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 Th38:
  for F be FinSequence of D,i,nt st not i in rng nt & [:rng nt,rng
  mt:] c= Indices A9 holds Segm(A9,nt,mt) = Segm(RLine(A9,i,F),nt,mt)
proof
  let F be FinSequence of D,i,nt such that
A1: not i in rng nt and
A2: [:rng nt,rng mt:] c= Indices A9;
  set S=Segm(A9,nt,mt);
  set R=RLine(A9,i,F);
  set SR=Segm(R,nt,mt);
  per cases;
  suppose
    len F<>width A9;
    hence thesis by MATRIX11:def 3;
  end;
  suppose
A3: len F=width A9;
A4: Indices SR = Indices S by MATRIX_0:26;
    now
A5:   dom nt=Seg n by FINSEQ_2:124;
      let k,m such that
A6:   [k,m] in Indices SR;
      Indices SR=[:Seg n,Seg width SR:] by MATRIX_0:25;
      then k in Seg n by A6,ZFMISC_1:87;
      then
A7:   i <> nt.k by A1,A5,FUNCT_1:def 3;
      reconsider K=k,M=m as Element of NAT by ORDINAL1:def 12;
      [nt.K,mt.M] in Indices A9 by A2,A4,A6,Th17;
      then
A8:   A9*(nt.K,mt.M)=R*(nt.K,mt.M) by A3,A7,MATRIX11:def 3;
      S*(K,M)=A9*(nt.K,mt.M) by A4,A6,Def1;
      hence SR*(k,m)=S*(k,m) by A6,A8,Def1;
    end;
    hence thesis by MATRIX_0:27;
  end;
end;
