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 Th53:
  for A be Matrix of n,k,K, B be Matrix of n,m,K st n > 0 holds x
  in Solutions_of(A,B) implies x is Matrix of k,m,K
proof
  let A be Matrix of n,k,K, B be Matrix of n,m,K;
  assume n > 0;
  then
A1: width A=k & width B=m by MATRIX_0:23;
  assume x in Solutions_of(A,B);
  then ex X st X=x & len X = k & width X = m & A * X = B by A1;
  hence thesis by MATRIX_0:51;
end;
