reserve i,j for Nat;

theorem Th46:
  for x1,x2 being FinSequence of REAL st len x1=len x2 & len x1>0
  holds ColVec2Mx (x1+x2)=ColVec2Mx (x1)+ColVec2Mx (x2)
proof
  let x1,x2 be FinSequence of REAL;
  assume that
A1: len x1=len x2 and
A2: len x1>0;
A3: width ColVec2Mx x1=1 by A2,Def9;
A4: Seg width ColVec2Mx x1=Seg 1 by A2,Def9;
A5: dom x1=dom x2 by A1,FINSEQ_3:29;
A6: len ColVec2Mx x1=len x1 by A2,Def9;
  then
A7: dom ColVec2Mx x1=dom x1 by FINSEQ_3:29;
  len ColVec2Mx x2=len x2 & width ColVec2Mx x2=1 by A1,A2,Def9;
  then
A8: Indices ColVec2Mx x2=Indices ColVec2Mx x1 by A1,A6,A3,MATRIX_4:55;
A9: len (x1+x2)=len x1 by A1,RVSUM_1:115;
  then
A10: dom (x1+x2)=dom x1 by FINSEQ_3:29;
A11: len ColVec2Mx (x1+x2)=len (x1+x2) & width ColVec2Mx (x1+x2)=1 by A2,A9
,Def9;
  then
A12: Indices ColVec2Mx (x1+x2)=Indices ColVec2Mx x1 by A1,A6,A3,MATRIX_4:55
,RVSUM_1:115;
  for i,j st [i,j] in Indices ColVec2Mx x1 holds (ColVec2Mx (x1+x2))*(i,j
  ) = (ColVec2Mx x1)*(i,j) + (ColVec2Mx x2)*(i,j)
  proof
    let i,j;
    thus [i,j] in Indices ColVec2Mx x1 implies (ColVec2Mx (x1+x2))*(i,j) = (
    ColVec2Mx x1)*(i,j) + (ColVec2Mx x2)*(i,j)
    proof
      assume
A13:  [i,j] in Indices ColVec2Mx x1;
      then consider q1 being FinSequence of REAL such that
A14:  q1 = (ColVec2Mx x1).i and
A15:  (ColVec2Mx x1)*(i,j)=q1.j by MATRIX_0:def 5;
      j in Seg 1 by A4,A13,ZFMISC_1:87;
      then 1<=j & j<=1 by FINSEQ_1:1;
      then
A16:  j=1 by XXREAL_0:1;
A17:  i in dom x1 by A7,A13,ZFMISC_1:87;
      then (ColVec2Mx x1).i=<* x1.i *> by A2,Def9;
      then
A18:  q1.j=x1.i by A16,A14;
      consider p being FinSequence of REAL such that
A19:  p = (ColVec2Mx (x1+x2)).i and
A20:  (ColVec2Mx (x1+x2))*(i,j) = p.j by A12,A13,MATRIX_0:def 5;
      consider q2 being FinSequence of REAL such that
A21:  q2 = (ColVec2Mx x2).i and
A22:  (ColVec2Mx x2)*(i,j)=q2.j by A8,A13,MATRIX_0:def 5;
      (ColVec2Mx x2).i=<* x2.i *> by A1,A2,A5,A17,Def9;
      then
A23:  q2.j=x2.i by A16,A21;
      (ColVec2Mx (x1+x2)).i=<* (x1+x2).i *> by A2,A9,A10,A17,Def9;
      then p.j=(x1+x2).i by A16,A19;
      hence thesis by A10,A17,A20,A15,A18,A22,A23,VALUED_1:def 1;
    end;
  end;
  hence thesis by A1,A6,A11,A3,Th26,RVSUM_1:115;
end;
