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 Th40:
  for F be FinSequence of D holds not i in Seg len A9 implies
  RLine(A9,i,F) = A9
proof
  let F be FinSequence of D;
  assume
A1: not i in Seg len A9;
  set R=RLine(A9,i,F);
  per cases;
  suppose
A2: len F=width A9;
A3: now
      let k such that
A4:   1 <= k and
A5:   k <= len A9;
A6:   k in Seg len A9 by A4,A5;
A7:   len A9=n9 by MATRIX_0:def 2;
      then
A8:   R.k=Line(R,k) by A6,MATRIX_0:52;
      Line(R,k)=Line(A9,k) by A1,A6,A7,MATRIX11:28;
      hence R.k=A9.k by A6,A7,A8,MATRIX_0:52;
    end;
    len A9=len R by A2,MATRIX11:def 3;
    hence thesis by A3;
  end;
  suppose
    len F<>width A9;
    hence thesis by MATRIX11:def 3;
  end;
end;
