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;

theorem Th23:
  n = 2 & m = 2 implies Segm(A,nt,mt) = ( A*(nt.1,mt.1),A*(nt.1,mt
  .2) )][( A*(nt.2,mt.1),A*(nt.2,mt.2) )
proof
  set S=Segm(A,nt,mt);
  set I=Indices S;
  assume that
A1: n = 2 and
A2: m = 2;
A3: I=[:Seg 2,Seg 2:] by A1,A2,MATRIX_0:24;
A4: 2 in Seg 2;
  then [2,2] in I by A3,ZFMISC_1:87;
  then
A5: S*(2,2)=A*(nt.2,mt.2) by Def1;
A6: 1 in Seg 2;
  then [1,1] in I by A3,ZFMISC_1:87;
  then
A7: S*(1,1)=A*(nt.1,mt.1) by Def1;
  [2,1] in I by A6,A4,A3,ZFMISC_1:87;
  then
A8: S*(2,1)=A*(nt.2,mt.1) by Def1;
  [1,2] in I by A6,A4,A3,ZFMISC_1:87;
  then S*(1,2)=A*(nt.1,mt.2) by Def1;
  hence thesis by A1,A2,A7,A8,A5,Th22;
end;
