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;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;

theorem Th59:
  for F,FQ be FinSequence of D st len F = width A9 &
  FQ = F * Sgm Q & [:P,Q:] c= Indices A9 holds
  RLine(Segm(A9,P,Q),i,FQ) = Segm(RLine(A9,Sgm P. i,F),P,Q)
proof
  let F,FQ be FinSequence of D such that
A1: len F = width A9 and
A2: FQ = F * Sgm Q and
A3: [:P,Q:] c= Indices A9;
  set SQ=Sgm Q;
  set SP=Sgm P;
A4: card P=len Segm(A9,P,Q) by MATRIX_0:def 2;
A5: rng SQ=Q by FINSEQ_1:def 14;
A7: rng SP=P by FINSEQ_1:def 14;
A8: SP is one-to-one by FINSEQ_3:92;
A9: dom SP=Seg card P by FINSEQ_3:40;
  per cases;
  suppose
    i in dom SP;
    then SP"{SP.i} = {i} by A8,FINSEQ_5:4;
    hence thesis by A1,A2,A3,A7,A5,Th37;
  end;
  suppose
A10: not i in dom SP;
A11: not 0 in Seg len A9;
    SP.i=0 by A10,FUNCT_1:def 2;
    hence Segm(RLine(A9,SP.i,F),P,Q) = Segm(A9,P,Q) by A11,Th40
      .= RLine(Segm(A9,P,Q),i,FQ) by A9,A4,A10,Th40;
  end;
end;
