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

theorem Th24:
  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,MATRIXR1:def 9;
A4: Seg width ColVec2Mx x1=Seg 1 by A2,MATRIXR1:def 9;
A5: dom x1=dom x2 by A1,FINSEQ_3:29;
A6: len (x1-x2)=len x1 by A1,RVSUM_1:116;
  then
A7: dom (x1-x2)=dom x1 by FINSEQ_3:29;
A8: len ColVec2Mx x1=len x1 by A2,MATRIXR1:def 9;
  then
A9: dom ColVec2Mx x1=dom x1 by FINSEQ_3:29;
A10: len ColVec2Mx x2=len x2 & width ColVec2Mx x2=1 by A1,A2,MATRIXR1:def 9;
  then
A11: Indices ColVec2Mx x2=Indices ColVec2Mx x1 by A1,A8,A3,MATRIX_4:55;
A12: len ColVec2Mx (x1-x2)=len (x1-x2) & width ColVec2Mx (x1-x2)=1 by A2,A6,
MATRIXR1:def 9;
  then
A13: Indices ColVec2Mx (x1-x2)=Indices ColVec2Mx x1 by A1,A8,A3,MATRIX_4:55
,RVSUM_1:116;
  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;
    assume
A14: [i,j] in Indices ColVec2Mx x1;
    then consider q1 being FinSequence of REAL such that
A15: q1 = (ColVec2Mx x1).i and
A16: (ColVec2Mx x1)*(i,j)=q1.j by MATRIX_0:def 5;
    j in Seg 1 by A4,A14,ZFMISC_1:87;
    then 1<=j & j<=1 by FINSEQ_1:1;
    then
A17: j=1 by XXREAL_0:1;
A18: i in dom x1 by A9,A14,ZFMISC_1:87;
    then (ColVec2Mx x1).i=<* x1.i *> by A2,MATRIXR1:def 9;
    then
A19: q1.j=x1.i by A17,A15;
    consider p being FinSequence of REAL such that
A20: p = (ColVec2Mx (x1-x2)).i and
A21: (ColVec2Mx (x1-x2))*(i,j) = p.j by A13,A14,MATRIX_0:def 5;
    consider q2 being FinSequence of REAL such that
A22: q2 = (ColVec2Mx x2).i and
A23: (ColVec2Mx x2)*(i,j)=q2.j by A11,A14,MATRIX_0:def 5;
    (ColVec2Mx x2).i=<* x2.i *> by A1,A2,A5,A18,MATRIXR1:def 9;
    then
A24: q2.j=x2.i by A17,A22;
    (ColVec2Mx (x1-x2)).i=<* (x1-x2).i *> by A2,A6,A7,A18,MATRIXR1:def 9;
    then p.j=(x1-x2).i by A17,A20;
    hence thesis by A1,A21,A16,A19,A23,A24,Lm1;
  end;
  hence thesis by A1,A6,A8,A12,A3,A10,Th22;
end;
