reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;

theorem Th1:
  n>=1 implies Seg n = {1} \/ {k where k is Element of NAT: 1<k & k <n} \/ {n}
proof
  assume
A1: n>=1;
A2: {1} \/ {k where k is Element of NAT: 1<k & k<n} \/ {n} c= Seg n
  proof
    let d be object;
A3: {k where k is Element of NAT: 1<k & k<n} c= Seg n
    proof
      let d be object;
      assume d in {k where k is Element of NAT: 1<k & k<n};
      then ex k being Element of NAT st d = k & 1<k & k<n;
      hence thesis;
    end;
    1 in Seg n by A1; then
    {1} c= Seg n by ZFMISC_1:31; then
A4: {1} \/ {k where k is Element of NAT: 1<k & k<n} c= Seg n by A3,XBOOLE_1:8;
    n in Seg n by A1; then
    {n} c= Seg n by ZFMISC_1:31; then
    {1} \/ {k where k is Element of NAT: 1<k & k<n} \/ {n} c= Seg n
      by A4,XBOOLE_1:8;
    hence thesis;
  end;
  Seg n c= {1} \/ {k where k is Element of NAT: 1<k & k<n} \/ {n}
  proof
    let d be object;
    assume
A5: d in Seg n;
    per cases by A1,XXREAL_0:1;
    suppose
A6:   n>1;
      reconsider l = d as Element of NAT by A5;
A7:   l <= n by A5,FINSEQ_1:1;
      1 <= l by A5,FINSEQ_1:1;
      then 1 = l or 1<l & l<n or l = n or 1 = n by A7,XXREAL_0:1;
      then
      d in {1} or d in {k where k is Element of NAT: 1<k & k<n} or d in {n
      } by A6,TARSKI:def 1;
      then
      d in {1} \/ {k where k is Element of NAT: 1<k & k<n} or d in {n} by
XBOOLE_0:def 3;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      n = 1;
      hence thesis by A5,FINSEQ_1:2,XBOOLE_0:def 3;
    end;
  end;
  hence thesis by A2;
end;
