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 Th72:
  [:P,Q:] c= Indices M & card P = card Q implies Det EqSegm(a*M,P,
  Q) = power(K).(a,card P) * Det EqSegm(M,P,Q)
proof
  assume that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q;
  EqSegm(a*M,P,Q) = Segm(a*M,P,Q) by A2,Def3
    .= a * Segm(M,P,Q) by A1,Th63
    .= a * EqSegm(M,P,Q) by A2,Def3;
  hence thesis by Th71;
end;
