reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;

theorem Th30:
  for M being Matrix of k,F_Real holds
  Det M = 0.F_Real iff the_rank_of M < k
  proof
    let M be Matrix of k,F_Real;
A1: Det M = 0.F_Real iff the_rank_of M <> k by MATRIX13:83;
A2: the_rank_of M <= len M by MATRIX13:74;
    len M = k by MATRIX_0:def 2;
    hence thesis by A1,A2,XXREAL_0:1;
  end;
