
theorem Th22:
  for P being Element of absolute holds tangent P /\ absolute = {P}
  proof
    let P be Element of absolute;
A1: {P} c= tangent P /\ absolute
    proof
      let x be object;
      assume x in {P};
      then x = P by TARSKI:def 1;
      then x in tangent P & x in absolute by Th21;
      hence x in tangent P /\ absolute by XBOOLE_0:def 4;
    end;
    tangent P /\ absolute c= {P}
    proof
      let x be object;
      assume
A2:   x in tangent P /\ absolute;
      then reconsider y = x as Element of real_projective_plane;
      consider p being Element of real_projective_plane such that
A3:   p = P and
A4:   tangent P = Line(p,pole_infty P) by Def04;
      y in Line(p,pole_infty P) by A2,A4,XBOOLE_0:def 4;
      then
A5:   p,pole_infty P,y are_collinear by COLLSP:11;
      consider u be Element of TOP-REAL 3 such that
A6:   u is non zero and
A7:   p = Dir u by ANPROJ_1:26;
      consider v be non zero Element of TOP-REAL 3 such that
A8:   P = Dir v & v.3 = 1 & (v.1)^2 + (v.2)^2 = 1 &
      pole_infty P = Dir |[- v.2,v.1,0]| by Def03;
      are_Prop u,v by A6,A7,A8,A3,ANPROJ_1:22;
      then consider a be Real such that
A9:   a <> 0 and
A10:  u = a * v by ANPROJ_1:1;
A11:  u`1 = u.1 by EUCLID_5:def 1
         .= a * v.1 by A10,RVSUM_1:44;
A12:  u`2 = u.2 by EUCLID_5:def 2
         .= a * v.2 by A10,RVSUM_1:44;
A13:  u`3 = u.3 by EUCLID_5:def 3
         .= a * 1 by A8,A10,RVSUM_1:44
         .= a;
      y is Element of absolute by A2,XBOOLE_0:def 4;
      then consider w be non zero Element of TOP-REAL 3 such that
A14:  (w.1)^2 + (w.2)^2 = 1 and
A15:  w.3 = 1 and
A16:  y = Dir w by BKMODEL1:89;
A17:  |[-v.2,v.1,0]|`1 = -v.2 & |[-v.2,v.1,0]|`2 = v.1 & |[-v.2,v.1,0]|`3 = 0
        by EUCLID_5:2;
      |[-v.2,v.1,0]| is non zero by A8,BKMODEL1:91;
      then 0 = |{ u,|[-v.2,v.1,0]|,w }| by A5,A6,A7,A8,A16,BKMODEL1:1
            .= u`1 * v.1 * w`3 - u`3 * v.1 *w`1 - u`1* 0 *w`2
                + u`2* 0 *w`1 - u`2*(-v.2)* w`3
                + u`3*(-v.2)* w`2 by A17,ANPROJ_8:27
            .= u`1 * v.1 * w.3 - u`3 * v.1 *w`1 - u`2*(-v.2)*w`3
                + u`3*(-v.2)* w`2 by EUCLID_5:def 3
            .= u`1 * v.1 * 1 - u`3 * v.1 *w`1 - u`2*(-v.2) * 1
                + u`3 *(-v.2)* w`2 by A15,EUCLID_5:def 3
            .= a * (v.1 * v.1 + v.2 * v.2 - v.1 * w`1 - v.2 * w`2)
                by A11,A12,A13
            .= a * (1 - v.1 * w`1 - v.2 * w`2) by A8;
      then 1 - v.1 * w`1 - v.2 * w`2 = 0 by A9; then
A18:  1 = v.1 * w`1 + v.2 * w`2
       .= v.1 * w.1 + v.2 * w`2 by EUCLID_5:def 1
       .= v.1 * w.1 + v.2 * w.2 by EUCLID_5:def 2;
      then
A19:  v.1 * w.2 = v.2 * w.1 by BKMODEL1:7,A14,A8;
      x = P
      proof
        per cases;
        suppose
A20:      v.2 = 0;
          then v.1 <> 0 by A8; then
A21:      w.2 = 0 by A19,A20;
          per cases by A20,A8,BKMODEL1:8;
          suppose
A22:        v.1 = 1;
            per cases by A21,A14,BKMODEL1:8;
            suppose w.1 = 1;
              then w`1 = 1 & w`2 = 0 & w`3 = 1 & v`1 = 1 & v`2 = 0 & v`3 = 1
                by A8,A20,A22,A19,A15,EUCLID_5:def 1,def 2,def 3;
              then v = |[w`1,w`2,w`3]| by EUCLID_5:3
                    .= w by EUCLID_5:3;
              hence x = P by A8,A16;
            end;
            suppose w.1 = -1;
              hence x = P by A18,A20,A22;
            end;
          end;
          suppose
A23:        v.1 = -1;
            per cases by A21,A14,BKMODEL1:8;
            suppose w.1 = 1;
              hence x = P by A18,A20,A23;
            end;
            suppose w.1 = -1;
              then w`1 = -1 & w`2 = 0 & w`3 = 1 & v`1 = -1 & v`2 = 0 & v`3 = 1
                by A23,A8,A20,A19,A15,EUCLID_5:def 1,def 2,def 3;
              then v = |[w`1,w`2,w`3]| by EUCLID_5:3
                   .= w by EUCLID_5:3;
              hence x = P by A8,A16;
            end;
          end;
        end;
        suppose
A24:      v.2 <> 0;
          per cases;
          suppose
A25:        v.1 = 0;
            then per cases by A8,BKMODEL1:8;
