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 Th41:
  X in Solutions_of(A,B) & i in dom A & Line(A,i) = width A |-> 0.
  K implies Line(B,i) = width B |-> 0.K
proof
  assume that
A1: X in Solutions_of(A,B) and
A2: i in dom A and
A3: Line(A,i) = width A|->0.K;
  set wB0=width B |->0.K;
  set LB=Line(B,i);
A4: len LB=width B by CARD_1:def 7;
A5: ex X1 be Matrix of K st X = X1 & len X1 = width A & width X1 = width B &
  A * X1 = B by A1;
A6: now
    let k such that
A7: 1 <=k & k <= len LB;
A8: k in Seg width B by A4,A7;
    len A=len B by A1,Th33;
    then dom A=Seg len B by FINSEQ_1:def 3;
    then i in dom B by A2,FINSEQ_1:def 3;
    then [i,k] in Indices B by A8,ZFMISC_1:87;
    then B*(i,k) = Line(A,i) "*" Col(X,k) by A5,MATRIX_3:def 4
      .= Sum(0.K*Col(X,k)) by A3,A5,FVSUM_1:66
      .= 0.K*Sum(Col(X,k)) by FVSUM_1:73
      .= 0.K
      .= wB0.k by A8,FINSEQ_2:57;
    hence wB0.k = LB.k by A8,MATRIX_0:def 7;
  end;
  len wB0=width B by CARD_1:def 7;
  hence thesis by A4,A6;
end;
