
theorem Th14:
  for N being invertible Matrix of 3,F_Real st
  N = <* <* 2/3,   0   ,-1/3 *>,
         <* 0, 1/sqrt 3, 0   *>,
         <* 1/3,   0   ,-2/3 *> *> holds
  homography(N).:absolute c= absolute
  proof
    let N be invertible Matrix of 3,F_Real;
    assume
A1: N = <* <* 2/3,   0   ,-1/3 *>,
           <* 0, 1/sqrt 3, 0   *>,
           <* 1/3,   0   ,-2/3 *> *>;
    reconsider 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;
    homography(N).:absolute c= absolute
    proof
      let x be object;
      assume x in homography(N).:absolute;
      then consider y be object such that
A2:   y in dom homography(N) and
A3:   y in absolute and
A4:   x = homography(N).y by FUNCT_1:def 6;
      reconsider y as Point of ProjectiveSpace TOP-REAL 3 by A2;
      consider yu be non zero Element of TOP-REAL 3 such that
A5:   (yu.1)^2 + (yu.2)^2 = 1 and
A6:   yu.3 = 1 and
A7:   y = Dir yu by A3,BKMODEL1:89;
A8:   yu`1 * yu`1 + yu`2 * yu`2 = yu.1 * yu`1 + yu`2 * yu`2
                                   by EUCLID_5:def 1
                               .= yu.1 * yu.1 + yu`2 * yu`2 by EUCLID_5:def 1
                               .= yu.1 * yu.1 + yu.2 * yu`2 by EUCLID_5:def 2
                               .= yu.1 * yu.1 + yu.2 * yu.2 by EUCLID_5:def 2
                               .= (yu.1)^2 + yu.2 * yu.2 by SQUARE_1:def 1
                               .= 1 by A5,SQUARE_1:def 1;
A9:   yu`3 * yu`3 = yu.3 * yu`3 by EUCLID_5:def 3
                 .= 1 by A6,EUCLID_5:def 3;
      consider u,v be Element of TOP-REAL 3,
                uf be FinSequence of F_Real,
                p be FinSequence of 1-tuples_on REAL such that
A10:  y = Dir u & u is not zero & u = uf & p = N * uf & v = M2F p &
        v is not zero & homography(N).y = Dir v by ANPROJ_8:def 4;
      are_Prop u,yu by A7,A10,ANPROJ_1:22;
      then consider l be Real such that l <> 0 and
A11:  u = l * yu by ANPROJ_1:1;
      reconsider u1 = l * yu`1,u2 = l * yu`2,u3 = l * yu`3 as Element of F_Real
        by XREAL_0:def 1;
      uf = <* u1,u2,u3 *> by A11,A10,EUCLID_5:7;
      then
      v = <* a*u1+b*u2+c*u3,d*u1+e*u2+f*u3,g*u1+h*u2+i*u3 *> by A1,A10,PASCAL:8
       .= <* (2/3) * u1 - (1/3) * u3, (1/sqrt 3) * u2,
             (1/3) * u1 - (2/3) * u3 *>;
       then v`1 = (2/3) * u1 - (1/3) * u3 & v`2 = (1 / sqrt 3) * u2 &
       v`3 = (1/3) * u1 - (2/3) * u3 by EUCLID_5:2;
       then
A12:   v.1 = (2/3) * u1 - (1/3) * u3 & v.2 = (1/sqrt 3) * u2 &
         v.3 = (1/3) * u1 - (2/3) * u3 by EUCLID_5:def 1,def 2,def 3;
       set k = (1/3)*(1/3);
       1 / sqrt 3 = sqrt 3 / 3 by BKMODEL3:10; then
A13:   (1 / sqrt 3) * (1 / sqrt 3) = k * ((sqrt 3) * sqrt 3)
                                  .= k * (sqrt (3 * 3)) by SQUARE_1:29
                                  .= k * sqrt (3^2) by SQUARE_1:def 1
                                  .= k * 3 by SQUARE_1:def 2;
A14:   v.2 * v.2 = (1/sqrt 3) * (1/sqrt 3) * u2 * u2 by A12
                .= k * 3 * u2 * u2 by A13;
        reconsider P = homography(N).y as Point of ProjectiveSpace TOP-REAL 3;
        qfconic(1,1,-1,0,0,0,v)=0
        proof
          qfconic(1,1,-1,0,0,0,v)
          = 1 * v.1 * v.1 + 1 * v.2 * v.2 + (-1) * v.3 * v.3
            + 0 * v.1 * v.2 + 0 * v.1 * v.3 + 0 * v.2 * v.3 by PASCAL:def 1
         .= k * (4 * u1 * u1 - 4 * u1 * u3 + u3 * u3) + k * 3 * u2 * u2
            - k * (u1 * u1 - 4 * u1 * u3 + 4 * u3 * u3) by A12,A14
         .= k * 3 * (l * l) * (yu`1 * yu`1 + yu`2 * yu`2 - yu`3 * yu`3)
         .= 0 by A8,A9;
         hence thesis;
       end;
       hence x in absolute by A4,A10,PASCAL:11;
     end;
     hence thesis;
   end;
