reserve k,t,i,j,m,n for Nat,
  x,y,y1,y2 for object,
  D for non empty set;
reserve K for Field,
  V for VectSp of K,
  a for Element of K,
  W for Element of V;
reserve KL1,KL2,KL3 for Linear_Combination of V,
  X for Subset of V;
reserve s for FinSequence,
  V1,V2,V3 for finite-dimensional VectSp of K,
  f,f1,f2 for Function of V1,V2,
  g for Function of V2,V3,
  b1 for OrdBasis of V1,
  b2 for OrdBasis of V2,
  b3 for OrdBasis of V3,
  v1,v2 for Vector of V2,
  v,w for Element of V1;
reserve p2,F for FinSequence of V1,
  p1,d for FinSequence of K,
  KL for Linear_Combination of V1;

theorem
  for M1 be Matrix of t,k,D, M2 be Matrix of m,k,D st n in dom M2 & i =
  len M1 + n holds Line(M1^M2,i) = Line(M2,n)
proof
  let M1 be Matrix of t,k,D;
  let M2 be Matrix of m,k,D;
  assume that
A1: n in dom M2 and
A2: i = len M1 + n;
  reconsider n1=n as Element of NAT by ORDINAL1:def 12;
  i in dom (M1^M2) by A1,A2,FINSEQ_1:28;
  hence Line(M1^M2,i) = (M1^M2).i by MATRIX_0:60
    .= M2.n1 by A1,A2,FINSEQ_1:def 7
    .= Line(M2,n) by A1,MATRIX_0:60;
end;
