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 Th63:
  for A st ( width A = 0 implies len A = 0 ) & the_rank_of A = 0
  holds Space_of_Solutions_of A = (width A)-VectSp_over K
proof
  let A such that
A1: width A = 0 implies len A = 0 and
A2: the_rank_of A = 0;
  set L=len A|->0.K;
  the carrier of (width A)-VectSp_over K c= Solutions_of(A,L)
  proof
    let x be object such that
A3: x in the carrier of (width A)-VectSp_over K;
    reconsider x9=x as Element of (width A)-tuples_on the carrier of K by A3,
MATRIX13:102;
A4: A=0.(K,len A,width A) & ColVec2Mx L=0.(K,len A,1) by A2,Th32,MATRIX13:95;
    per cases;
    suppose
A5:   len A=0;
      then Solutions_of(A,ColVec2Mx L)={{}} by A4,Th51;
      then
A6:   {} in Solutions_of(A,ColVec2Mx L) by TARSKI:def 1;
      then consider f such that
A7:   {} = ColVec2Mx f and
A8:   len f = width A by Th58;
      width A=0 by A5,MATRIX_0:def 3;
      then
      the carrier of (width A)-VectSp_over K = 0-tuples_on the carrier of
      K by MATRIX13:102
        .= {<*>the carrier of K} by FINSEQ_2:94;
      then x=<*>the carrier of K by A3,TARSKI:def 1;
      then f=x by A5,A8,MATRIX_0:def 3;
      hence thesis by A6,A7;
    end;
    suppose
A9:   len A>0;
A10:  len x9=width A by CARD_1:def 7;
      Solutions_of(A,ColVec2Mx L)= the set of all
X where X is Matrix of width A,1,K by A1,A4,A9,Th54;
      then ColVec2Mx x9 in Solutions_of(A,ColVec2Mx L) by A10;
      hence thesis;
    end;
  end;
  then the carrier of (width A)-VectSp_over K = Solutions_of(A,L)
    .= the carrier of Space_of_Solutions_of A by A1,Def5;
  hence thesis by VECTSP_4:31;
end;
