bkmodel2.miz



    begin

    definition

      :: BKMODEL2:def1

      func BK_model -> non empty Subset of ( ProjectiveSpace ( TOP-REAL 3)) equals ( negative_conic (1,1,( - 1), 0 , 0 , 0 ));

      coherence

      proof

        reconsider u = |[ 0 , 0 , 1]| as non zero Element of ( TOP-REAL 3) by ANPROJ_9: 10;

        (u `1 ) = 0 & (u `2 ) = 0 & (u `3 ) = 1 by EUCLID_5: 2;

        then

         A1: (u . 1) = 0 & (u . 2) = 0 & (u . 3) = 1 by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

        reconsider P = ( Dir u) as Element of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

        ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative

        proof

          ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) = (((((((1 * (u . 1)) * (u . 1)) + ((1 * (u . 2)) * (u . 2))) + ((( - 1) * (u . 3)) * (u . 3))) + (( 0 * (u . 1)) * (u . 2))) + (( 0 * (u . 1)) * (u . 3))) + (( 0 * (u . 2)) * (u . 3))) by PASCAL:def 1

          .= ( - 1) by A1;

          hence thesis;

        end;

        then for v be Element of ( TOP-REAL 3) st v is non zero & P = ( Dir v) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,v)) is negative by BKMODEL1: 81;

        then P in { P where P be Point of ( ProjectiveSpace ( TOP-REAL 3)) : for u be Element of ( TOP-REAL 3) st u is non zero & P = ( Dir u) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative };

        hence thesis;

      end;

    end

    theorem :: BKMODEL2:1

    

     Th01: BK_model misses absolute

    proof

      assume not BK_model misses absolute ;

      then

      consider x be object such that

       A1: x in ( BK_model /\ absolute ) by XBOOLE_0:def 1;

      

       A2: x in BK_model & x in absolute by A1, XBOOLE_0:def 4;

      x in { P where P be Point of ( ProjectiveSpace ( TOP-REAL 3)) : for u be Element of ( TOP-REAL 3) st u is non zero & P = ( Dir u) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) = 0 } by A2, PASCAL:def 2;

      then

      consider P be Point of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A3: x = P and

       A4: for u be Element of ( TOP-REAL 3) st u is non zero & P = ( Dir u) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) = 0 ;

      consider u be Element of ( TOP-REAL 3) such that

       A5: u is non zero and

       A6: P = ( Dir u) by ANPROJ_1: 26;

      consider Q be Point of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A7: x = Q and

       A8: for u be Element of ( TOP-REAL 3) st u is non zero & Q = ( Dir u) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative by A2;

      ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) = 0 & ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative by A3, A4, A5, A6, A7, A8;

      hence contradiction;

    end;

    theorem :: BKMODEL2:2

    for P be Element of ( ProjectiveSpace ( TOP-REAL 3)) holds for u be non zero Element of ( TOP-REAL 3) st P = ( Dir u) & P in BK_model holds (u . 3) <> 0

    proof

      let P be Element of ( ProjectiveSpace ( TOP-REAL 3));

      let u be non zero Element of ( TOP-REAL 3);

      assume that

       A1: P = ( Dir u) and

       A2: P in BK_model ;

      assume

       A3: (u . 3) = 0 ;

      consider Q be Point of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A4: P = Q and

       A5: for u be Element of ( TOP-REAL 3) st u is non zero & Q = ( Dir u) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative by A2;

      

       A6: ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative by A1, A4, A5;

      ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) = (((((((1 * (u . 1)) * (u . 1)) + ((1 * (u . 2)) * (u . 2))) + ((( - 1) * (u . 3)) * (u . 3))) + (( 0 * (u . 1)) * (u . 2))) + (( 0 * (u . 1)) * (u . 3))) + (( 0 * (u . 2)) * (u . 3))) by PASCAL:def 1

      .= (((u . 1) ^2 ) + ((u . 2) ^2 )) by A3;

      then (u . 1) = 0 & (u . 2) = 0 by A6, BKMODEL1: 19;

      then (u `1 ) = 0 & (u `2 ) = 0 & (u `3 ) = 0 by A3, EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

      hence contradiction by EUCLID_5: 3, EUCLID_5: 4;

    end;

    definition

      let P be Element of BK_model ;

      :: BKMODEL2:def2

      func BK_to_REAL2 P -> Element of ( inside_of_circle ( 0 , 0 ,1)) means

      : Def01: ex u be non zero Element of ( TOP-REAL 3) st ( Dir u) = P & (u . 3) = 1 & it = |[(u . 1), (u . 2)]|;

      existence

      proof

        P in { P where P be Point of ( ProjectiveSpace ( TOP-REAL 3)) : for u be Element of ( TOP-REAL 3) st u is non zero & P = ( Dir u) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative };

        then

        consider Q be Point of ( ProjectiveSpace ( TOP-REAL 3)) such that

         A1: P = Q and

         A2: for u be Element of ( TOP-REAL 3) st u is non zero & Q = ( Dir u) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative;

        consider u be Element of ( TOP-REAL 3) such that

         A3: not u is zero and

         A4: P = ( Dir u) by ANPROJ_1: 26;

        reconsider u1 = (u . 1), u2 = (u . 2), u3 = (u . 3) as Real;

        ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) = (((((((1 * (u . 1)) * (u . 1)) + ((1 * (u . 2)) * (u . 2))) + ((( - 1) * (u . 3)) * (u . 3))) + (( 0 * (u . 1)) * (u . 2))) + (( 0 * (u . 1)) * (u . 2))) + (( 0 * (u . 2)) * (u . 3))) by PASCAL:def 1

        .= ((((u . 1) ^2 ) + ((u . 2) ^2 )) - ((u . 3) ^2 ));

        then (((u1 ^2 ) + (u2 ^2 )) - (u3 ^2 )) < 0 by A1, A3, A4, A2;

        then

         A5: ((((u1 ^2 ) + (u2 ^2 )) - (u3 ^2 )) + (u3 ^2 )) < ( 0 + (u3 ^2 )) by XREAL_1: 6;

        

         A6: (u . 3) <> 0

        proof

          assume

           A7: (u . 3) = 0 ;

          

           A8: ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) = (((((((1 * (u . 1)) * (u . 1)) + ((1 * (u . 2)) * (u . 2))) + ((( - 1) * (u . 3)) * (u . 3))) + (( 0 * (u . 1)) * (u . 2))) + (( 0 * (u . 1)) * (u . 2))) + (( 0 * (u . 2)) * (u . 3))) by PASCAL:def 1

          .= ((u1 ^2 ) + (u2 ^2 )) by A7;

          ( 0 ^2 ) = 0 ;

          then 0 <= (u1 ^2 ) & 0 <= (u2 ^2 ) by SQUARE_1: 12;

          hence contradiction by A1, A3, A4, A2, A8;

        end;

        reconsider k = (1 / (u3 ^2 )) as Real;

         0 < (u3 ^2 ) by A6, SQUARE_1: 12;

        then

         A10: (((u1 ^2 ) + (u2 ^2 )) * k) < ((u3 ^2 ) * k) by A5, XREAL_1: 68;

        

         A11: ((u1 ^2 ) * k) = ((u1 / u3) ^2 ) & ((u2 ^2 ) * k) = ((u2 / u3) ^2 ) by BKMODEL1: 12;

         |[((u . 1) / (u . 3)), ((u . 2) / (u . 3)), 1]| <> ( 0. ( TOP-REAL 3))

        proof

          assume |[((u . 1) / (u . 3)), ((u . 2) / (u . 3)), 1]| = ( 0. ( TOP-REAL 3));

          then 1 = ( |[ 0 , 0 , 0 ]| `3 ) by EUCLID_5: 2, EUCLID_5: 4;

          hence contradiction by EUCLID_5: 2;

        end;

        then |[((u . 1) / (u . 3)), ((u . 2) / (u . 3)), 1]| is non zero;

        then

        reconsider v = |[((u . 1) / (u . 3)), ((u . 2) / (u . 3)), 1]| as non zero Element of ( TOP-REAL 3);

        

         A12: (v . 1) = (v `1 ) by EUCLID_5:def 1

        .= ((u . 1) / (u . 3)) by EUCLID_5: 2;

        

         A13: (v . 2) = (v `2 ) by EUCLID_5:def 2

        .= ((u . 2) / (u . 3)) by EUCLID_5: 2;

        

         A14: ((u . 3) * ((u . 1) / (u . 3))) = (u . 1) & ((u . 3) * ((u . 2) / (u . 3))) = (u . 2) by A6, XCMPLX_1: 87;

        ((u . 3) * v) = |[((u . 3) * ((u . 1) / (u . 3))), ((u . 3) * ((u . 2) / (u . 3))), ((u . 3) * 1)]| by EUCLID_5: 8

