
theorem Th12:
  for N1,N2 being Matrix of 3,F_Real st
  N2 = <* <* 2,   0,   -1 *>,
          <* 0, sqrt 3, 0 *>,
          <* 1,   0,   -2 *> *> &
  N1 = <* <* 2/3,   0,     -1/3 *>,
          <* 0,  1/sqrt 3,   0  *>,
          <* 1/3,   0,     -2/3 *> *> holds
  N1 * N2 = <* <* 1,0,0 *>,
               <* 0,1,0 *>,
               <* 0,0,1 *> *>
  proof
    let N1,N2 be Matrix of 3,F_Real;
    assume that
A1: N2 = <* <* 2,   0,    -1 *>,
            <* 0, sqrt 3, 0  *>,
            <* 1,   0,    -2 *> *> and
A2: N1 = <* <* 2/3,    0,     -1/3 *>,
            <* 0,   1/sqrt 3, 0    *>,
            <* 1/3,    0,     -2/3 *> *>;
    reconsider a9 = 2,b9 = 0,c9 = -1,d9 = 0, e9 = sqrt 3,f9 = 0,g9 = 1,h9=0,
      i9 = -2,a = 2/3,b=0,c = - 1/3,d=0,e = 1/sqrt 3,f=0,g = 1/3,h=0,i = -2/3
    as Element of F_Real by XREAL_0:def 1;
    reconsider n11 = a*a9+b*d9+c*g9, n12 = a*b9+b*e9+c*h9,
    n13 = a*c9+b*f9+c*i9, n21 = d*a9+e*d9+f*g9,
    n22 = d*b9+e*e9+f*h9, n23 = d*c9+e*f9+f*i9,
    n31 = g*a9+h*d9+i*g9,n32 = g*b9+h*e9+i*h9,n33 = g*c9+h*f9+i*i9
      as Element of F_Real;
    n11 = 1 & n12 = 0 & n13 = 0 & n21 = 0 &
      n22 = 1 & n23 = 0 & n31 = 0 & n32 = 0 & n33 = 1
      by SQUARE_1:24,XCMPLX_1:106;
    hence thesis by A1,A2,Th10;
  end;
