theorem Th32:
  for M be Matrix of len b1,len b2,K st len b1 > 0 holds
  LineVec2Mx(Mx2Tran(M,b1,b2).v1|--b2) = LineVec2Mx(v1|--b1) * M
proof
  set L=LineVec2Mx(v1|--b1);
A1: width L=len (v1|--b1) & len (v1|--b1)=len b1 by MATRIX_0:23,MATRLIN:def 7;
  let M be Matrix of len b1,len b2,K such that
A2: len b1 > 0;
A3: len M=len b1 by A2,MATRIX_0:23;
  set LM=L*M;
  width M=len b2 by A2,MATRIX_0:23;
  then width LM=len b2 by A1,A3,MATRIX_3:def 4;
  then len Line(LM,1)=len b2 by CARD_1:def 7;
  then
A4: Sum lmlt (Line(LM,1),b2) |--b2=Line(LM,1) by MATRLIN:36;
  len L=1 by MATRIX_0:23;
  then len LM=1 by A1,A3,MATRIX_3:def 4;
  hence LM = LineVec2Mx(Sum lmlt (Line(LM,1),b2) |--b2) by A4,MATRIX15:25
    .= LineVec2Mx(Mx2Tran(M,b1,b2).v1|--b2) by Def3;
end;
