reserve n for Nat;
reserve K for Field;
reserve a,b,c,d,e,f,g,h,i,a1,b1,c1,d1,e1,f1,g1,h1,i1 for Element of K;
reserve M,N for Matrix of 3,K;
reserve p for FinSequence of REAL;
reserve a,b,c,d,e,f for Real;
reserve u,u1,u2 for non zero Element of TOP-REAL 3;
reserve P for Element of ProjectiveSpace TOP-REAL 3;

theorem Th12:
  symmetric_3(a,b,c,d,e,f) is symmetric
  proof
    reconsider a,b,c,d,e,f as Element of F_Real by XREAL_0:def 1;
    (symmetric_3(a,b,c,d,e,f))@ = <* <* a, d, e *>,
                                     <* d, b, f *>,
                                     <* e, f, c *> *> by Th05;
    hence thesis by MATRIX_6:def 5;
  end;
