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 Th34:
  for f be Function of Seg m,Seg m st mt1 = mt * f holds Segm(A,nt
  ,mt1)@ = Segm(A,nt,mt)@ * f
proof
  let f be Function of Seg m,Seg m such that
A1: mt1 = mt * f;
  set S=Segm(A,nt,mt);
  set S1=Segm(A,nt,mt1);
  per cases;
  suppose
A2: m=0 or n=0;
    len S=n by MATRIX_0:def 2;
    then width S=0 by A2,Th1,MATRIX_0:def 3;
    then len (S@)=0 by MATRIX_0:def 6;
    then
A3: S@={};
    len S1=n by MATRIX_0:def 2;
    then width S1=0 by A2,Th1,MATRIX_0:def 3;
    then len (S1@)=0 by MATRIX_0:def 6;
    then S1@={};
    hence thesis by A3;
  end;
  suppose
A4: m>0 & n>0;
    then
A5: width S1=m by Th1;
    then
A6: len (S1@)=m by A4,MATRIX_0:54;
    len S1=n by A4,Th1;
    then
A7: width (S1@)=n by A4,A5,MATRIX_0:54;
A8: width S=m by A4,Th1;
    len S=n by A4,Th1;
    then
A9: width (S@)=n by A4,A8,MATRIX_0:54;
    len (S@)=m by A4,A8,MATRIX_0:54;
    then reconsider S9=S@,S19=S1@ as Matrix of m,n,D by A4,A9,A7,A6,MATRIX_0:20
;
    set Sf=S9*f;
    now
      let i,j such that
A10:  [i,j] in Indices S19;
A11:  [j,i] in Indices S1 by A10,MATRIX_0:def 6;
      then
A12:  S19*(i,j) = S1*(j,i) by MATRIX_0:def 6;
      Indices S19=[:Seg m,Seg n:] by A7,MATRIX_0:25;
      then
A13:  i in Seg m by A10,ZFMISC_1:87;
      Indices S19=Indices S9 by MATRIX_0:26;
      then consider k such that
A14:  f.i=k and
A15:  [k,j] in Indices S9 and
A16:  Sf*(i,j)=S9*(k,j) by A10,MATRIX11:37;
      reconsider i9=i,j9=j,k9=k as Element of NAT by ORDINAL1:def 12;
      Seg m=dom mt1 by FINSEQ_2:124;
      then mt1.i9=mt.(f.i) by A1,A13,FUNCT_1:12;
      then
A17:  S1*(j9,i9) = A*(nt.j9,mt.k9) by A14,A11,Def1;
A18:  [j,k] in Indices S by A15,MATRIX_0:def 6;
      then S9*(k,j) = S*(j,k) by MATRIX_0:def 6;
      hence S19*(i,j)=Sf*(i,j) by A16,A18,A12,A17,Def1;
    end;
    hence thesis by MATRIX_0:27;
  end;
end;
