reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);
reserve D for non empty set,
  bD for FinSequence of D,
  b,f,g for FinSequence of K,
  MD for Matrix of D;

theorem Th65:
  for i,j st j in Seg m & n>0 & (i = j implies a <> -1_K) holds
Space_of_Solutions_of A9 = Space_of_Solutions_of RLine(A9,i,Line(A9,i) + a*Line
  (A9,j))
proof
  let i,j such that
A1: j in Seg m and
A2: n>0 and
A3: i = j implies a <> -1_K;
  set L=len A9|->0.K;
  set R=RLine(A9,i,Line(A9,i) + a*Line(A9,j));
A4: m<>0 by A1;
  then
A5: width R=n by MATRIX_0:23;
 len L=len A9 by CARD_1:def 7;
  then len L = m by MATRIX_0:def 2;
  then reconsider C=ColVec2Mx L as Matrix of m,1,K;
  set RC=RLine(C,i,Line(C,i) + a*Line(C,j));
A6: C=0.(K,len A9,1) by Th32;
  now
    let i9,j9 be Nat such that
A7: [i9,j9] in Indices C;
    reconsider I=i9,J=j9 as Element of NAT by ORDINAL1:def 12;
A8: len (Line(C,i) + a*Line(C,j))=width C by CARD_1:def 7;
    now
      per cases;
      suppose
A9:     i9=i;
A10:    1=width C by A4,MATRIX_0:23;
        then
A11:    j9 in Seg 1 by A7,ZFMISC_1:87;
        then Line(C,j).j9=C*(j,j9) by A10,MATRIX_0:def 7;
        then
A12:    (a*Line(C,j)).j9=a*(C*(j,j9)) by A10,A11,FVSUM_1:51;
        Indices C=[:Seg m,Seg 1:] by A4,MATRIX_0:23;
        then
A13:    [j,j9] in Indices C by A1,A11,ZFMISC_1:87;
        Line(C,i).j9=C*(i,j9) by A10,A11,MATRIX_0:def 7;
        then (Line(C,i) + a*Line(C,j)).j9 = C*(i,j9)+a*(C*(j,j9)) by A10,A11
,A12,FVSUM_1:18
          .= 0.K +a*(C*(j,j9)) by A6,A7,A9,MATRIX_3:1
          .= 0.K +a*0.K by A6,A13,MATRIX_3:1
          .= 0.K +0.K
          .= 0.K by RLVECT_1:def 4
          .= C*(i9,j9) by A6,A7,MATRIX_3:1;
        hence C*(I,J)=RC*(I,J) by A7,A8,A9,MATRIX11:def 3;
      end;
      suppose
        i<>i9;
        hence C*(I,J)=RC*(I,J) by A7,A8,MATRIX11:def 3;
      end;
    end;
    hence C*(i9,j9)=RC*(i9,j9);
  end;
  then RC=C by MATRIX_0:27;
  then
A14: Solutions_of(A9,C)=Solutions_of(R,C) by A1,A3,Th40;
  set SR=Space_of_Solutions_of R;
  len A9=m & len R=m by A4,MATRIX_0:23;
  then
A15: the carrier of SR = Solutions_of(R,L) by A2,A5,Def5;
  set SA=Space_of_Solutions_of A9;
A16: width A9=n by A4,MATRIX_0:23;
  then the carrier of SA = Solutions_of(A9,L) by A2,Def5;
  hence thesis by A16,A5,A14,A15,VECTSP_4:29;
end;
