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 Th15:
  for a1,a2,a3,a4,a5,a6,b1,b2,b3,b4,b5,b6 being Real st
  symmetric_3(a1,a2,a3,a4,a5,a6) = symmetric_3(b1,b2,b3,b4,b5,b6) holds
  a1 = b1 & a2 = b2 & a3 = b3 & a4 = b4 & a5 = b5 & a6 = b6
  proof
    let a1,a2,a3,a4,a5,a6,b1,b2,b3,b4,b5,b6 being Real;
    assume
A1: symmetric_3(a1,a2,a3,a4,a5,a6) = symmetric_3(b1,b2,b3,b4,b5,b6);
    reconsider fa1 = a1,fa2 = a2, fa3 = a3,
               fa4 = a4,fa5 = a5, fa6 = a6,
               fb1 = b1,fb2 = b2, fb3 = b3,
               fb4 = b4,fb5 = b5, fb6 = b6 as Element of F_Real
      by XREAL_0:def 1;
    <* <* fa1, fa4, fa5 *>,
       <* fa4, fa2, fa6 *>,
       <* fa5, fa6, fa3 *> *> =
    <* <* fb1, fb4, fb5 *>,
       <* fb4, fb2, fb6 *>,
       <* fb5, fb6, fb3 *> *> by A1;
    hence thesis by Th02;
  end;