        .= |[(u `1 ), (u . 2), (u . 3)]| by A14, EUCLID_5:def 1

        .= |[(u `1 ), (u `2 ), (u . 3)]| by EUCLID_5:def 2

        .= |[(u `1 ), (u `2 ), (u `3 )]| by EUCLID_5:def 3

        .= u by EUCLID_5: 3;

        then are_Prop (v,u) by A6, ANPROJ_1: 1;

        then

         A15: P = ( Dir v) by A3, A4, ANPROJ_1: 22;

         |[(v . 1), (v . 2)]| in ( inside_of_circle ( 0 , 0 ,1))

        proof

          reconsider t = |[(v . 1), (v . 2)]| as Element of ( TOP-REAL 2);

           |.(t - |[ 0 , 0 ]|).| < 1

          proof

            

             A16: |.(t - |[ 0 , 0 ]|).| = |. |[((v . 1) - 0 ), ((v . 2) - 0 )]|.| by EUCLID: 62

            .= |.t.|;

            

             A17: (v . 1) = (t `1 ) & (v . 2) = (t `2 ) by EUCLID: 52;

            ( |.t.| ^2 ) = (((u1 ^2 ) * k) + ((u2 ^2 ) * k)) by A17, JGRAPH_1: 29, A12, A13, A11;

            then ( |.t.| ^2 ) < (1 ^2 ) by A10, A6, XCMPLX_1: 106;

            hence thesis by A16, SQUARE_1: 48;

          end;

          hence thesis;

        end;

        then

        reconsider w = |[(v . 1), (v . 2)]| as Element of ( inside_of_circle ( 0 , 0 ,1));

        take w;

        (v . 3) = (v `3 ) by EUCLID_5:def 3

        .= 1 by EUCLID_5: 2;

        hence thesis by A15;

      end;

      uniqueness

      proof

        let P1,P2 be Element of ( inside_of_circle ( 0 , 0 ,1)) such that

         A18: ex u be non zero Element of ( TOP-REAL 3) st ( Dir u) = P & (u . 3) = 1 & P1 = |[(u . 1), (u . 2)]| and

         A19: ex u be non zero Element of ( TOP-REAL 3) st ( Dir u) = P & (u . 3) = 1 & P2 = |[(u . 1), (u . 2)]|;

        consider u be non zero Element of ( TOP-REAL 3) such that

         A20: ( Dir u) = P & (u . 3) = 1 & P1 = |[(u . 1), (u . 2)]| by A18;

        consider v be non zero Element of ( TOP-REAL 3) such that

         A21: ( Dir v) = P & (v . 3) = 1 & P2 = |[(v . 1), (v . 2)]| by A19;

         are_Prop (u,v) by A20, A21, ANPROJ_1: 22;

        then

        consider a be Real such that a <> 0 and

         A22: u = (a * v) by ANPROJ_1: 1;

        1 = (a * (v . 3)) by A20, A22, RVSUM_1: 44

        .= a by A21;

        then u = v by A22, RVSUM_1: 52;

        hence thesis by A20, A21;

      end;

    end

    definition

      let Q be Element of ( inside_of_circle ( 0 , 0 ,1));

      :: BKMODEL2:def3

      func REAL2_to_BK Q -> Element of BK_model means

      : Def02: ex P be Element of ( TOP-REAL 2) st P = Q & it = ( Dir |[(P `1 ), (P `2 ), 1]|);

      existence

      proof

        reconsider P = Q as Element of ( TOP-REAL 2);

        

         A1: |.(P - |[ 0 , 0 ]|).| = |.( |[(P `1 ), (P `2 )]| - |[ 0 , 0 ]|).| by EUCLID: 53

        .= |. |[((P `1 ) - 0 ), ((P `2 ) - 0 )]|.| by EUCLID: 62

        .= |.P.| by EUCLID: 53;

        (1 ^2 ) = 1;

        then ( |.P.| ^2 ) < 1 by TOPREAL9: 45, A1, SQUARE_1: 16;

        then (((P `1 ) ^2 ) + ((P `2 ) ^2 )) < 1 by JGRAPH_3: 1;

        then

         A2: ((((P `1 ) ^2 ) + ((P `2 ) ^2 )) - 1) < (1 - 1) by XREAL_1: 14;

        

         A3: |[(P `1 ), (P `2 ), 1]| is non zero by EUCLID_5: 4, FINSEQ_1: 78;

        then

        reconsider R = ( Dir |[(P `1 ), (P `2 ), 1]|) as Element of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

        for u be Element of ( TOP-REAL 3) st u is non zero & R = ( Dir u) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative

        proof

          let u be Element of ( TOP-REAL 3);

          assume that

           A4: u is non zero and

           A5: R = ( Dir u);

           are_Prop (u, |[(P `1 ), (P `2 ), 1]|) by A3, A4, A5, ANPROJ_1: 22;

          then

          consider k be Real such that

           A6: k <> 0 and

           A7: u = (k * |[(P `1 ), (P `2 ), 1]|) by ANPROJ_1: 1;

          ( |[(P `1 ), (P `2 ), 1]| . 1) = ( |[(P `1 ), (P `2 ), 1]| `1 ) by EUCLID_5:def 1

          .= (P `1 ) by EUCLID_5: 2;

          then

           A8: (u . 1) = (k * (P `1 )) by A7, RVSUM_1: 44;

          ( |[(P `1 ), (P `2 ), 1]| . 2) = ( |[(P `1 ), (P `2 ), 1]| `2 ) by EUCLID_5:def 2

          .= (P `2 ) by EUCLID_5: 2;

          then

           A9: (u . 2) = (k * (P `2 )) by A7, RVSUM_1: 44;

          ( |[(P `1 ), (P `2 ), 1]| . 3) = ( |[(P `1 ), (P `2 ), 1]| `3 ) by EUCLID_5:def 3

          .= 1 by EUCLID_5: 2;

          then

           A10: (u . 3) = (k * 1) by A7, RVSUM_1: 44;

          

           A11: ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) = (((((((1 * (u . 1)) * (u . 1)) + ((1 * (u . 2)) * (u . 2))) + ((( - 1) * (u . 3)) * (u . 3))) + (( 0 * (u . 1)) * (u . 2))) + (( 0 * (u . 1)) * (u . 3))) + (( 0 * (u . 2)) * 1)) by PASCAL:def 1

          .= ((k ^2 ) * ((((P `1 ) ^2 ) + ((P `2 ) ^2 )) - 1)) by A8, A9, A10;

           0 < (k ^2 ) by A6, SQUARE_1: 12;

          hence thesis by A11, A2;

        end;

        then R in { P where P be Point of ( ProjectiveSpace ( TOP-REAL 3)) : for u be Element of ( TOP-REAL 3) st u is non zero & P = ( Dir u) holds ( qfconic (1,1,( - 1), 0 , 0 , 0 ,u)) is negative };

        hence thesis;

      end;

      uniqueness ;

    end

    theorem :: BKMODEL2:3

    for P be Element of BK_model holds ( REAL2_to_BK ( BK_to_REAL2 P)) = P

    proof

      let P be Element of BK_model ;

      consider u be non zero Element of ( TOP-REAL 3) such that

       A1: ( Dir u) = P and

       A2: (u . 3) = 1 and

       A3: ( BK_to_REAL2 P) = |[(u . 1), (u . 2)]| by Def01;

      consider Q be Element of ( TOP-REAL 2) such that

       A4: Q = ( BK_to_REAL2 P) and

       A5: ( REAL2_to_BK ( BK_to_REAL2 P)) = ( Dir |[(Q `1 ), (Q `2 ), 1]|) by Def02;

      

       A6: |[(Q `1 ), (Q `2 ), 1]| is non zero by EUCLID_5: 4, FINSEQ_1: 78;

       are_Prop ( |[(Q `1 ), (Q `2 ), 1]|,u)

      proof

        

         A7: (Q `1 ) = (u . 1) & (Q `2 ) = (u . 2) by A3, A4, EUCLID: 52;

        u = |[(u `1 ), (u `2 ), (u `3 )]| by EUCLID_5: 3

        .= |[(u . 1), (u `2 ), (u `3 )]| by EUCLID_5:def 1

        .= |[(u . 1), (u . 2), (u `3 )]| by EUCLID_5:def 2

        .= |[(u . 1), (u . 2), (u . 3)]| by EUCLID_5:def 3;

        hence thesis by A2, A7;

      end;

      hence thesis by A1, A5, A6, ANPROJ_1: 22;

    end;

    theorem :: BKMODEL2:4

    

     Th02: for P,Q be Element of BK_model holds P = Q iff ( BK_to_REAL2 P) = ( BK_to_REAL2 Q)

    proof

      let P,Q be Element of BK_model ;

      thus P = Q implies ( BK_to_REAL2 P) = ( BK_to_REAL2 Q);

      assume

       A1: ( BK_to_REAL2 P) = ( BK_to_REAL2 Q);

      consider u be non zero Element of ( TOP-REAL 3) such that

       A2: ( Dir u) = P & (u . 3) = 1 & ( BK_to_REAL2 P) = |[(u . 1), (u . 2)]| by Def01;

      consider v be non zero Element of ( TOP-REAL 3) such that

       A3: ( Dir v) = Q & (v . 3) = 1 & ( BK_to_REAL2 Q) = |[(v . 1), (v . 2)]| by Def01;

