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 Th42:
  for nt be Element of n-tuples_on NAT st rng nt c= dom A & n>0
holds Solutions_of(A,B) c= Solutions_of(Segm(A,nt,Sgm Seg width A), Segm(B,nt,
  Sgm Seg width B))
proof
  let nt be Element of n-tuples_on NAT such that
A1: rng nt c= dom A and
A2: n>0;
  set SA=Segm(A,nt,Sgm Seg width A);
A3: len SA=n by A2,MATRIX_0:23;
  width SA=card Seg width A by A2,MATRIX_0:23;
  then
A4: width SA=width A by FINSEQ_1:57;
  set SB=Segm(B,nt,Sgm Seg width B);
A5: len SB=n by A2,MATRIX_0:23;
  width SB=card Seg width B by A2,MATRIX_0:23;
  then
A6: width SB=width B by FINSEQ_1:57;
  let x be object such that
A7: x in Solutions_of(A,B);
  consider X be Matrix of K such that
A8: x = X and
A9: len X = width A and
A10: width X = width B and
A11: A * X = B by A7;
  set SX=SA*X;
A12: len A = len B by A7,Th33;
A13: now
A14: len SX=len SA by A9,A4,MATRIX_3:def 4;
    let j,k such that
A15: [j,k] in Indices SX;
    j in dom SX by A15,ZFMISC_1:87;
    then
A16: j in Seg n by A3,A14,FINSEQ_1:def 3;
    width SX=width X by A9,A4,MATRIX_3:def 4;
    then
A17: k in Seg width B by A10,A15,ZFMISC_1:87;
    dom nt=Seg n by FINSEQ_2:124;
    then nt.j in rng nt by A16,FUNCT_1:def 3;
    then
A18: nt.j in dom A by A1;
    dom A=Seg len B by A12,FINSEQ_1:def 3;
    then nt.j in dom B by A18,FINSEQ_1:def 3;
    then
A19: [nt.j,k] in Indices B by A17,ZFMISC_1:87;
    reconsider j9=j,k9=k as Element of NAT by ORDINAL1:def 12;
    Sgm Seg width B=idseq width B by FINSEQ_3:48;
    then
A20: (Sgm Seg width B).k9=k by A17,FINSEQ_2:49;
    j in dom SB by A5,A16,FINSEQ_1:def 3;
    then
A21: [j,k] in Indices SB by A6,A17,ZFMISC_1:87;
    Line(SA,j)=Line(A,nt.j) by A16,Lm6;
    hence SX*(j,k) = Line(A,nt.j) "*" Col(X,k) by A9,A4,A15,MATRIX_3:def 4
      .= B*(nt.j9,k) by A9,A11,A19,MATRIX_3:def 4
      .= SB*(j,k) by A21,A20,MATRIX13:def 1;
  end;
  len SX=len SA & width SX=width X by A9,A4,MATRIX_3:def 4;
  then SX=SB by A10,A3,A5,A6,A13,MATRIX_0:21;
  hence thesis by A8,A9,A10,A4,A6;
end;
