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 Th66:
  for N st N c= dom A & N is non empty & width A > 0 & for i st i
in (dom A) \ N holds Line(A,i) = width A |-> 0.K holds Space_of_Solutions_of A
  = Space_of_Solutions_of Segm(A,N,Seg width A)
proof
  let N such that
A1: N c= dom A and
A2: N is non empty and
A3: width A>0 and
A4: for i st i in (dom A) \ N holds Line(A,i) = width A |-> 0.K;
  set L=len A|->0.K;
  set C=ColVec2Mx L;
A5: len L=len A by CARD_1:def 7;
  set S=Segm(A,N,Seg width A);
A6: width S=card Seg width A by A2,MATRIX_0:23;
  then
A7: width A=width S by FINSEQ_1:57;
  set SS=Space_of_Solutions_of S;
  len S=card N by MATRIX_0:def 2;
  then
A8: the carrier of SS = Solutions_of(S,card N|->0.K) by A3,A6,Def5;
  set SA=Space_of_Solutions_of A;
A9: the carrier of SA = Solutions_of(A,L) by A3,Def5;
A10: C=0.(K,len A,1) by Th32;
  len C=len L by MATRIX_0:def 2;
  then
A11: dom C=dom A by A5,FINSEQ_3:29;
A12: dom A=Seg len A by FINSEQ_1:def 3;
  then
A13: Seg len A<>{} by A1,A2;
  then
A14: width C=1 by Th26;
  then
A15: card Seg width C=1 by FINSEQ_1:57;
  now
A16: rng Sgm Seg 1=Seg 1 by FINSEQ_1:def 14;
A17: Indices Segm(C,N,Seg width C)=Indices 0.(K,card N,1) by A15,MATRIX_0:26;
    let k,l such that
A18: [k,l] in Indices Segm(C,N,Seg width C);
    reconsider kk=k,ll=l as Element of NAT by ORDINAL1:def 12;
    [:N,Seg width C:] c= Indices C & rng Sgm N=N
    by A1,A11,FINSEQ_1:def 14,ZFMISC_1:95;
    then
A19: [Sgm N.kk,Sgm Seg width C.ll] in Indices C by A14,A18,A16,MATRIX13:17;
    thus Segm(C,N,Seg width C)*(k,l) = C*(Sgm N.kk,Sgm Seg width C.ll)
    by A18,MATRIX13:def 1
      .= 0.K by A10,A19,MATRIX_3:1
      .=0.(K,card N,1)*(k,l) by A18,A17,MATRIX_3:1;
  end;
  then
A20: Segm(C,N,Seg width C) = 0.(K,card N,1) by A15,MATRIX_0:27
    .= ColVec2Mx (card N|->0.K) by Th32;
  now
    let i such that
A21: i in (dom A) \ N;
A22: i in dom A by A21,XBOOLE_0:def 5;
    then Line(C,i) = C.i by A5,A12,MATRIX_0:52
      .= (len A|->(width C|->0.K)).i by A10,A13,Th26
      .= width C|->0.K by A12,A22,FINSEQ_2:57;
    hence Line(A,i)=width A|->0.K & Line(C,i)=width C|->0.K by A4,A21;
  end;
  then
  Solutions_of(A,C)=Solutions_of(Segm(A,N,Seg width A),Segm(C,N,Seg width
  C)) by A1,A2,A11,Th45;
  hence thesis by A20,A7,A9,A8,VECTSP_4:29;
end;
