reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th23:
  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;
A1: width LineVec2Mx x1=len x1 & len LineVec2Mx x1=1 by MATRIXR1:def 10;
A2: Seg width LineVec2Mx x1=Seg len x1 by MATRIXR1:def 10
    .= dom x1 by FINSEQ_1:def 3;
A3: dom LineVec2Mx x1=Seg len LineVec2Mx x1 by FINSEQ_1:def 3
    .=Seg 1 by MATRIXR1:def 10;
  assume
A4: len x1=len x2;
  then
A5: dom x1 = dom x2 by FINSEQ_3:29;
A6: width LineVec2Mx x2=len x2 & len LineVec2Mx x2=1 by MATRIXR1:def 10;
  then
A7: Indices LineVec2Mx x2=Indices LineVec2Mx x1 by A4,A1,MATRIX_4:55;
A8: len (x1-x2)=len x1 by A4,RVSUM_1:116;
  then
A9: dom (x1-x2)=dom x1 by FINSEQ_3:29;
A10: width LineVec2Mx (x1-x2)=len (x1-x2) & len LineVec2Mx (x1-x2)=1 by
MATRIXR1:def 10;
  then
A11: Indices LineVec2Mx (x1-x2)=Indices LineVec2Mx x1 by A4,A1,MATRIX_4:55
,RVSUM_1:116;
  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
A12: [i,j] in Indices LineVec2Mx x1;
    then consider q1 being FinSequence of REAL such that
    q1 = (LineVec2Mx x1).i and
A13: (LineVec2Mx x1)*(i,j) = q1.j by MATRIX_0:def 5;
    i in Seg 1 by A3,A12,ZFMISC_1:87;
    then 1<=i & i<=1 by FINSEQ_1:1;
    then
A14: i=1 by XXREAL_0:1;
A15: j in dom x1 by A2,A12,ZFMISC_1:87;
    then
A16: q1.j=x1.j by A14,A13,MATRIXR1:def 10;
    consider p being FinSequence of REAL such that
    p = (LineVec2Mx (x1-x2)).i and
A17: (LineVec2Mx (x1-x2))*(i,j) = p.j by A11,A12,MATRIX_0:def 5;
    consider q2 being FinSequence of REAL such that
    q2 = (LineVec2Mx x2).i and
A18: (LineVec2Mx x2)*(i,j) =q2.j by A7,A12,MATRIX_0:def 5;
A19: q2.j=x2.j by A5,A15,A14,A18,MATRIXR1:def 10;
    p.j=(x1-x2).j by A9,A15,A14,A17,MATRIXR1:def 10;
    hence thesis by A4,A17,A13,A16,A18,A19,Lm1;
  end;
  hence thesis by A4,A8,A10,A1,A6,Th22;
end;