            suppose
A26:          v.2 = 1;
A27:          v.2 * w.1 = 0 * w.2 by A25,A18,BKMODEL1:7,A14,A8
                       .= 0;
              then w.1 = 0 by A24;
              then per cases by A14,BKMODEL1:8;
              suppose w.2 = 1;
                then
A28:            w`1 = 0 & w`2 = 1 & w`3 = 1 & v`1 = 0 & v`2 = 1 & v`3 = 1
                  by A25,A26,A27,A8,A15,EUCLID_5:def 1,def 2,def 3;
                v = |[w`1,w`2,w`3]| by A28,EUCLID_5:3
                 .= w by EUCLID_5:3;
                hence x = P by A8,A16;
              end;
              suppose w.2 = -1;
                hence x = P by A18,A25,A26;
              end;
            end;
            suppose
A29:          v.2 = -1;
A30:          v.2 * w.1 = 0 * w.2 by A25,A18,BKMODEL1:7,A14,A8
                       .= 0;
              then w.1 = 0 by A24;
              then per cases by A14,BKMODEL1:8;
              suppose w.2 = 1;
                hence x = P by A18,A25,A29;
              end;
              suppose w.2 = -1;
                then
A31:            w`1 = 0 & w`2 = -1 & w`3 = 1 & v`1 = 0 & v`2 = -1 & v`3 = 1
                  by A29,A25,A30,A8,A15,EUCLID_5:def 1,def 2,def 3;
                v = |[w`1,w`2,w`3]| by A31,EUCLID_5:3
                 .= w by EUCLID_5:3;
                hence x = P by A8,A16;
              end;
            end;
          end;
          suppose v.1 <> 0;
            then reconsider l = v.1 / v.2 as non zero Real by A24;
A32:        l * v.2 = v.1 by XCMPLX_1:87,A24;
A33:        l * w.2 = (v.1 * w.2) / v.2 by XCMPLX_1:74
                   .= (v.2 * w.1) / v.2 by A18,BKMODEL1:7,A14,A8
                   .= w.1 by XCMPLX_1:89,A24;
            per cases by A32,A8,BKMODEL1:10;
            suppose
A34:          v.2 = 1 / sqrt(1+l^2);
              per cases by A33,A14,BKMODEL1:10;
              suppose w.2 = 1 / sqrt(1+l^2);
                then w`1 = v.1 & w`2 = v.2 & w`3 = v.3
                  by A8,A15,A34,A32,A33,EUCLID_5:def 1,def 2,def 3;
                then w`1 = v`1 & w`2 = v`2 & w`3 = v`3
                   by EUCLID_5:def 1,def 2,def 3;
                then v = |[w`1,w`2,w`3]| by EUCLID_5:3
                 .= w by EUCLID_5:3;
                hence x = P by A8,A16;
              end;
              suppose
A35:            w.2 = (-1)/sqrt(1+l^2);
                0 <= l^2 by SQUARE_1:12;
                then
A36:            (sqrt(1+l^2))^2 = 1 + l*l by SQUARE_1:def 2;
                v.1 * w.1 + v.2 * w.2 = l * (1/sqrt(1+l^2)) * l
                                         * ((-1)/sqrt(1+l^2)) + v.2 * w.2
                                          by A34,A35,A32,A33
                                     .= -1 by A34,A35,A36,BKMODEL1:11;
                hence x = P by A18;
              end;
            end;
            suppose
A37:          v.2 = (-1) / sqrt(1+l^2);
              per cases by A33,A14,BKMODEL1:10;
              suppose
A38:            w.2 = 1 / sqrt(1+l^2);
                0 <= l^2 by SQUARE_1:12; then
A39:            (sqrt(1+l^2))^2 = 1 + l*l by SQUARE_1:def 2;
                v.1 * w.1 + v.2 * w.2 = l * (1/sqrt(1+l^2)) * l
                                         * ((-1)/sqrt(1+l^2)) +
                                         (1/sqrt(1+l^2)) * ((-1)/sqrt(1+l^2))
                                         by A37,A38,A32,A33
                                     .= -1 by A39,BKMODEL1:11;
                hence x = P by A18;
              end;
              suppose w.2 = (-1) / sqrt(1+l^2);
                then w`1 = v.1 & w`2 = v.2 & w`3 = v.3
                  by A8,A15,A37,A32,A33,EUCLID_5:def 1,def 2,def 3;
                then w`1 = v`1 & w`2 = v`2 & w`3 = v`3
                  by EUCLID_5:def 1,def 2,def 3;
                then v = |[w`1,w`2,w`3]| by EUCLID_5:3
                 .= w by EUCLID_5:3;
                hence x = P by A8,A16;
              end;
            end;
          end;
        end;
      end;
      hence thesis by TARSKI:def 1;
    end;
    hence thesis by A1;
  end;
