theorem Th33:
  for M be Matrix of len b1,len B2,K st len b1 = 0 holds Mx2Tran(M
  ,b1,B2).v1 = 0.V2
proof
  let M be Matrix of len b1,len B2,K such that
A1: len b1 = 0;
  set L=LineVec2Mx(v1|--b1);
  set LM=L*M;
  set LL=Line(LM,1);
A2: width L=len (v1|--b1) & len (v1|--b1)=len b1 by MATRIX_0:23,MATRLIN:def 7;
A3: len M=len b1 by MATRIX_0:def 2;
  then width M=0 by A1,MATRIX_0:def 3;
  then width LM=0 by A2,A3,MATRIX_3:def 4;
  then
A4: dom LL={};
  dom lmlt(LL,B2)=dom LL/\dom B2 by Lm1;
  then lmlt(LL,B2)=<*>(the carrier of V2) by A4;
  then Sum lmlt(LL,B2)=0.V2 by RLVECT_1:43;
  hence thesis by Def3;
end;
