reserve i,j for Nat;

theorem Th50:
  for x1,x2 being FinSequence of REAL st len x1=len x2 holds
  LineVec2Mx (x1+x2)=LineVec2Mx (x1)+LineVec2Mx (x2)
proof
  let x1,x2 be FinSequence of REAL;
  assume
A1: len x1=len x2;
  then
A2: dom x1 = dom x2 by FINSEQ_3:29;
  len (x1+x2)=len x1 by A1,RVSUM_1:115;
  then
A3: dom (x1+x2)=dom x1 by FINSEQ_3:29;
A4: width LineVec2Mx x1=len x1 & len LineVec2Mx x1=1 by Def10;
  width LineVec2Mx x2=len x2 & len LineVec2Mx x2=1 by Def10;
  then
A5: Indices LineVec2Mx x2=Indices LineVec2Mx x1 by A1,A4,MATRIX_4:55;
A6: Seg width LineVec2Mx x1=Seg len x1 by Def10
    .= dom x1 by FINSEQ_1:def 3;
A7: dom LineVec2Mx x1=Seg len LineVec2Mx x1 by FINSEQ_1:def 3
    .=Seg 1 by Def10;
A8: width LineVec2Mx (x1+x2)=len (x1+x2) & len LineVec2Mx (x1+x2)=1 by Def10;
  then
A9: Indices LineVec2Mx (x1+x2)=Indices LineVec2Mx x1 by A1,A4,MATRIX_4:55
,RVSUM_1:115;
  for i,j holds [i,j] in Indices LineVec2Mx x1 implies (LineVec2Mx (x1+x2
  ))*(i,j) = (LineVec2Mx x1)*(i,j) + (LineVec2Mx x2)*(i,j)
  proof
    let i,j;
    assume
A10: [i,j] in Indices LineVec2Mx x1;
    then consider q1 being FinSequence of REAL such that
    q1 = (LineVec2Mx x1).i and
A11: (LineVec2Mx x1)*(i,j) = q1.j by MATRIX_0:def 5;
    consider p being FinSequence of REAL such that
    p = (LineVec2Mx (x1+x2)).i and
A12: (LineVec2Mx (x1+x2))*(i,j) = p.j by A9,A10,MATRIX_0:def 5;
    consider q2 being FinSequence of REAL such that
    q2 = (LineVec2Mx x2).i and
A13: (LineVec2Mx x2)*(i,j) =q2.j by A5,A10,MATRIX_0:def 5;
A14: j in dom x1 by A6,A10,ZFMISC_1:87;
    i in Seg 1 by A7,A10,ZFMISC_1:87;
    then 1<=i & i<=1 by FINSEQ_1:1;
    then
A15: i=1 by XXREAL_0:1;
    then
A16: q1.j=x1.j by A14,A11,Def10;
A17: q2.j=x2.j by A2,A14,A15,A13,Def10;
    p.j=(x1+x2).j by A3,A14,A15,A12,Def10;
    hence thesis by A3,A14,A12,A11,A16,A13,A17,VALUED_1:def 1;
  end;
  hence thesis by A1,A8,A4,Th26,RVSUM_1:115;
end;
