theorem Th17:
  v1 + w1 |-- b1 = (v1 |-- b1) + (w1 |-- b1)
proof
  set vb=v1 |-- b1;
  set wb=w1 |-- b1;
  set vwb=v1+w1 |-- b1;
  consider L1 be Linear_Combination of V1 such that
A1: v1 = Sum(L1) & Carrier L1 c= rng b1 and
A2: for k st 1<=k & k<=len vb holds vb/.k = L1.(b1/.k) by MATRLIN:def 7;
  consider L3 be Linear_Combination of V1 such that
A3: v1+w1 = Sum(L3) & Carrier L3 c= rng b1 and
A4: for k st 1<=k & k<=len vwb holds vwb/.k = L3.(b1/.k) by MATRLIN:def 7;
A5: len wb=len b1 by MATRLIN:def 7;
  reconsider rb1=rng b1 as Basis of V1 by MATRLIN:def 2;
  consider L2 be Linear_Combination of V1 such that
A6: w1 = Sum(L2) & Carrier L2 c= rng b1 and
A7: for k st 1<=k & k<=len wb holds wb/.k = L2.(b1/.k) by MATRLIN:def 7;
A8: len vb=len b1 by MATRLIN:def 7;
A9: len vwb=len b1 by MATRLIN:def 7;
  then reconsider
  vb,wb,vwb as Element of (len b1)-tuples_on the carrier of K by A8,A5,
FINSEQ_2:92;
  rb1 is linearly-independent by VECTSP_7:def 3;
  then
A10: L3=L1+L2 by A1,A6,A3,MATRLIN:6;
  now
A11: dom b1=Seg len b1 by FINSEQ_1:def 3;
    let i such that
A12: i in Seg len b1;
A13: 1<=i & i<=len b1 by A12,FINSEQ_1:1;
    dom wb=dom b1 by A5,FINSEQ_3:29;
    then
A14: wb.i=wb/.i by A12,A11,PARTFUN1:def 6;
    dom vb=dom b1 by A8,FINSEQ_3:29;
    then
A15: vb.i=vb/.i by A12,A11,PARTFUN1:def 6;
    dom vwb=dom b1 by A9,FINSEQ_3:29;
    then vwb.i=vwb/.i by A12,A11,PARTFUN1:def 6;
    hence vwb.i = (L1+L2).(b1/.i) by A4,A9,A10,A13
      .= L1.(b1/.i)+L2.(b1/.i) by VECTSP_6:22
      .= vb/.i +L2.(b1/.i) by A2,A8,A13
      .= vb/.i +wb/.i by A7,A5,A13
      .= (vb+wb).i by A12,A15,A14,FVSUM_1:18;
  end;
  hence thesis by FINSEQ_2:119;
end;
