reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;

theorem
  M = symmetric_3(0,0,0,0,0,0) implies Det M = 0.F_Real
  proof
    assume
A1: M = symmetric_3(0,0,0,0,0,0);
    reconsider z = 0 as Element of F_Real;
    M = <* <* z,z,z *>,
           <* z,z,z *>,
           <* z,z,z *> *> by A1,PASCAL:def 3;
    then Det M = z * z * z - z * z * z - z * z * z + z * z * z
                   - z * z * z + z * z * z by MATRIX_9:46;
    hence thesis;
  end;
