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;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;

theorem Th53:
  Sgm P " X is without_zero finite Subset of NAT
proof
A1: Sgm P " X c= dom (Sgm P) by RELAT_1:132;
  dom (Sgm P) = Seg card P by FINSEQ_3:40;
  then not 0 in Sgm P"X by A1;
  hence thesis by A1,MEASURE6:def 2,XBOOLE_1:1;
end;
