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 Th55:
  [:Sgm P " X,Sgm Q " Y:] c= Indices Segm(A,P,Q)
proof
  set SP=Sgm P;
  set SQ=Sgm Q;
  set I=Indices Segm(A,P,Q);
A1: now
    per cases;
    suppose
A2:   card P=0;
      then Seg card P = {};
      then [:Seg card P,Seg card Q:]={} by ZFMISC_1:90;
      hence [:Seg card P,Seg card Q:]=I by A2,MATRIX_0:22;
    end;
    suppose
      card P>0;
      hence [:Seg card P,Seg card Q:]=I by MATRIX_0:23;
    end;
  end;
  dom SQ=Seg card Q by FINSEQ_3:40;
  then
A3: SQ " Y c= Seg card Q by RELAT_1:132;
  dom SP=Seg card P by FINSEQ_3:40;
  then SP " X c= Seg card P by RELAT_1:132;
  hence thesis by A3,A1,ZFMISC_1:96;
end;
