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 Th67:
  card P=card Q implies ([:P,Q:] c= Indices A iff P c= Seg len A &
  Q c= Seg width A)
proof
A1: Indices A=[:Seg len A,Seg width A:] by FINSEQ_1:def 3;
  assume
A2: card P = card Q;
  thus [:P,Q:] c= Indices A implies P c= Seg len A & Q c= Seg width A
  proof
    assume
A3: [:P,Q:] c= Indices A;
    per cases;
    suppose
      [:P,Q:]<>{};
      hence thesis by A1,A3,ZFMISC_1:114;
    end;
    suppose
A4:   [:P,Q:]={};
      then
A5:   Q={} by A2,CARD_1:27,ZFMISC_1:90;
      P={} by A2,A4,CARD_1:27,ZFMISC_1:90;
      hence thesis by A5;
    end;
  end;
  thus thesis by A1,ZFMISC_1:96;
end;
