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 Th45:
  for N st N c= dom A & N is non empty & dom A = dom B & for i st
i in (dom A) \ N holds Line(A,i) = width A |-> 0.K & Line(B,i) = width B |-> 0.
  K holds Solutions_of(A,B) = Solutions_of(Segm(A,N,Seg width A), Segm(B,N,Seg
  width B))
proof
  let N such that
A1: N c= dom A and
A2: N is non empty & dom A=dom B & for i st i in (dom A) \ N holds Line(
  A,i) = width A |-> 0.K & Line(B,i) = width B |-> 0.K;
  rng Sgm N = N by FINSEQ_1:def 14;
  hence thesis by A1,A2,Th43;
end;
