
theorem LM100:
  for x being real-valued FinSequence,
  I being set
  st x is positive & I c= dom x & I <> {}
  holds
  Seq (x, I) is positive & Seq (x, I) <> {}
  proof
    let x be real-valued FinSequence;
    let I be set;
    assume that
    A2: x is positive and
    A3: I c= dom x and
    A4: I <> {};
    A5: dom x meets I by A4,A3,XBOOLE_1:28;
    for j being Nat st j in dom (Seq (x, I)) holds 0 < Seq (x, I) . j
    proof
      let j be Nat;
      assume A6: j in dom (Seq (x, I));
      reconsider sfi = Seq (x | I) as real-valued FinSequence;
      A9: (Sgm (dom (x | I))) . j in dom (x | I) by A6,FUNCT_1:11;
      A10: dom (x | I) c= dom x by RELAT_1:60;
      Seq (x, I) . j = (x | I) . ((Sgm (dom (x | I))) . j) by A6,FUNCT_1:12
      .= x . ((Sgm (dom (x | I))) . j) by A9,FUNCT_1:47;
      hence thesis by A10,A2,A9;
    end;
    hence thesis by A5,RELAT_1:66,FINSEQ_1:97;
  end;
