
theorem IncFinNAT:
  for n being Nat holds
    (Seg 1) --> n is increasing FinSequence of NAT
  proof
    let n be Nat;
    set f = (Seg 1) --> n;
B1: {n} c= NAT by ZFMISC_1:31,ORDINAL1:def 12;
a4: rng f c= {n} by FUNCOP_1:13;
    dom f = Seg 1 by FUNCOP_1:13; then
    f is FinSequence by FINSEQ_1:def 2; then
    reconsider f as FinSequence of NAT
      by a4,FINSEQ_1:def 4,B1,XBOOLE_1:1;
    for m,n being Nat st m in dom f & n in dom f & m < n holds f.m < f.n
    proof
      let m,n be Nat;
A0:   dom f = {1} by FUNCOP_1:13,FINSEQ_1:2;
      assume
A1:   m in dom f & n in dom f & m < n; then
      m = 1 & n = 1 by A0,TARSKI:def 1;
      hence thesis by A1;
    end;
    hence thesis by SEQM_3:def 1;
  end;
