
theorem
  for N being invertible Matrix of 3,F_Real st
  N = <* <* 2,   0   , -1 *>,
         <* 0, sqrt 3,  0 *>,
         <* 1,   0   , -2 *> *>
  holds
  homography(N).:absolute = absolute
  proof
    let N be invertible Matrix of 3,F_Real;
    assume
A1: N = <* <* 2,   0   ,-1 *>,
           <* 0, sqrt 3, 0 *>,
           <* 1,   0   ,-2 *> *>;
    reconsider a=2,b=0,c = -1,d=0,e=sqrt 3,f=0,g=1,h=0,i=-2
      as Element of F_Real by XREAL_0:def 1;
A2: homography(N).:absolute c= absolute
    proof
      let x be object;
      assume x in homography(N).:absolute;
      then consider y be object such that
A3:   y in dom homography(N) and
A4:   y in absolute and
A5:   x = homography(N).y by FUNCT_1:def 6;
      reconsider y as Point of ProjectiveSpace TOP-REAL 3 by A3;
      consider yu be non zero Element of TOP-REAL 3 such that
A6:   (yu.1)^2 + (yu.2)^2 = 1 and
A7:   yu.3 = 1 and
A8:   y = Dir yu by A4,BKMODEL1:89;
A9:   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 A6,SQUARE_1:def 1;
A10:  yu`3 * yu`3 = yu.3 * yu`3 by EUCLID_5:def 3
                 .= 1 by A7,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
A11:  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 A8,A11,ANPROJ_1:22;
      then consider l be Real such that l <> 0 and
A12:  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 A12,EUCLID_5:7,A11;
      then v = <* a*u1+b*u2+c*u3,d*u1+e*u2+f*u3,g*u1+h*u2+i*u3 *>
                by A1,A11,PASCAL:8
            .= <* 2 * u1 - u3, (sqrt 3) * u2,u1 - 2 * u3 *>;
      then v`1 = 2 * u1 - u3 & v`2 = (sqrt 3) * u2 & v`3 = u1 - 2 * u3
        by EUCLID_5:2;
      then
A13:  v.1 = 2 * u1 - u3 & v.2 = (sqrt 3) * u2 & v.3 = u1 - 2 * u3
             by EUCLID_5:def 1,def 2,def 3;
A14:  v.1 * v.1 = (2 * u1 - u3)^2 by A13,SQUARE_1:def 1
               .= (2 * u1)^2 - 2 * (2 * u1) * u3 + u3^2 by SQUARE_1:5
               .= (2 * u1) * (2 * u1) - 2 * (2 * u1) * u3 + u3^2
                 by SQUARE_1:def 1
               .= 4 * u1 * u1 - 4 * u1 * u3 + u3 * u3 by SQUARE_1:def 1;
A15:  v.2 * v.2 = (sqrt 3) * (sqrt 3) * u2 * u2 by A13
               .= sqrt (3 * 3) * u2 * u2 by SQUARE_1:29
               .= sqrt (3^2) * u2 * u2 by SQUARE_1:def 1
               .= 3 * u2 * u2 by SQUARE_1:def 2;
A16:  v.3 * v.3 = (u1 - 2 * u3)^2 by A13,SQUARE_1:def 1
               .= u1^2 - 2 * u1 * (2 * u3) + (2 * u3)^2 by SQUARE_1:5
               .= u1 * u1 - 2 * 2 * u1 * u3 + (2 * u3)^2 by SQUARE_1:def 1
               .= u1 * u1 - 2 * 2 * u1 * u3 + (2 * u3) * (2 * u3)
                 by SQUARE_1:def 1
               .= u1 * u1 - 4 * u1 * u3 + 4 * u3 * u3;
      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
                               .= (4 * u1 * u1 - 4 * u1 * u3 + u3 * u3) +
                                   3 * u2 * u2 - (u1 * u1
                                   - 4 * u1 * u3 + 4 * u3 * u3) by A14,A15,A16
                               .= 3 * (l * l) * (yu`1 * yu`1 + yu`2 * yu`2
                                   - yu`3 * yu`3)
                               .= 0 by A9,A10;
        hence thesis;
      end;
      hence x in absolute by A5,A11,PASCAL:11;
    end;
    absolute c= homography(N).:absolute
    proof
      let x be object;
      assume
A17:  x in absolute;
      reconsider N1 = <* <* 2/3,   0   ,-1/3 *>,
                         <* 0, 1/sqrt 3, 0 *>,
                         <* 1/3,   0   ,-2/3 *> *> as Matrix of 3,F_Real
                         by ANPROJ_8:19;
      N1 is_reverse_of N by A1,Th13;
      then reconsider N1 as invertible Matrix of 3,F_Real by MATRIX_6:def 3;
A18:  homography(N1).:absolute c= absolute by Th14;
      dom homography(N1) = the carrier of ProjectiveSpace TOP-REAL 3
        by FUNCT_2:def 1;
      then homography(N1).x in homography(N1).:absolute by A17,FUNCT_1:108;
      then reconsider y = homography(N1).x as Element of absolute by A18;
A19:  N * N1 = 1.(F_Real,3) by A1,Th11,ANPROJ_9:1;
      now
        dom homography(N) = the carrier of ProjectiveSpace TOP-REAL 3
          by FUNCT_2:def 1;
        hence y in dom homography(N);
        thus y in absolute;
        reconsider P = x as Point of ProjectiveSpace TOP-REAL 3 by A17;
        thus homography(N).y = homography(1.(F_Real,3)).P by A19,ANPROJ_9:13
                            .= x by ANPROJ_9:14;
      end;
      hence x in homography(N).:absolute by FUNCT_1:108;
    end;
    hence thesis by A2;
  end;
