theorem Th20:
  0.V1|-- b1 = len b1 |-> 0.K
proof
  per cases;
  suppose
A1: dom b1={};
    then
A2: len b1=0 by CARD_1:27,RELAT_1:41;
    len (0.V1|--b1)=len b1 by MATRLIN:def 7;
    hence 0.V1|-- b1 = {} by A1,CARD_1:27,RELAT_1:41
      .= len b1 |-> 0.K by A2;
  end;
  suppose
    dom b1<>{};
    then consider x being object such that
A3: x in dom b1 by XBOOLE_0:def 1;
A4: width 1.(K,len b1)=len b1 by MATRIX_0:24;
    reconsider x as Nat by A3;
    0.V1 = b1/.x-b1/.x by VECTSP_1:16
      .= b1/.x+(-1_K)*(b1/.x) by VECTSP_1:14;
    hence 0.V1|-- b1 = (b1/.x |-- b1) + ((-1_K)*(b1/.x) |--b1) by Th17
      .= (b1/.x |-- b1) + (-1_K)*((b1/.x) |--b1) by Th18
      .= Line(1.(K,len b1),x)+(-1_K)*((b1/.x) |--b1) by A3,Th19
      .= Line(1.(K,len b1),x)+(-1_K)*Line(1.(K,len b1),x) by A3,Th19
      .= Line(1.(K,len b1),x)+-Line(1.(K,len b1),x) by FVSUM_1:59
      .= len b1|->0.K by A4,FVSUM_1:26;
  end;
end;