      (u . 1) = (v . 1) & (u . 2) = (v . 2) & (u . 3) = (v . 3) by A1, A2, A3, FINSEQ_1: 77;

      then (u `1 ) = (v . 1) & (u `2 ) = (v . 2) & (u `3 ) = (v . 3) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

      then

       A4: (u `1 ) = (v `1 ) & (u `2 ) = (v `2 ) & (u `3 ) = (v `3 ) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

      u = |[(u `1 ), (u `2 ), (u `3 )]| & v = |[(v `1 ), (v `2 ), (v `3 )]| by EUCLID_5: 3;

      hence thesis by A2, A3, A4;

    end;

    theorem :: BKMODEL2:5

    for Q be Element of ( inside_of_circle ( 0 , 0 ,1)) holds ( BK_to_REAL2 ( REAL2_to_BK Q)) = Q

    proof

      let Q be Element of ( inside_of_circle ( 0 , 0 ,1));

      consider P be Element of ( TOP-REAL 2) such that

       A1: P = Q and

       A2: ( REAL2_to_BK Q) = ( Dir |[(P `1 ), (P `2 ), 1]|) by Def02;

      consider u be non zero Element of ( TOP-REAL 3) such that

       A3: ( Dir u) = ( REAL2_to_BK Q) and

       A4: (u . 3) = 1 and

       A5: ( BK_to_REAL2 ( REAL2_to_BK Q)) = |[(u . 1), (u . 2)]| by Def01;

       |[(P `1 ), (P `2 ), 1]| is non zero by EUCLID_5: 4, FINSEQ_1: 78;

      then are_Prop ( |[(P `1 ), (P `2 ), 1]|,u) by A2, A3, ANPROJ_1: 22;

      then

      consider a be Real such that a <> 0 and

       A6: |[(P `1 ), (P `2 ), 1]| = (a * u) by ANPROJ_1: 1;

      

       A7: a = (a * (u . 3)) by A4

      .= ((a * u) . 3) by RVSUM_1: 44

      .= ( |[(P `1 ), (P `2 ), 1]| `3 ) by A6, EUCLID_5:def 3

      .= 1 by EUCLID_5: 2;

      

       A8: |[(P `1 ), (P `2 ), 1]| = u by A7, RVSUM_1: 52, A6;

      

      then

       A9: (P `1 ) = (u `1 ) by EUCLID_5: 2

      .= (u . 1) by EUCLID_5:def 1;

      (P `2 ) = (u `2 ) by A8, EUCLID_5: 2

      .= (u . 2) by EUCLID_5:def 2;

      hence thesis by A9, A5, A1, EUCLID: 53;

    end;

    theorem :: BKMODEL2:6

    for P,Q be Element of BK_model holds for P1,P2,P3 be Element of absolute st P <> Q & P1 <> P2 & (P,Q,P1) are_collinear & (P,Q,P2) are_collinear & (P,Q,P3) are_collinear holds P3 = P1 or P3 = P2

    proof

      let P,Q be Element of BK_model ;

      let P1,P2,P3 be Element of absolute ;

      assume that

       A1: P <> Q and

       A2: P1 <> P2 and

       A3: (P,Q,P1) are_collinear and

       A4: (P,Q,P2) are_collinear and

       A5: (P,Q,P3) are_collinear ;

      P3 = P1 or P3 = P2

      proof

        assume P3 <> P1 & P3 <> P2;

        then (P1,P2,P3) are_mutually_distinct by A2;

        hence contradiction by A1, A3, A4, A5, COLLSP: 3, BKMODEL1: 92;

      end;

      hence thesis;

    end;

    theorem :: BKMODEL2:7

    

     Th03: for P be Element of BK_model holds for Q be Element of ( ProjectiveSpace ( TOP-REAL 3)) holds for v be non zero Element of ( TOP-REAL 3) st P <> Q & Q = ( Dir v) & (v . 3) = 1 holds ex P1 be Element of absolute st (P,Q,P1) are_collinear

    proof

      let P be Element of BK_model ;

      let Q be Element of ( ProjectiveSpace ( TOP-REAL 3));

      let v be non zero Element of ( TOP-REAL 3);

      assume that

       A1: P <> Q and

       A2: Q = ( Dir v) and

       A3: (v . 3) = 1;

      consider u be non zero Element of ( TOP-REAL 3) such that

       A4: ( Dir u) = P & (u . 3) = 1 & ( BK_to_REAL2 P) = |[(u . 1), (u . 2)]| by Def01;

      reconsider s = |[(u . 1), (u . 2)]|, t = |[(v . 1), (v . 2)]| as Point of ( TOP-REAL 2);

      set a = 0 , b = 0 , r = 1;

      reconsider S = s, T = t, X = |[a, b]| as Element of ( REAL 2) by EUCLID: 22;

      reconsider w1 = ((( - (2 * |((t - s), (s - |[a, b]|))|)) + ( sqrt ( delta (( Sum ( sqr (T - S))),(2 * |((t - s), (s - |[a, b]|))|),(( Sum ( sqr (S - X))) - (r ^2 )))))) / (2 * ( Sum ( sqr (T - S))))) as Real;

      

       A5: s <> t

      proof

        assume s = t;

        then (u . 1) = (v . 1) & (u . 2) = (v . 2) by FINSEQ_1: 77;

        then (u `1 ) = (v . 1) & (u `2 ) = (v . 2) & (u `3 ) = (v . 3) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3, A4, A3;

        then

         A6: (u `1 ) = (v `1 ) & (u `2 ) = (v `2 ) & (u `3 ) = (v `3 ) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

        u = |[(u `1 ), (u `2 ), (u `3 )]| by EUCLID_5: 3

        .= v by A6, EUCLID_5: 3;

        hence contradiction by A4, A1, A2;

      end;

      consider e1 be Point of ( TOP-REAL 2) such that

       A7: ( {e1} = (( halfline (s,t)) /\ ( circle (a,b,r))) & e1 = (((1 - w1) * s) + (w1 * t))) by A5, A4, TOPREAL9: 58;

      reconsider f = |[(e1 `1 ), (e1 `2 ), 1]| as Element of ( TOP-REAL 3);

      f is non zero by FINSEQ_1: 78, EUCLID_5: 4;

      then

      reconsider ee1 = f as non zero Element of ( TOP-REAL 3);

       |[(s `1 ), (s `2 )]| = |[(u . 1), (u . 2)]| & |[(t `1 ), (t `2 )]| = |[(v . 1), (v . 2)]| by EUCLID: 53;

      then (s `1 ) = (u . 1) & (s `2 ) = (u . 2) & (t `1 ) = (v . 1) & (t `2 ) = (v . 2) by FINSEQ_1: 77;

      then

       A8: (s . 1) = (u . 1) & (s . 2) = (u . 2) & (t . 1) = (v . 1) & (t . 2) = (v . 2) by EUCLID:def 9, EUCLID:def 10;

      reconsider P1 = ( Dir ee1) as Point of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

      (((1 * ee1) + (( - (1 - w1)) * u)) + (( - w1) * v)) = ( 0. ( TOP-REAL 3))

      proof

        

         A9: (1 * ee1) = |[(1 * (ee1 `1 )), (1 * (ee1 `2 )), (1 * (ee1 `3 ))]| by EUCLID_5: 7

        .= ee1 by EUCLID_5: 3;

        ee1 = (((1 - w1) * u) + (w1 * v))

        proof

          

           A10: (((1 - w1) * s) + (w1 * t)) = |[((((1 - w1) * s) + (w1 * t)) `1 ), ((((1 - w1) * s) + (w1 * t)) `2 )]| by EUCLID: 53;

          (((1 - w1) * s) + (w1 * t)) = |[((((1 - w1) * s) `1 ) + ((w1 * t) `1 )), ((((1 - w1) * s) `2 ) + ((w1 * t) `2 ))]| by EUCLID: 55

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) `1 )), ((((1 - w1) * s) `2 ) + ((w1 * t) `2 ))]| by EUCLID:def 9

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) . 1)), ((((1 - w1) * s) `2 ) + ((w1 * t) `2 ))]| by EUCLID:def 9

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) . 1)), ((((1 - w1) * s) . 2) + ((w1 * t) `2 ))]| by EUCLID:def 10

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) . 1)), ((((1 - w1) * s) . 2) + ((w1 * t) . 2))]| by EUCLID:def 10

