <> {} by A3,A6;
  end;
