theorem Th18:
  a * v1 |-- b1 = a * (v1 |-- b1)
proof
  set vb=v1 |-- b1;
  set avb=(a*v1) |-- 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;
A3: len vb=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
A4: a*v1 = Sum(L2) and
A5: Carrier L2 c= rng b1 and
A6: for k st 1<=k & k<=len avb holds avb/.k = L2.(b1/.k) by MATRLIN:def 7;
A7: len avb=len b1 by MATRLIN:def 7;
  len (a*vb)=len vb by MATRIXR1:16;
  then reconsider
  vb9=vb,avb,Avb=a*vb as Element of (len b1)-tuples_on the carrier
  of K by A3,A7,FINSEQ_2:92;
A8: rb1 is linearly-independent by VECTSP_7:def 3;
  now
    let i such that
A9: i in Seg len b1;
A10: 1<=i & i<=len b1 by A9,FINSEQ_1:1;
A11: now
      per cases;
      suppose
        a<>0.K;
        then a*L1=L2 by A1,A4,A5,A8,MATRLIN:7;
        hence L2.(b1/.i) = a*L1.(b1/.i) by VECTSP_6:def 9
          .= a*(vb9/.i) by A2,A3,A10;
      end;
      suppose
A12:    a=0.K;
        then
A13:    a*v1=0.V1 by VECTSP_1:14;
        L2 is Linear_Combination of Carrier L2 & Carrier L2 is
        linearly-independent by A5,A8,VECTSP_6:7,VECTSP_7:1;
        then not b1/.i in Carrier L2 by A4,A13;
        hence L2.(b1/.i) = 0.K .= a*(vb9/.i) by A12;
      end;
    end;
A14: dom b1=Seg len b1 by FINSEQ_1:def 3;
    dom vb=dom b1 by A3,FINSEQ_3:29;
    then
A15: vb.i=vb/.i by A9,A14,PARTFUN1:def 6;
    dom avb=dom b1 by A7,FINSEQ_3:29;
    then avb.i=avb/.i by A9,A14,PARTFUN1:def 6;
    hence avb.i = L2.(b1/.i) by A6,A7,A10
      .= Avb.i by A9,A15,A11,FVSUM_1:51;
  end;
  hence thesis by FINSEQ_2:119;
end;
