
theorem Th38:
  for h being Element of SubGroupK-isometry
  for N being invertible Matrix of 3,F_Real
  for n11,n12,n13,n21,n22,n23,n31,n32,n33 being Element of F_Real
  for u2 being Element of TOP-REAL 2
  st h = homography(N) & N = <* <* n11,n12,n13 *>,
                                <* n21,n22,n23 *>,
                                <* n31,n32,n33 *> *> &
  u2 in closed_inside_of_circle(0,0,1) holds n31 * u2.1 + n32 * u2.2 + n33 <> 0
  proof
    let h be Element of SubGroupK-isometry;
    let N be invertible Matrix of 3,F_Real;
    let n11,n12,n13,n21,n22,n23,n31,n32,n33 be Element of F_Real;
    let u2 be Element of TOP-REAL 2;
    assume that
A1: h = homography(N) and
A2: N = <* <* n11,n12,n13 *>, <* n21,n22,n23 *>, <* n31,n32,n33 *> *> and
A3: u2 in closed_inside_of_circle(0,0,1);
    u2 in inside_of_circle(0,0,1) \/ circle(0,0,1) by A3,TOPREAL9:55;
    then per cases by XBOOLE_0:def 3;
    suppose
      u2 in inside_of_circle(0,0,1);
      then reconsider uic = u2 as Element of inside_of_circle(0,0,1);
      consider Q be Element of TOP-REAL 2 such that
A4:   Q = uic & REAL2_to_BK uic = Dir |[Q`1,Q`2,1]| by BKMODEL2:def 3;
      reconsider P = REAL2_to_BK uic as Element of BK_model;
      reconsider v = |[Q`1,Q`2,1]| as non zero Element of TOP-REAL 3;
A5:   v.1 = v`1 by EUCLID_5:def 1
         .= u2.1 by A4,EUCLID_5:2;
A6:   v.2 = v`2 by EUCLID_5:def 2
         .= u2.2 by A4,EUCLID_5:2;
      now
        thus P = Dir v by A4;
        thus v.3 = v`3 by EUCLID_5:def 3
                .= 1 by EUCLID_5:2;
      end;
      hence thesis by A5,A6,A1,A2,Th20;
    end;
    suppose
A7:   u2 in circle(0,0,1);
      then reconsider u2 as non zero Element of TOP-REAL 2 by Th37;
      reconsider P = REAL2_to_absolute u2 as Element of absolute;
      reconsider v = |[u2.1,u2.2,1]| as non zero Element of TOP-REAL 3;
A8:   v.1 = v`1 by EUCLID_5:def 1
         .= u2.1 by EUCLID_5:2;
A9:   v.2 = v`2 by EUCLID_5:def 2
         .= u2.2 by EUCLID_5:2;
      now
        thus P is Element of absolute;
        thus P = Dir v by A7,BKMODEL1:def 9;
        thus v.3 = v`3 by EUCLID_5:def 3
                .= 1 by EUCLID_5:2;
      end;
      hence thesis by A8,A9,A1,A2,Th22;
    end;
  end;
