reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

theorem Th35:
  for A,B be Matrix of K st i in dom A & width A = len B holds
  LineVec2Mx(Line(A,i)) * B = LineVec2Mx(Line(A*B,i))
proof
  let A,B be Matrix of K such that
A1: i in dom A and
A2: width A = len B;
A3: width (A*B)=width B by A2,MATRIX_3:def 4;
  set LAB=LineVec2Mx(Line(A*B,i));
A4: width LAB=len Line(A*B,i) & len Line(A*B,i) =width (A*B) by CARD_1:def 7
,MATRIX_0:23;
  set L=LineVec2Mx(Line(A,i));
A5: width L=len Line(A,i) & len Line(A,i)=width A by CARD_1:def 7,MATRIX_0:23;
  then
A6: width (L*B)=width B by A2,MATRIX_3:def 4;
  len L=1 by CARD_1:def 7;
  then
A7: len (L*B)=1 by A2,A5,MATRIX_3:def 4;
  len (A*B)=len A by A2,MATRIX_3:def 4;
  then
A8: dom A=dom (A*B) by FINSEQ_3:29;
A9: now
    let j,k such that
A10: [j,k] in Indices (L*B);
A11: k in Seg width (A*B) by A3,A6,A10,ZFMISC_1:87;
    then
A12: [i,k] in Indices (A*B) by A1,A8,ZFMISC_1:87;
    Indices (L*B)=[:Seg 1,Seg width B:] by A7,A6,FINSEQ_1:def 3;
    then j in Seg 1 by A10,ZFMISC_1:87;
    then
A13: j=1 by FINSEQ_1:2,TARSKI:def 1;
    hence (L*B)*(j,k) = Line(L,1)"*"Col(B,k) by A2,A5,A10,MATRIX_3:def 4
      .= Line(A,i)"*"Col(B,k) by MATRIX15:25
      .= (A*B)*(i,k) by A2,A12,MATRIX_3:def 4
      .= Line(A*B,i).k by A11,MATRIX_0:def 7
      .= Line(LAB,j).k by A13,MATRIX15:25
      .= LAB*(j,k) by A4,A11,MATRIX_0:def 7;
  end;
  len LAB=1 by CARD_1:def 7;
  hence thesis by A4,A3,A7,A6,A9,MATRIX_0:21;
end;
