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 Th48:
  i in Seg card P implies Line(Segm(A,P,Seg width A),i) = Line(A, Sgm P.i)
proof
  assume
A1: i in Seg card P;
  set S=Seg width A;
  set sP=Sgm P;
  len Line(A,sP.i)=width A by MATRIX_0:def 7;
  then
A2: dom Line(A,sP.i)=S by FINSEQ_1:def 3;
  Sgm S = idseq width A by FINSEQ_3:48;
  then Line(A,sP.i) * Sgm S = Line(A,sP.i) by A2,RELAT_1:52;
  hence thesis by A1,Th47;
end;
