reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);

theorem Th15:
  for D be non empty set, A be Matrix of n,m,D, B be Matrix of n,k
  ,D for i st i in Seg n holds Line(A^^B,i)=Line(A,i)^Line(B,i)
proof
  let D be non empty set, A be Matrix of n,m,D, B be Matrix of n,k,D;
  set AB=A^^B;
A1: len AB=n & dom AB=Seg len AB by FINSEQ_1:def 3,MATRIX_0:def 2;
  let i such that
A2: i in Seg n;
  Line(A,i)=A.i & Line(B,i)=B.i by A2,MATRIX_0:52;
  hence Line(A,i)^Line(B,i) = AB.i by A2,A1,PRE_POLY:def 4
    .= Line(AB,i) by A2,MATRIX_0:52;
end;
