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);
reserve D for non empty set,
  bD for FinSequence of D,
  b,f,g for FinSequence of K,
  MD for Matrix of D;

theorem
  len f = len g implies (LineVec2Mx f) + (LineVec2Mx g) = LineVec2Mx (f + g)
proof
  set Lf=LineVec2Mx f;
  set Lg=LineVec2Mx g;
A1: len Lf=1 by CARD_1:def 7;
  assume
A2: len f = len g;
  then reconsider F=f,G=g as Element of (len f)-tuples_on the carrier of K by
FINSEQ_2:92;
A3: width Lg=len f by A2,MATRIX_0:23;
  set FG=F+G;
  set Lfg=LineVec2Mx FG;
A6: len (Lf+Lg)=len Lf by MATRIX_3:def 3;
    dom Lf=Seg 1 & 1 in Seg 1 by A1,FINSEQ_1:def 3; then
    1 in dom Lf; then
    Line(Lf+Lg,1) = Line(Lf,1)+Line(Lg,1) by A3,MATRIX_0:23,MATRIX_4:59
      .= f+Line(Lg,1) by Th25
      .= f+g by Th25;
    hence thesis by A1,A6,Th25;
end;
