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 Th73:
  for P,Q st [:P,Q:] c= Indices M & card P = card Q holds card P
  <= len M & card Q <= width M
proof
  let P,Q such that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q;
  Q c= Seg width M by A1,A2,Th67;
  then
A3: card Q <=card Seg width M by NAT_1:43;
  P c= Seg len M by A1,A2,Th67;
  then card P<=card Seg len M by NAT_1:43;
  hence thesis by A3,FINSEQ_1:57;
end;
