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;
reserve v,v1,v2,u,w for Vector of n-VectSp_over K,
  t,t1,t2 for Element of n -tuples_on the carrier of K,

  L for Linear_Combination of n-VectSp_over K,
  M,M1 for Matrix of m,n,K;

theorem Th118:
  P c= Seg m implies lines Segm(M,P,Seg n) c= lines M
proof
  set S=Segm(M,P,Seg n);
  assume
A1: P c= Seg m;
A2: rng Sgm P = P by FINSEQ_1:def 14;
  let x be object;
  assume x in lines S;
  then consider i such that
A3: i in Seg card P and
A4: x = Line(S,i) by Th103;
  Seg m <> {} by A1,A3;
  then m <> 0;
  then width M=n by Th1;
  then
A5: Line(S,i)=Line(M,Sgm P.i) by A3,Th48;
  dom Sgm P=Seg card P by FINSEQ_3:40;
  then Sgm P.i in rng Sgm P by A3,FUNCT_1:def 3;
  hence thesis by A1,A4,A2,A5,Th103;
end;
