reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;

theorem
  N c= Segm n & N <> {n-1} implies min* N < n-1
proof
  assume that
A1: N c= Segm n and
A2: N<>{n-1} and
A3: min* N >=n-1;
  now
    let k be object such that
A4: k in N;
    reconsider k9=k as Element of NAT by A4;
    min* N <=k9 by A4,NAT_1:def 1;
    then
A5: n-1 <=k9 by A3,XXREAL_0:2;
    k9 <= n-1 by A1,A4,Th10;
    then n-1 = k by A5,XXREAL_0:1;
    hence k in {n-1} by TARSKI:def 1;
  end;
  then N c= {n-1};
  hence thesis by A2,ZFMISC_1:33;
end;