          .= |[(((1 - w1) * (s . 1)) + ((w1 * t) . 1)), ((((1 - w1) * s) . 2) + ((w1 * t) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w1) * (s . 1)) + (w1 * (t . 1))), ((((1 - w1) * s) . 2) + ((w1 * t) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w1) * (s . 1)) + (w1 * (t . 1))), (((1 - w1) * (s . 2)) + ((w1 * t) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w1) * (u . 1)) + (w1 * (v . 1))), (((1 - w1) * (u . 2)) + (w1 * (v . 2)))]| by A8, RVSUM_1: 44;

          then

           A11: (e1 `1 ) = (((1 - w1) * (u . 1)) + (w1 * (v . 1))) & (e1 `2 ) = (((1 - w1) * (u . 2)) + (w1 * (v . 2))) by A7, A10, FINSEQ_1: 77;

          (((1 - w1) * u) + (w1 * v)) = |[(((1 - w1) * (u . 1)) + (w1 * (v . 1))), (((1 - w1) * (u . 2)) + (w1 * (v . 2))), (((1 - w1) * (u . 3)) + (w1 * (v . 3)))]|

          proof

            (((1 - w1) * u) `1 ) = ((1 - w1) * (u `1 )) by EUCLID_5: 9

            .= ((1 - w1) * (u . 1)) by EUCLID_5:def 1;

            

            then

             A12: ((((1 - w1) * u) `1 ) + ((w1 * v) `1 )) = (((1 - w1) * (u . 1)) + ((w1 * v) . 1)) by EUCLID_5:def 1

            .= (((1 - w1) * (u . 1)) + (w1 * (v . 1))) by RVSUM_1: 44;

            (((1 - w1) * u) `2 ) = ((1 - w1) * (u `2 )) by EUCLID_5: 9

            .= ((1 - w1) * (u . 2)) by EUCLID_5:def 2;

            

            then

             A13: ((((1 - w1) * u) `2 ) + ((w1 * v) `2 )) = (((1 - w1) * (u . 2)) + ((w1 * v) . 2)) by EUCLID_5:def 2

            .= (((1 - w1) * (u . 2)) + (w1 * (v . 2))) by RVSUM_1: 44;

            (((1 - w1) * u) `3 ) = ((1 - w1) * (u `3 )) by EUCLID_5: 9

            .= ((1 - w1) * (u . 3)) by EUCLID_5:def 3;

            

            then ((((1 - w1) * u) `3 ) + ((w1 * v) `3 )) = (((1 - w1) * (u . 3)) + ((w1 * v) . 3)) by EUCLID_5:def 3

            .= (((1 - w1) * (u . 3)) + (w1 * (v . 3))) by RVSUM_1: 44;

            hence thesis by A12, A13, EUCLID_5: 5;

          end;

          hence thesis by A11, A4, A3;

        end;

        

        then ((ee1 + (( - (1 - w1)) * u)) + (( - w1) * v)) = ((((1 - w1) * u) + (w1 * v)) + ((( - (1 - w1)) * u) + (( - w1) * v))) by RVSUM_1: 15

        .= (((1 - w1) * u) + ((w1 * v) + ((( - (1 - w1)) * u) + (( - w1) * v)))) by RVSUM_1: 15

        .= (((1 - w1) * u) + ((( - (1 - w1)) * u) + ((w1 * v) + (( - w1) * v)))) by RVSUM_1: 15

        .= ((((1 - w1) * u) + (( - (1 - w1)) * u)) + ((w1 * v) + (( - w1) * v))) by RVSUM_1: 15

        .= (( 0. ( TOP-REAL 3)) + ((w1 * v) + (( - w1) * v))) by BKMODEL1: 4

        .= (( 0. ( TOP-REAL 3)) + ( 0. ( TOP-REAL 3))) by BKMODEL1: 4

        .= |[( 0 + 0 ), ( 0 + 0 ), ( 0 + 0 )]| by EUCLID_5: 4, EUCLID_5: 6

        .= ( 0. ( TOP-REAL 3)) by EUCLID_5: 4;

        hence thesis by A9;

      end;

      then

       A14: (P1,P,Q) are_collinear by A4, A2, ANPROJ_8: 11;

      e1 in {e1} by TARSKI:def 1;

      then

       A15: e1 in ( circle ( 0 , 0 ,1)) by A7, XBOOLE_0:def 4;

      now

        

         A16: (ee1 `1 ) = (e1 `1 ) & (ee1 `2 ) = (e1 `2 ) & (ee1 `3 ) = 1 by EUCLID_5: 2;

        then (ee1 . 1) = (e1 `1 ) & (ee1 . 2) = (e1 `2 ) by EUCLID_5:def 1, EUCLID_5:def 2;

        hence |[(ee1 . 1), (ee1 . 2)]| in ( circle ( 0 , 0 ,1)) by A15, EUCLID: 53;

        thus (ee1 . 3) = 1 by A16, EUCLID_5:def 3;

      end;

      then P1 is Element of absolute by BKMODEL1: 86;

      hence thesis by A14, COLLSP: 8;

    end;

    theorem :: BKMODEL2:8

    for P be Element of BK_model holds for L be LINE of ( IncProjSp_of real_projective_plane ) holds ex Q be Element of ( ProjectiveSpace ( TOP-REAL 3)) st P <> Q & Q in L

    proof

      let P be Element of BK_model ;

      let L be LINE of ( IncProjSp_of real_projective_plane );

      consider p,q be Point of real_projective_plane such that

       A2: p <> q and

       A3: L = ( Line (p,q)) by BKMODEL1: 73;

      P <> p or P <> q by A2;

      hence thesis by A3, COLLSP: 10;

    end;

    theorem :: BKMODEL2:9

    

     Th04: for a,b,c,d,e be Real holds for u,v,w be Element of ( TOP-REAL 3) st u = |[a, b, e]| & v = |[c, d, 0 ]| & w = |[(a + c), (b + d), e]| holds |{u, v, w}| = 0

    proof

      let a,b,c,d,e be Real;

      let u,v,w be Element of ( TOP-REAL 3);

      assume that

       A1: u = |[a, b, e]| and

       A2: v = |[c, d, 0 ]| and

       A3: w = |[(a + c), (b + d), e]|;

      

       A4: (u `1 ) = a & (u `2 ) = b & (u `3 ) = e & (v `1 ) = c & (v `2 ) = d & (v `3 ) = 0 & (w `1 ) = (a + c) & (w `2 ) = (b + d) & (w `3 ) = e by A1, A2, A3, EUCLID_5: 2;

       |{u, v, w}| = ((((((((u `1 ) * (v `2 )) * (w `3 )) - (((u `3 ) * (v `2 )) * (w `1 ))) - (((u `1 ) * (v `3 )) * (w `2 ))) + (((u `2 ) * (v `3 )) * (w `1 ))) - (((u `2 ) * (v `1 )) * (w `3 ))) + (((u `3 ) * (v `1 )) * (w `2 ))) by ANPROJ_8: 27

      .= (((((a * d) * e) - ((e * d) * (a + c))) - ((b * c) * e)) + ((e * c) * (b + d))) by A4;

      hence thesis;

    end;

    theorem :: BKMODEL2:10

    

     Th05: for a,b be Real holds for c be non zero Real holds |[a, b, c]| is non zero Element of ( TOP-REAL 3)

    proof

      let a,b be Real;

      let c be non zero Real;

       |[a, b, c]| is non zero by EUCLID_5: 4, FINSEQ_1: 78;

      hence thesis;

    end;

    theorem :: BKMODEL2:11

    

     Th06: for u,v be Element of ( TOP-REAL 3) holds for a,b,c,d,e be Real st u = |[a, b, c]| & v = |[d, e, 0 ]| & are_Prop (u,v) holds c = 0

    proof

      let u,v be Element of ( TOP-REAL 3);

      let a,b,c,d,e be Real;

      assume that

       A1: u = |[a, b, c]| and

       A2: v = |[d, e, 0 ]| and

       A3: are_Prop (u,v);

      consider f be Real such that f <> 0 and

       A5: |[a, b, c]| = (f * |[d, e, 0 ]|) by A1, A2, A3, ANPROJ_1: 1;

      (f * |[d, e, 0 ]|) = |[(f * d), (f * e), (f * 0 )]| by EUCLID_5: 8;

      

      then c = ( |[(f * d), (f * e), (f * 0 )]| `3 ) by A5, EUCLID_5: 2

      .= (f * 0 ) by EUCLID_5: 2;

      hence thesis;

    end;

    theorem :: BKMODEL2:12

    for P,Q,R be Element of ( ProjectiveSpace ( TOP-REAL 3)) holds for u,v,w be non zero Element of ( TOP-REAL 3) st P = ( Dir u) & Q = ( Dir v) & R = ( Dir w) & (u `3 ) <> 0 & (v `3 ) = 0 & w = |[((u `1 ) + (v `1 )), ((u `2 ) + (v `2 )), (u `3 )]| holds R <> P & R <> Q

    proof

      let P,Q,R be Element of ( ProjectiveSpace ( TOP-REAL 3));

      let u,v,w be non zero Element of ( TOP-REAL 3);

      assume that

       A1: P = ( Dir u) and

       A2: Q = ( Dir v) and

       A3: R = ( Dir w) and

       A4: (u `3 ) <> 0 and

       A5: (v `3 ) = 0 and

       A6: w = |[((u `1 ) + (v `1 )), ((u `2 ) + (v `2 )), (u `3 )]|;

      hereby

        assume R = P;

        then are_Prop (u,w) by A1, A3, ANPROJ_1: 22;

        then

        consider a be Real such that a <> 0 and

         A7: u = (a * w) by ANPROJ_1: 1;

