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 Th41:
  for a being non zero Real holds
  symmetric_3(a,a,-a,0,0,0) * symmetric_3(1/a,1/a,-1/a,0,0,0) = 1.(F_Real,3)
  proof
    let a be non zero Real;
    reconsider z1 = 0, z2 = a,z3 = -a as Element of F_Real by XREAL_0:def 1;
A1: symmetric_3(a,a,-a,0,0,0) =  <* <* z2, z1, z1 *>,
                                    <* z1, z2, z1 *>,
                                    <* z1, z1, z3 *> *> by PASCAL:def 3;
A2: z2 * (1/z2) = 1 & z3 * (1/z3) = 1 by XCMPLX_1:106;
    reconsider y1 = z1,y2 = 1/z2, y3 = 1/z3 as Element of F_Real
      by XREAL_0:def 1;
A3: symmetric_3(y2,y2,y3,y1,y1,y1) =  <* <* 1/z2, z1, z1 *>,
                                                <* z1, 1/z2, z1 *>,
                                                <* z1, z1, 1/z3 *> *>
      by PASCAL:def 3;
    symmetric_3(a,a,-a,0,0,0) * symmetric_3(1/a,1/a,-1/a,0,0,0)
     = symmetric_3(z2,z2,z3,z1,z1,z1) * symmetric_3(y2,y2,y3,y1,y1,y1)
      by XCMPLX_1:188
    .= <* <* z2 * y2 + z1 * y1 + z1 * y1,
             z2 * y1 + z1 * y2 + z1 * y1,
             z2 * y1 + z1 * y1 + z1 * y3 *>,
          <* z1 * y2 + z2 * y1 + z1 * y1,
             z1 * y1 + z2 * y2 + z1 * y1,
             z1 * y1 + z2 * y1 + z1 * y3 *>,
          <* z1 * y2 + z1 * y1 + z3 * y1,
             z1 * y1 + z1 * y2 + z3 * y1,
             z1 * y1 + z1 * y1 + z3 * y3 *> *> by A1,A3,ANPROJ_9:6
    .= <* <* 1,0,0 *>,
          <* 0,1,0 *>,
          <* 0,0,1 *> *> by A2;
    hence thesis by ANPROJ_9:1;
  end;
