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 Th20:
  for A be Matrix of 1,D holds A = <*<* A*(1,1) *>*>
proof
  let A be Matrix of 1,D;
  reconsider AA=<*<* A*(1,1) *>*> as Matrix of 1,D by MATRIX_0:15;
  now
A1: Indices A=[:Seg 1,Seg 1:] by MATRIX_0:24;
    let i,j such that
A2: [i,j] in Indices A;
    j in {1} by A2,A1,FINSEQ_1:2,ZFMISC_1:87;
    then
A3: j=1 by TARSKI:def 1;
    i in {1} by A2,A1,FINSEQ_1:2,ZFMISC_1:87;
    then i=1 by TARSKI:def 1;
    hence AA*(i,j)=A*(i,j) by A3,MATRIX_0:49;
  end;
  hence thesis by MATRIX_0:27;
end;