        

         A8: |[(u `1 ), (u `2 ), (u `3 )]| = u by EUCLID_5: 3

        .= |[(a * (w `1 )), (a * (w `2 )), (a * (w `3 ))]| by A7, EUCLID_5: 7;

        then |[(u `1 ), (u `2 ), (u `3 )]| = |[(a * (w `1 )), (a * (w `2 )), (a * (u `3 ))]| by A6, EUCLID_5: 2;

        then (u `3 ) = (a * (u `3 )) by FINSEQ_1: 78;

        then

         A9: a = 1 by A4, XCMPLX_1: 7;

        (w `1 ) = ((u `1 ) + (v `1 )) & (w `2 ) = ((u `2 ) + (v `2 )) & (w `3 ) = (u `3 ) by A6, EUCLID_5: 2;

        then (u `1 ) = ((u `1 ) + (v `1 )) & (u `2 ) = ((u `2 ) + (v `2 )) by A8, A9, FINSEQ_1: 78;

        hence contradiction by A5, EUCLID_5: 3, EUCLID_5: 4;

      end;

      hereby

        assume R = Q;

        then are_Prop (v,w) by A2, A3, ANPROJ_1: 22;

        then

        consider b be Real such that

         A11: b <> 0 and

         A12: v = (b * w) by ANPROJ_1: 1;

         |[(v `1 ), (v `2 ), (v `3 )]| = v by EUCLID_5: 3

        .= |[(b * (w `1 )), (b * (w `2 )), (b * (w `3 ))]| by A12, EUCLID_5: 7;

        then |[(v `1 ), (v `2 ), (v `3 )]| = |[(b * (w `1 )), (b * (w `2 )), (b * (u `3 ))]| by A6, EUCLID_5: 2;

        hence contradiction by A4, A11, A5, FINSEQ_1: 78;

      end;

    end;

    theorem :: BKMODEL2:13

    

     Th07: for L be LINE of ( IncProjSp_of real_projective_plane ) holds for P,Q be Element of ( ProjectiveSpace ( TOP-REAL 3)) st P <> Q & P in L & Q in L holds L = ( Line (P,Q))

    proof

      let L be LINE of ( IncProjSp_of real_projective_plane );

      let P,Q be Element of ( ProjectiveSpace ( TOP-REAL 3));

      assume that

       A1: P <> Q and

       A2: P in L and

       A3: Q in L;

      reconsider L9 = L as LINE of real_projective_plane by INCPROJ: 4;

      L9 = ( Line (P,Q)) by A1, A2, A3, COLLSP: 19;

      hence thesis;

    end;

    theorem :: BKMODEL2:14

    for L be LINE of ( IncProjSp_of real_projective_plane ) holds for P,Q be Element of ( ProjectiveSpace ( TOP-REAL 3)) holds for u,v be non zero Element of ( TOP-REAL 3) st P in L & Q in L & P = ( Dir u) & Q = ( Dir v) & (u `3 ) <> 0 & (v `3 ) = 0 holds P <> Q & ( Dir |[((u `1 ) + (v `1 )), ((u `2 ) + (v `2 )), (u `3 )]|) in L

    proof

      let L be LINE of ( IncProjSp_of real_projective_plane );

      let P,Q be Element of ( ProjectiveSpace ( TOP-REAL 3));

      let u,v be non zero Element of ( TOP-REAL 3);

      assume that

       A1: P in L and

       A2: Q in L and

       A3: P = ( Dir u) and

       A4: Q = ( Dir v) and

       A5: (u `3 ) <> 0 and

       A6: (v `3 ) = 0 ;

      thus

       A7: P <> Q

      proof

        assume P = Q;

        then

         A8: are_Prop (u,v) by A3, A4, ANPROJ_1: 22;

        u = |[(u `1 ), (u `2 ), (u `3 )]| & v = |[(v `1 ), (v `2 ), 0 ]| by A6, EUCLID_5: 3;

        hence contradiction by A5, A8, Th06;

      end;

      reconsider w = |[((u `1 ) + (v `1 )), ((u `2 ) + (v `2 )), (u `3 )]| as non zero Element of ( TOP-REAL 3) by A5, Th05;

      reconsider R = ( Dir w) as Element of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

      u = |[(u `1 ), (u `2 ), (u `3 )]| & v = |[(v `1 ), (v `2 ), 0 ]| by A6, EUCLID_5: 3;

      then |{u, v, w}| = 0 by Th04;

      then (P,Q,R) are_collinear by A3, A4, BKMODEL1: 1;

      then R in ( Line (P,Q)) by COLLSP: 11;

      hence thesis by A1, A2, A7, Th07;

    end;

    theorem :: BKMODEL2:15

    

     Th08: for u,v,w be Element of ( TOP-REAL 3) st (v `3 ) = 0 & w = |[((u `1 ) + (v `1 )), ((u `2 ) + (v `2 )), (u `3 )]| holds |{u, v, w}| = 0

    proof

      let u,v,w be Element of ( TOP-REAL 3);

      assume that

       A2: (v `3 ) = 0 and

       A3: w = |[((u `1 ) + (v `1 )), ((u `2 ) + (v `2 )), (u `3 )]|;

      

       A4: |{u, v, w}| = ((((((((u `1 ) * (v `2 )) * (w `3 )) - (((u `3 ) * (v `2 )) * (w `1 ))) - (((u `1 ) * (v `3 )) * (w `2 ))) + (((u `2 ) * (v `3 )) * (w `1 ))) - (((u `2 ) * (v `1 )) * (w `3 ))) + (((u `3 ) * (v `1 )) * (w `2 ))) by ANPROJ_8: 27

      .= ((((((u `1 ) * (v `2 )) * (w `3 )) - (((u `3 ) * (v `2 )) * (w `1 ))) - (((u `2 ) * (v `1 )) * (w `3 ))) + (((u `3 ) * (v `1 )) * (w `2 ))) by A2;

      (w `1 ) = ((u `1 ) + (v `1 )) & (w `2 ) = ((u `2 ) + (v `2 )) & (w `3 ) = (u `3 ) by A3, EUCLID_5: 2;

      hence thesis by A4;

    end;

    theorem :: BKMODEL2:16

    

     Th09: for L be LINE of ( IncProjSp_of real_projective_plane ) holds for P be Element of ( ProjectiveSpace ( TOP-REAL 3)) holds for u be non zero Element of ( TOP-REAL 3) st P = ( Dir u) & P in L & (u . 3) <> 0 holds ex Q be Element of ( ProjectiveSpace ( TOP-REAL 3)) st (ex v be non zero Element of ( TOP-REAL 3) st Q = ( Dir v) & Q in L & P <> Q & (v . 3) <> 0 )

    proof

      let L be LINE of ( IncProjSp_of real_projective_plane );

      let P be Element of ( ProjectiveSpace ( TOP-REAL 3));

      let u be non zero Element of ( TOP-REAL 3);

      assume that

       A1: P = ( Dir u) and

       A2: P in L and

       A3: (u . 3) <> 0 ;

      assume

       A4: not ex Q be Element of ( ProjectiveSpace ( TOP-REAL 3)) st (ex v be non zero Element of ( TOP-REAL 3) st Q = ( Dir v) & Q in L & P <> Q & (v . 3) <> 0 );

      consider p,q be Element of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A5: p <> q and

       A6: L = ( Line (p,q)) by BKMODEL1: 73;

      consider up be Element of ( TOP-REAL 3) such that

       A7: not up is zero and

       A8: p = ( Dir up) by ANPROJ_1: 26;

      consider vp be Element of ( TOP-REAL 3) such that

       A9: not vp is zero and

       A10: q = ( Dir vp) by ANPROJ_1: 26;

      reconsider P9 = P as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 3;

      reconsider L9 = L as LINE of real_projective_plane by INCPROJ: 4;

      per cases ;

        suppose

         A11: (up `3 ) = 0 & (vp `3 ) = 0 ;

        per cases by A5;

          suppose

           A12: P <> p;

          

           A13: (u `3 ) <> 0 by A3, EUCLID_5:def 3;

           |[((u `1 ) + (up `1 )), ((u `2 ) + (up `2 )), (u `3 )]| is non zero

          proof

            assume |[((u `1 ) + (up `1 )), ((u `2 ) + (up `2 )), (u `3 )]| is zero;

            then (u `3 ) = 0 by EUCLID_5: 4, FINSEQ_1: 78;

            hence contradiction by A3, EUCLID_5:def 3;

          end;

          then

          reconsider wp = |[((u `1 ) + (up `1 )), ((u `2 ) + (up `2 )), (u `3 )]| as non zero Element of ( TOP-REAL 3);

          reconsider R = ( Dir wp) as Element of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

