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;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;

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;
