theorem Th46:
  for M be OrdBasis of (len b2)-VectSp_over K st 
  M = MX2FinS 1.(K,len b2) 
  for v1 be Vector of (len b2)-VectSp_over K holds v1|--M = v1
proof
  let M be OrdBasis of (len b2)-VectSp_over K such that
A1: M = MX2FinS 1.(K,len b2);
  let v1 be Vector of (len b2)-VectSp_over K;
  set vM=v1|--M;
  consider KL be Linear_Combination of (len b2)-VectSp_over K such that
A2: v1 = Sum(KL) & Carrier KL c= rng M and
A3: for k st 1<=k & k<=len (v1|--M) holds vM/.k=KL.(M/.k) by MATRLIN:def 7;
  reconsider t1=v1 as Element of (len b2)-tuples_on the carrier of K by
MATRIX13:102;
A4: len t1=len b2 by CARD_1:def 7;
A5: len M=dim ((len b2)-VectSp_over K) & dim ((len b2)-VectSp_over K) =len
  b2 by Th21,MATRIX13:112;
A6: len vM=len M by MATRLIN:def 7;
  now
A7: dom M= dom vM by A6,FINSEQ_3:29;
A8: the_rank_of 1.(K,len b2)=len b2 by Lm6;
    set F=FinS2MX(KL (#) M);
A9: Indices 1.(K,len b2)=[:Seg len b2,Seg len b2:] by MATRIX_0:24;
    let i such that
A10: 1<=i & i<=len t1;
A11: i in Seg len b2 by A4,A10;
    then
A12: [i,i] in [:Seg len b2,Seg len b2:] by ZFMISC_1:87;
A13: width 1.(K,len b2)=len b2 by MATRIX_0:24;
    then
A14: Line(1.(K,len b2),i).i = 1.(K,len b2)*(i,i) by A11,MATRIX_0:def 7
      .= 1_K by A9,A12,MATRIX_1:def 3;
A15: len Col(F,i)=len F by CARD_1:def 7;
    then
A16: dom Col(F,i)=dom F by FINSEQ_3:29;
A17: len F=len M by VECTSP_6:def 5;
    then
A18: dom F= dom M by FINSEQ_3:29;
A19: i in dom Col(F,i) by A4,A5,A10,A17,A15,FINSEQ_3:25;
A20: width F=len b2 by A5,A17,MATRIX_0:24;
    now
      let j such that
A21:  j in dom Col(F,i) and
A22:  j<>i;
A23:  dom Col(F,i)=Seg len b2 by A5,A17,A15,FINSEQ_1:def 3;
      then
A24:  [j,i] in [:Seg len b2,Seg len b2:] by A11,A21,ZFMISC_1:87;
A25:  Line(F,j) = (KL (#) M).j by A5,A17,A21,A23,MATRIX_0:52
        .= KL.(M/.j) * M/.j by A16,A21,VECTSP_6:def 5;
A26:  Col(F,i).j = F*(j,i) by A16,A21,MATRIX_0:def 8
        .= Line(F,j).i by A11,A20,MATRIX_0:def 7;
A27:  Line(1.(K,len b2),j).i = 1.(K,len b2)*(j,i) by A11,A13,MATRIX_0:def 7
        .= 0.K by A9,A22,A24,MATRIX_1:def 3;
      M/.j = M.j by A16,A18,A21,PARTFUN1:def 6
        .= Line(1.(K,len b2),j) by A1,A21,A23,MATRIX_0:52;
      hence Col(F,i).j = (KL.(M/.j) * Line(1.(K,len b2),j)).i by A13,A26,A25,
MATRIX13:102
        .= KL.(M/.j)*0.K by A11,A13,A27,FVSUM_1:51
        .= 0.K;
    end;
    then
A28: Col(F,i).i = Sum Col(F,i) by A19,MATRIX_3:12
      .= v1.i by A1,A2,A11,A8,MATRIX13:105,107;
A29: Line(F,i) = (KL (#) M).i by A5,A11,A17,MATRIX_0:52
      .= KL.(M/.i) * M/.i by A19,A16,VECTSP_6:def 5;
A30: Col(F,i).i = F*(i,i) by A19,A16,MATRIX_0:def 8
      .= Line(F,i).i by A11,A20,MATRIX_0:def 7;
    M/.i = M.i by A19,A16,A18,PARTFUN1:def 6
      .= Line(1.(K,len b2),i) by A1,A11,MATRIX_0:52;
    then Col(F,i).i = (KL.(M/.i) * Line(1.(K,len b2),i)).i by A30,A13,A29,
MATRIX13:102
      .= KL.(M/.i)*1_K by A11,A13,A14,FVSUM_1:51
      .= KL.(M/.i);
    hence t1.i = vM/.i by A3,A4,A6,A5,A10,A28
      .= vM.i by A19,A16,A18,A7,PARTFUN1:def 6;
  end;
  hence thesis by A4,A6,A5,FINSEQ_1:14;
end;