          

           A14: |{u, up, wp}| = 0 by A11, Th08;

          now

            thus R <> P

            proof

              assume R = P;

              then are_Prop (wp,u) by A1, ANPROJ_1: 22;

              then

              consider a be Real such that a <> 0 and

               A15: wp = (a * u) by ANPROJ_1: 1;

              a = 1 & (up `1 ) = 0 & (up `2 ) = 0

              proof

                

                 A16: |[(a * (u `1 )), (a * (u `2 )), (a * (u `3 ))]| = |[((u `1 ) + (up `1 )), ((u `2 ) + (up `2 )), (u `3 )]| by A15, EUCLID_5: 7;

                then (a * (u `1 )) = ((u `1 ) + (up `1 )) & (a * (u `2 )) = ((u `2 ) + (up `2 )) & (a * (u `3 )) = (u `3 ) by FINSEQ_1: 78;

                hence a = 1 by XCMPLX_1: 7, A13;

                then (u `1 ) = ((u `1 ) + (up `1 )) & (u `2 ) = ((u `2 ) + (up `2 )) by A16, FINSEQ_1: 78;

                hence thesis;

              end;

              hence contradiction by A7, A11, EUCLID_5: 3, EUCLID_5: 4;

            end;

            reconsider R2 = R as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 2;

            now

              L = ( Line (P,p))

              proof

                P in L & p in L & P <> p & L is LINE of real_projective_plane by A6, A2, A12, COLLSP: 10, INCPROJ: 4;

                hence thesis by COLLSP: 19;

              end;

              hence R2 on L by A1, A7, A8, A14, BKMODEL1: 77;

              thus L is LINE of real_projective_plane by INCPROJ: 4;

            end;

            hence R in L by INCPROJ: 5;

            (wp `3 ) = (u `3 ) by EUCLID_5: 2;

            hence (wp . 3) <> 0 by A13, EUCLID_5:def 3;

          end;

          hence contradiction by A4;

        end;

          suppose

           A17: P <> q;

          

           A18: (u `3 ) <> 0 by A3, EUCLID_5:def 3;

           |[((u `1 ) + (vp `1 )), ((u `2 ) + (vp `2 )), (u `3 )]| is non zero

          proof

            assume |[((u `1 ) + (vp `1 )), ((u `2 ) + (vp `2 )), (u `3 )]| is zero;

            then (u `3 ) = 0 by EUCLID_5: 4, FINSEQ_1: 78;

            hence contradiction by A3, EUCLID_5:def 3;

          end;

          then

          reconsider wp = |[((u `1 ) + (vp `1 )), ((u `2 ) + (vp `2 )), (u `3 )]| as non zero Element of ( TOP-REAL 3);

          reconsider R = ( Dir wp) as Element of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

          

           A19: |{u, vp, wp}| = 0 by A11, Th08;

          now

            thus R <> P

            proof

              assume R = P;

              then are_Prop (wp,u) by A1, ANPROJ_1: 22;

              then

              consider a be Real such that a <> 0 and

               A20: wp = (a * u) by ANPROJ_1: 1;

              a = 1 & (vp `1 ) = 0 & (vp `2 ) = 0

              proof

                 |[(a * (u `1 )), (a * (u `2 )), (a * (u `3 ))]| = |[((u `1 ) + (vp `1 )), ((u `2 ) + (vp `2 )), (u `3 )]| by A20, EUCLID_5: 7;

                then

                 A21: (a * (u `1 )) = ((u `1 ) + (vp `1 )) & (a * (u `2 )) = ((u `2 ) + (vp `2 )) & (a * (u `3 )) = (u `3 ) by FINSEQ_1: 78;

                hence a = 1 by XCMPLX_1: 7, A18;

                hence thesis by A21;

              end;

              hence contradiction by A11, A9, EUCLID_5: 3, EUCLID_5: 4;

            end;

            reconsider R2 = R as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 2;

            now

              L = ( Line (P,q))

              proof

                P in L & q in L & P <> q & L is LINE of real_projective_plane by A6, A2, COLLSP: 10, A17, INCPROJ: 4;

                hence thesis by COLLSP: 19;

              end;

              hence R2 on L by A1, A9, A10, A19, BKMODEL1: 77;

              thus L is LINE of real_projective_plane by INCPROJ: 4;

            end;

            hence R in L by INCPROJ: 5;

            (wp `3 ) = (u `3 ) by EUCLID_5: 2;

            hence (wp . 3) <> 0 by A18, EUCLID_5:def 3;

          end;

          hence contradiction by A4;

        end;

      end;

        suppose (up `3 ) <> 0 or (vp `3 ) <> 0 ;

        per cases ;

          suppose

           A22: (up `3 ) <> 0 ;

          per cases ;

            suppose

             A23: P = p;

            per cases ;

              suppose

               A24: (vp `3 ) <> 0 ;

              (vp . 3) = 0 by A9, A10, A23, A5, A4, A6, COLLSP: 10;

              hence contradiction by A24, EUCLID_5:def 3;

            end;

              suppose

               A25: (vp `3 ) = 0 ;

              

               A26: (u `3 ) <> 0 by A3, EUCLID_5:def 3;

               |[((u `1 ) + (vp `1 )), ((u `2 ) + (vp `2 )), (u `3 )]| is non zero

              proof

                assume |[((u `1 ) + (vp `1 )), ((u `2 ) + (vp `2 )), (u `3 )]| is zero;

                then (u `3 ) = 0 by EUCLID_5: 4, FINSEQ_1: 78;

                hence contradiction by A3, EUCLID_5:def 3;

              end;

              then

              reconsider wp = |[((u `1 ) + (vp `1 )), ((u `2 ) + (vp `2 )), (u `3 )]| as non zero Element of ( TOP-REAL 3);

              reconsider R = ( Dir wp) as Element of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

              

               A27: |{u, vp, wp}| = 0 by A25, Th08;

              now

                thus R <> P

                proof

                  assume R = P;

                  then are_Prop (wp,u) by A1, ANPROJ_1: 22;

                  then

                  consider a be Real such that a <> 0 and

                   A28: wp = (a * u) by ANPROJ_1: 1;

                  a = 1 & (vp `1 ) = 0 & (vp `2 ) = 0

                  proof

                     |[(a * (u `1 )), (a * (u `2 )), (a * (u `3 ))]| = |[((u `1 ) + (vp `1 )), ((u `2 ) + (vp `2 )), (u `3 )]| by A28, EUCLID_5: 7;

                    then

                     A29: (a * (u `1 )) = ((u `1 ) + (vp `1 )) & (a * (u `2 )) = ((u `2 ) + (vp `2 )) & (a * (u `3 )) = (u `3 ) by FINSEQ_1: 78;

                    hence a = 1 by XCMPLX_1: 7, A26;

                    hence thesis by A29;

                  end;

                  hence contradiction by A25, EUCLID_5: 3, EUCLID_5: 4, A9;

                end;

                reconsider R2 = R as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 2;

                R2 on L & L is LINE of real_projective_plane by A6, A23, A1, A9, A10, A27, BKMODEL1: 77, INCPROJ: 4;

                hence R in L by INCPROJ: 5;

                (wp `3 ) = (u `3 ) by EUCLID_5: 2;

                hence (wp . 3) <> 0 by A26, EUCLID_5:def 3;

              end;

              hence contradiction by A4;

            end;

          end;

            suppose P <> p;

            then (up . 3) = 0 by A8, A6, A4, A7, COLLSP: 10;

            hence contradiction by A22, EUCLID_5:def 3;

          end;

        end;

          suppose

           A30: (vp `3 ) <> 0 ;

          per cases ;

            suppose

             A31: P = q;

            per cases ;

              suppose

               A32: (up `3 ) <> 0 ;

              (up . 3) = 0 by A7, A8, A31, A5, A4, A6, COLLSP: 10;

              hence contradiction by A32, EUCLID_5:def 3;

            end;

              suppose

               A33: (up `3 ) = 0 ;

              

               A34: (u `3 ) <> 0 by A3, EUCLID_5:def 3;

               |[((u `1 ) + (up `1 )), ((u `2 ) + (up `2 )), (u `3 )]| is non zero

              proof

                assume |[((u `1 ) + (up `1 )), ((u `2 ) + (up `2 )), (u `3 )]| is zero;

                then (u `3 ) = 0 by EUCLID_5: 4, FINSEQ_1: 78;

                hence contradiction by A3, EUCLID_5:def 3;

              end;

              then

              reconsider wp = |[((u `1 ) + (up `1 )), ((u `2 ) + (up `2 )), (u `3 )]| as non zero Element of ( TOP-REAL 3);

              reconsider R = ( Dir wp) as Element of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

