reserve i,j for Nat;

theorem
  for a being Real, x being FinSequence of REAL
    holds LineVec2Mx (a*x)=a*LineVec2Mx x
proof
  let a be Real, x be FinSequence of REAL;
A1: len (a*LineVec2Mx x)=len LineVec2Mx x by Th27
    .=1 by Def10;
A2: len (a*x)=len x by RVSUM_1:117;
  then
A3: dom (a*x)=dom x by FINSEQ_3:29;
A4: for j st j in dom (a*x) holds (a*LineVec2Mx x)*(1,j) = (a*x).j
  proof
    len LineVec2Mx x=1 by Def10;
    then 1 in Seg len LineVec2Mx x by FINSEQ_1:1;
    then
A5: 1 in dom LineVec2Mx x by FINSEQ_1:def 3;
    1 in Seg len (a*LineVec2Mx x) by A1,FINSEQ_1:1;
    then 1 in dom (a*LineVec2Mx x) by FINSEQ_1:def 3;
    then (a*LineVec2Mx x).1 in rng (a*LineVec2Mx x) by FUNCT_1:def 3;
    then reconsider q=(a*LineVec2Mx x).1 as FinSequence of REAL by
FINSEQ_1:def 11;
    let j;
A6: width LineVec2Mx x=len x by Def10;
A7: Indices (a*LineVec2Mx x)=Indices LineVec2Mx x by Th28;
    assume
A8: j in dom (a*x);
    then j in Seg len x by A2,FINSEQ_1:def 3;
    then
A9: [1,j] in Indices (a*LineVec2Mx x) by A6,A5,A7,ZFMISC_1:87;
    then
A10: ex p being FinSequence of REAL st p = (a*LineVec2Mx x).1 & (a*
    LineVec2Mx x)*(1,j) =p.j by MATRIX_0:def 5;
    reconsider j as Element of NAT by ORDINAL1:def 12;
A11:  q.j in REAL by XREAL_0:def 1;
    q.j = a*((LineVec2Mx x)*(1,j)) by A7,A9,A10,Th29
      .= a*(x.j) by A3,A8,Def10
      .= (a*x).j by RVSUM_1:44;
    hence thesis by A9,MATRIX_0:def 5,A11;
  end;
  width (a*LineVec2Mx x)=width LineVec2Mx x by Th27
    .=len x by Def10;
  hence thesis by A1,A2,A4,Def10;
end;
