theorem Th48:
  symmetric_3(a,a,-a,0,0,0) = a * (symmetric_3(1,1,-1,0,0,0))
  proof
    reconsider z0 = 0, z1 = 1, z2 = -1 as Element of F_Real by XREAL_0:def 1;
    symmetric_3(1,1,-1,0,0,0) = <* <* z1,z0,z0 *>,
                                   <* z0,z1,z0 *>,
                                   <* z0,z0,z2 *> *> by PASCAL:def 3;
    then a * (symmetric_3(1,1,-1,0,0,0)) = <* <* a * z1, a * z0 ,a  * z0 *>,
                                              <* a * z0,a * z1, a * z0 *>,
                                              <* a * z0,a * z0,a * z2 *> *>
                                                by Th39;
    hence thesis by PASCAL:def 3;
  end;
