reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;

theorem Th43:
  for N be finite without_zero Subset of NAT holds N is included_in_Seg
proof
  let N be finite without_zero Subset of NAT;
  consider k be Nat such that
A1: for n be Nat st n in N holds n<=k by STIRL2_1:56;
  take k;
  N c= Seg k
  proof
    let x be object;
    assume
A2: x in N;
    then consider n be Element of NAT such that
A3: n=x;
A4: n>=1 by A2,A3,NAT_1:14;
    n<=k by A1,A2,A3;
    hence thesis by A3,A4;
  end;
  hence thesis;
end;
