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 Th106:
  i in Seg len M & a = L.(M.i) implies Line(FinS2MX(L (#) MX2FinS
  M),i) = a * Line(M,i)
proof
  assume that
A1: i in Seg len M and
A2: a = L.(M.i);
  set MX=MX2FinS M;
  set LM=L (#) MX;
  i in dom M by A1,FINSEQ_1:def 3;
  then
A3: M.i=MX/.i by PARTFUN1:def 6;
  len M=m by MATRIX_0:def 2;
  then
A4: Line(M,i)=M.i by A1,MATRIX_0:52;
  then reconsider
  L=Line(M,i) as Element of n-tuples_on the carrier of K by A3,Th102;
  set FLM = FinS2MX LM;
A5: len LM=len M by VECTSP_6:def 5;
  then
A6: i in dom FLM by A1,FINSEQ_1:def 3;
  Line(FLM,i)=FLM.i by A1,A5,MATRIX_0:52;
  hence Line(FLM,i) = a *MX/.i by A2,A3,A6,VECTSP_6:def 5
    .= a*L by A3,A4,Th102
    .= a*Line(M,i);
end;
