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 Th33:
  for f be Function of Seg n,Seg n st nt1 = nt * f holds Segm(A,
  nt1,mt) = Segm(A,nt,mt) * f
proof
  let f be Function of Seg n,Seg n such that
A1: nt1 = nt * f;
  set S=Segm(A,nt,mt);
  set S1=Segm(A,nt1,mt);
  set Sf=S*f;
  now
    let i,j such that
A2: [i,j] in Indices S1;
    Indices S1=[:Seg n,Seg width S1:] by MATRIX_0:25;
    then
A3: i in Seg n by A2,ZFMISC_1:87;
    Indices S1=Indices S by MATRIX_0:26;
    then consider k such that
A4: f.i = k and
A5: [k,j] in Indices S and
A6: Sf*(i,j) = S*(k,j) by A2,MATRIX11:37;
    reconsider i9=i,j9=j,k9=k as Element of NAT by ORDINAL1:def 12;
    Seg n=dom nt1 by FINSEQ_2:124;
    then nt1.i9=nt.(f.i) by A1,A3,FUNCT_1:12;
    hence S1*(i,j) = A*(nt.k9,mt.j9) by A2,A4,Def1
      .= Sf*(i,j) by A5,A6,Def1;
  end;
  hence thesis by MATRIX_0:27;
end;
