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;
