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 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;