              

               A35: |{u, up, wp}| = 0 by A33, Th08;

              now

                thus R <> P

                proof

                  assume R = P;

                  then are_Prop (wp,u) by A1, ANPROJ_1: 22;

                  then

                  consider a be Real such that a <> 0 and

                   A36: wp = (a * u) by ANPROJ_1: 1;

                  a = 1 & (up `1 ) = 0 & (up `2 ) = 0

                  proof

                     |[(a * (u `1 )), (a * (u `2 )), (a * (u `3 ))]| = |[((u `1 ) + (up `1 )), ((u `2 ) + (up `2 )), (u `3 )]| by A36, EUCLID_5: 7;

                    then

                     A37: (a * (u `1 )) = ((u `1 ) + (up `1 )) & (a * (u `2 )) = ((u `2 ) + (up `2 )) & (a * (u `3 )) = (u `3 ) by FINSEQ_1: 78;

                    hence a = 1 by XCMPLX_1: 7, A34;

                    hence thesis by A37;

                  end;

                  hence contradiction by A33, EUCLID_5: 3, EUCLID_5: 4, A7;

                end;

                reconsider R2 = R as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 2;

                now

                  L = ( Line (P,p))

                  proof

                    P in L & p in L & P <> p & L is LINE of real_projective_plane by A6, A31, A5, COLLSP: 10, INCPROJ: 4;

                    hence thesis by COLLSP: 19;

                  end;

                  hence R2 on L by A1, A7, A8, A35, BKMODEL1: 77;

                  thus L is LINE of real_projective_plane by INCPROJ: 4;

                end;

                hence R in L by INCPROJ: 5;

                (wp `3 ) = (u `3 ) by EUCLID_5: 2;

                hence (wp . 3) <> 0 by A34, EUCLID_5:def 3;

              end;

              hence contradiction by A4;

            end;

          end;

            suppose P <> q;

            then (vp . 3) = 0 by A10, A6, COLLSP: 10, A4, A9;

            hence contradiction by A30, EUCLID_5:def 3;

          end;

        end;

      end;

    end;

    theorem :: BKMODEL2:17

    

     Th10: for P be Element of BK_model holds for L be LINE of ( IncProjSp_of real_projective_plane ) st P in L holds ex Q be Element of ( ProjectiveSpace ( TOP-REAL 3)) st P <> Q & Q in L & for u be non zero Element of ( TOP-REAL 3) st Q = ( Dir u) holds (u . 3) <> 0

    proof

      let P be Element of BK_model ;

      let L be LINE of ( IncProjSp_of real_projective_plane );

      assume

       A1: P in L;

      consider u be non zero Element of ( TOP-REAL 3) such that

       A2: P = ( Dir u) & (u . 3) = 1 and ( BK_to_REAL2 P) = |[(u . 1), (u . 2)]| by Def01;

      consider Q be Element of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A3: (ex v be non zero Element of ( TOP-REAL 3) st Q = ( Dir v) & Q in L & P <> Q & (v . 3) <> 0 ) by A1, A2, Th09;

      consider v be non zero Element of ( TOP-REAL 3) such that

       A4: Q = ( Dir v) & Q in L & P <> Q & (v . 3) <> 0 by A3;

      take Q;

      now

        thus P <> Q & Q in L by A3;

        thus for u be non zero Element of ( TOP-REAL 3) st Q = ( Dir u) holds (u . 3) <> 0

        proof

          let w be non zero Element of ( TOP-REAL 3);

          assume Q = ( Dir w);

          then are_Prop (v,w) by A4, ANPROJ_1: 22;

          then

          consider a be Real such that

           A5: a <> 0 and

           A6: v = (a * w) by ANPROJ_1: 1;

          (a * w) = |[(a * (w `1 )), (a * (w `2 )), (a * (w `3 ))]| by EUCLID_5: 7;

          then (v `3 ) = (a * (w `3 )) by A6, EUCLID_5: 2;

          

          then (w `3 ) = ((v `3 ) / a) by A5, XCMPLX_1: 89

          .= ((v . 3) / a) by EUCLID_5:def 3;

          hence (w . 3) <> 0 by A5, A4, EUCLID_5:def 3;

        end;

      end;

      hence thesis;

    end;

    theorem :: BKMODEL2:18

    

     Th11: for u,v be non zero Element of ( TOP-REAL 3) holds for k be non zero Real st u = (k * v) holds ( Dir u) = ( Dir v)

    proof

      let u,v be non zero Element of ( TOP-REAL 3);

      let k be non zero Real;

      assume u = (k * v);

      then are_Prop (u,v) by ANPROJ_1: 1;

      hence thesis by ANPROJ_1: 22;

    end;

    theorem :: BKMODEL2:19

    for P be Element of BK_model holds for Q be Element of ( ProjectiveSpace ( TOP-REAL 3)) st P <> Q holds ex P1 be Element of absolute st (P,Q,P1) are_collinear

    proof

      let P be Element of BK_model ;

      let Q be Element of ( ProjectiveSpace ( TOP-REAL 3));

      assume P <> Q;

      then ( Line (P,Q)) is LINE of real_projective_plane by COLLSP:def 7;

      then

      reconsider L = ( Line (P,Q)) as LINE of ( IncProjSp_of real_projective_plane ) by INCPROJ: 4;

      consider R be Element of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A1: P <> R and

       A2: R in L and

       A3: for u be non zero Element of ( TOP-REAL 3) st R = ( Dir u) holds (u . 3) <> 0 by COLLSP: 10, Th10;

      consider u be non zero Element of ( TOP-REAL 3) such that

       A4: ( Dir u) = P & (u . 3) = 1 & ( BK_to_REAL2 P) = |[(u . 1), (u . 2)]| by Def01;

      consider v9 be Element of ( TOP-REAL 3) such that

       A5: v9 is non zero and

       A6: ( Dir v9) = R by ANPROJ_1: 26;

      

       A7: (v9 . 3) <> 0 by A5, A6, A3;

      then

       A8: (v9 `3 ) <> 0 by EUCLID_5:def 3;

      then

      reconsider k = (1 / (v9 `3 )) as non zero Real;

      (k * v9) is non zero

      proof

        assume (k * v9) is zero;

        then |[ 0 , 0 , 0 ]| = |[(k * (v9 `1 )), (k * (v9 `2 )), (k * (v9 `3 ))]| by EUCLID_5: 4, EUCLID_5: 7;

        then (v9 `3 ) = 0 by FINSEQ_1: 78;

        hence contradiction by A7, EUCLID_5:def 3;

      end;

      then

      reconsider v = (k * v9) as non zero Element of ( TOP-REAL 3);

      

       A9: ( Dir v) = R & (v . 3) = 1

      proof

        thus ( Dir v) = R by A6, A5, Th11;

        

         A10: |[(v `1 ), (v `2 ), (v `3 )]| = v by EUCLID_5: 3

        .= |[(k * (v9 `1 )), (k * (v9 `2 )), (k * (v9 `3 ))]| by EUCLID_5: 7;

        

        thus (v . 3) = (v `3 ) by EUCLID_5:def 3

        .= (k * (v9 `3 )) by A10, FINSEQ_1: 78

        .= 1 by A8, XCMPLX_1: 106;

      end;

      reconsider s = |[(u . 1), (u . 2)]|, t = |[(v . 1), (v . 2)]| as Point of ( TOP-REAL 2);

      set a = 0 , b = 0 , r = 1;

      reconsider S = s, T = t, X = |[a, b]| as Element of ( REAL 2) by EUCLID: 22;

      reconsider w1 = ((( - (2 * |((t - s), (s - |[a, b]|))|)) + ( sqrt ( delta (( Sum ( sqr (T - S))),(2 * |((t - s), (s - |[a, b]|))|),(( Sum ( sqr (S - X))) - (r ^2 )))))) / (2 * ( Sum ( sqr (T - S))))) as Real;

      s <> t

      proof

        assume s = t;

        then (u . 1) = (v . 1) & (u . 2) = (v . 2) & (u . 3) = (v . 3) by A4, A9, FINSEQ_1: 77;

        then (u `1 ) = (v . 1) & (u `2 ) = (v . 2) & (u `3 ) = (v . 3) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

        then

         A11: (u `1 ) = (v `1 ) & (u `2 ) = (v `2 ) & (u `3 ) = (v `3 ) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

        u = |[(u `1 ), (u `2 ), (u `3 )]| by EUCLID_5: 3

        .= v by A11, EUCLID_5: 3;

        hence contradiction by A4, A9, A1;

      end;

      then

      consider e1 be Point of ( TOP-REAL 2) such that

       A12: ( {e1} = (( halfline (s,t)) /\ ( circle (a,b,r))) & e1 = (((1 - w1) * s) + (w1 * t))) by A4, TOPREAL9: 58;

      reconsider w2 = ((( - (2 * |((s - t), (t - |[a, b]|))|)) + ( sqrt ( delta (( Sum ( sqr (S - T))),(2 * |((s - t), (t - |[a, b]|))|),(( Sum ( sqr (T - X))) - (r ^2 )))))) / (2 * ( Sum ( sqr (S - T))))) as Real;

      reconsider f = |[(e1 `1 ), (e1 `2 ), 1]| as Element of ( TOP-REAL 3);

      f is non zero by FINSEQ_1: 78, EUCLID_5: 4;

      then

      reconsider ee1 = f as non zero Element of ( TOP-REAL 3);

       |[(s `1 ), (s `2 )]| = |[(u . 1), (u . 2)]| & |[(t `1 ), (t `2 )]| = |[(v . 1), (v . 2)]| by EUCLID: 53;

      then (s `1 ) = (u . 1) & (s `2 ) = (u . 2) & (t `1 ) = (v . 1) & (t `2 ) = (v . 2) by FINSEQ_1: 77;

      then

       A13: (s . 1) = (u . 1) & (s . 2) = (u . 2) & (t . 1) = (v . 1) & (t . 2) = (v . 2) by EUCLID:def 9, EUCLID:def 10;

      reconsider P1 = ( Dir ee1) as Point of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

