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
  for A st Space_of_Solutions_of A = (width A)-VectSp_over K holds
  the_rank_of A = 0
proof
  let A such that
A1: Space_of_Solutions_of A = (width A)-VectSp_over K;
  assume the_rank_of A <> 0;
  then consider i,j such that
A2: [i,j] in Indices A and
A3: A*(i,j) <> 0.K by MATRIX13:94;
A4: j in Seg width A by A2,ZFMISC_1:87;
then A5: width A <> 0;
  set L=Line(1.(K,width A),j);
A6: width 1.(K,width A)=width A by MATRIX_0:24;
  then
A7: dom L=Seg width A by FINSEQ_2:124;
A8: Indices 1.(K,width A)=[:Seg width A,Seg width A:] by MATRIX_0:24;
A9: now
    let k such that
A10: k in dom L and
A11: k<>j;
    [j,k] in Indices 1.(K,width A) by A4,A7,A8,A10,ZFMISC_1:87;
    hence 0.K = 1.(K,width A)*(j,k) by A11,MATRIX_1:def 3
      .= L.k by A6,A7,A10,MATRIX_0:def 7;
  end;
A12: dom Line(A,i)=Seg width A by FINSEQ_2:124;
  [j,j] in Indices 1.(K,width A) by A4,A8,ZFMISC_1:87;
  then 1_K = 1.(K,width A)*(j,j) by MATRIX_1:def 3
    .= L.j by A4,A6,MATRIX_0:def 7;
  then
A13: Sum(mlt(L,Line(A,i))) = Line(A,i).j by A4,A7,A12,A9,MATRIX_3:17
    .= A*(i,j) by A4,MATRIX_0:def 7;
A14: ColVec2Mx (len A|->0.K)=0.(K,len A,1) by Th32;
A15: i in dom A by A2,ZFMISC_1:87;
  L in (width A)-tuples_on the carrier of K by A6;
  then L in the carrier of Space_of_Solutions_of A by A1,MATRIX13:102;
  then L in Solutions_of(A,len A|->0.K) by Def5,A5;
  then consider f such that
A16: f=L and
A17: ColVec2Mx f in Solutions_of(A,ColVec2Mx (len A|->0.K));
  consider X such that
A18: X=ColVec2Mx f and
A19: len X = width A and
  width X = width ColVec2Mx (len A|->0.K) and
A20: A * X = ColVec2Mx (len A|->0.K) by A17;
A21: 1 in Seg 1;
A22: dom A=Seg len A by FINSEQ_1:def 3;
  then len A<>0 by A2,ZFMISC_1:87;
  then Indices ColVec2Mx (len A|->0.K)=[:Seg len A,Seg 1:] by A14,MATRIX_0:23;
  then
A23: [i,1] in Indices ColVec2Mx (len A|->0.K) by A15,A22,A21,ZFMISC_1:87;
  then Line(A,i)"*"Col(X,1) = 0.(K,len A,1)*(i,1) by A19,A20,A14,MATRIX_3:def 4
    .= 0.K by A14,A23,MATRIX_3:1;
  then 0.K = Col(X,1)"*"Line(A,i) by FVSUM_1:90
    .= Sum(mlt(f,Line(A,i))) by A18,A19,Th26,A5;
  hence thesis by A3,A16,A13;
end;