      (((1 * ee1) + (( - (1 - w1)) * u)) + (( - w1) * v)) = ( 0. ( TOP-REAL 3))

      proof

        

         A14: (1 * ee1) = |[(1 * (ee1 `1 )), (1 * (ee1 `2 )), (1 * (ee1 `3 ))]| by EUCLID_5: 7

        .= ee1 by EUCLID_5: 3;

        ee1 = (((1 - w1) * u) + (w1 * v))

        proof

          

           A15: (((1 - w1) * s) + (w1 * t)) = |[((((1 - w1) * s) + (w1 * t)) `1 ), ((((1 - w1) * s) + (w1 * t)) `2 )]| by EUCLID: 53;

          (((1 - w1) * s) + (w1 * t)) = |[((((1 - w1) * s) `1 ) + ((w1 * t) `1 )), ((((1 - w1) * s) `2 ) + ((w1 * t) `2 ))]| by EUCLID: 55

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) `1 )), ((((1 - w1) * s) `2 ) + ((w1 * t) `2 ))]| by EUCLID:def 9

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) . 1)), ((((1 - w1) * s) `2 ) + ((w1 * t) `2 ))]| by EUCLID:def 9

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) . 1)), ((((1 - w1) * s) . 2) + ((w1 * t) `2 ))]| by EUCLID:def 10

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) . 1)), ((((1 - w1) * s) . 2) + ((w1 * t) . 2))]| by EUCLID:def 10

          .= |[(((1 - w1) * (s . 1)) + ((w1 * t) . 1)), ((((1 - w1) * s) . 2) + ((w1 * t) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w1) * (s . 1)) + (w1 * (t . 1))), ((((1 - w1) * s) . 2) + ((w1 * t) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w1) * (s . 1)) + (w1 * (t . 1))), (((1 - w1) * (s . 2)) + ((w1 * t) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w1) * (u . 1)) + (w1 * (v . 1))), (((1 - w1) * (u . 2)) + (w1 * (v . 2)))]| by A13, RVSUM_1: 44;

          then

           A16: (e1 `1 ) = (((1 - w1) * (u . 1)) + (w1 * (v . 1))) & (e1 `2 ) = (((1 - w1) * (u . 2)) + (w1 * (v . 2))) by A12, A15, FINSEQ_1: 77;

          (((1 - w1) * u) + (w1 * v)) = |[(((1 - w1) * (u . 1)) + (w1 * (v . 1))), (((1 - w1) * (u . 2)) + (w1 * (v . 2))), (((1 - w1) * (u . 3)) + (w1 * (v . 3)))]|

          proof

            (((1 - w1) * u) `1 ) = ((1 - w1) * (u `1 )) by EUCLID_5: 9

            .= ((1 - w1) * (u . 1)) by EUCLID_5:def 1;

            

            then

             A17: ((((1 - w1) * u) `1 ) + ((w1 * v) `1 )) = (((1 - w1) * (u . 1)) + ((w1 * v) . 1)) by EUCLID_5:def 1

            .= (((1 - w1) * (u . 1)) + (w1 * (v . 1))) by RVSUM_1: 44;

            (((1 - w1) * u) `2 ) = ((1 - w1) * (u `2 )) by EUCLID_5: 9

            .= ((1 - w1) * (u . 2)) by EUCLID_5:def 2;

            

            then

             A18: ((((1 - w1) * u) `2 ) + ((w1 * v) `2 )) = (((1 - w1) * (u . 2)) + ((w1 * v) . 2)) by EUCLID_5:def 2

            .= (((1 - w1) * (u . 2)) + (w1 * (v . 2))) by RVSUM_1: 44;

            (((1 - w1) * u) `3 ) = ((1 - w1) * (u `3 )) by EUCLID_5: 9

            .= ((1 - w1) * (u . 3)) by EUCLID_5:def 3;

            

            then ((((1 - w1) * u) `3 ) + ((w1 * v) `3 )) = (((1 - w1) * (u . 3)) + ((w1 * v) . 3)) by EUCLID_5:def 3

            .= (((1 - w1) * (u . 3)) + (w1 * (v . 3))) by RVSUM_1: 44;

            hence thesis by A17, A18, EUCLID_5: 5;

          end;

          hence thesis by A16, A4, A9;

        end;

        

        then ((ee1 + (( - (1 - w1)) * u)) + (( - w1) * v)) = ((((1 - w1) * u) + (w1 * v)) + ((( - (1 - w1)) * u) + (( - w1) * v))) by RVSUM_1: 15

        .= (((1 - w1) * u) + ((w1 * v) + ((( - (1 - w1)) * u) + (( - w1) * v)))) by RVSUM_1: 15

        .= (((1 - w1) * u) + ((( - (1 - w1)) * u) + ((w1 * v) + (( - w1) * v)))) by RVSUM_1: 15

        .= ((((1 - w1) * u) + (( - (1 - w1)) * u)) + ((w1 * v) + (( - w1) * v))) by RVSUM_1: 15

        .= (( 0. ( TOP-REAL 3)) + ((w1 * v) + (( - w1) * v))) by BKMODEL1: 4

        .= ( |[ 0 , 0 , 0 ]| + |[ 0 , 0 , 0 ]|) by BKMODEL1: 4, EUCLID_5: 4

        .= |[( 0 + 0 ), ( 0 + 0 ), ( 0 + 0 )]| by EUCLID_5: 6

        .= ( 0. ( TOP-REAL 3)) by EUCLID_5: 4;

        hence thesis by A14;

      end;

      then

       A19: (P1,P,R) are_collinear by A4, A9, ANPROJ_8: 11;

      e1 in {e1} by TARSKI:def 1;

      then

       A20: e1 in ( circle ( 0 , 0 ,1)) by A12, XBOOLE_0:def 4;

      now

        

         A21: (ee1 `1 ) = (e1 `1 ) & (ee1 `2 ) = (e1 `2 ) & (ee1 `3 ) = 1 by EUCLID_5: 2;

        then (ee1 . 1) = (e1 `1 ) & (ee1 . 2) = (e1 `2 ) by EUCLID_5:def 1, EUCLID_5:def 2;

        hence |[(ee1 . 1), (ee1 . 2)]| in ( circle ( 0 , 0 ,1)) by A20, EUCLID: 53;

        thus (ee1 . 3) = 1 by A21, EUCLID_5:def 3;

      end;

      then

       A22: P1 is Element of absolute by BKMODEL1: 86;

      

       A23: (P,R,P1) are_collinear by COLLSP: 8, A19;

      (P,Q,R) are_collinear by A2, COLLSP: 11;

      then (P,R,Q) are_collinear by ANPROJ_8: 57, HESSENBE: 1;

      hence thesis by A22, A23, A1, HESSENBE: 2, ANPROJ_8: 57;

    end;

    theorem :: BKMODEL2:20

    

     Th12: for P,Q be Element of BK_model st P <> Q holds ex P1,P2 be Element of absolute st P1 <> P2 & (P,Q,P1) are_collinear & (P,Q,P2) are_collinear

    proof

      let P,Q be Element of BK_model ;

      assume

       A1: P <> Q;

      consider u be non zero Element of ( TOP-REAL 3) such that

       A2: ( Dir u) = P & (u . 3) = 1 & ( BK_to_REAL2 P) = |[(u . 1), (u . 2)]| by Def01;

      consider v be non zero Element of ( TOP-REAL 3) such that

       A3: ( Dir v) = Q & (v . 3) = 1 & ( BK_to_REAL2 Q) = |[(v . 1), (v . 2)]| by Def01;

      reconsider s = |[(u . 1), (u . 2)]|, t = |[(v . 1), (v . 2)]| as Point of ( TOP-REAL 2);

      set a = 0 , b = 0 , r = 1;

      reconsider S = s, T = t, X = |[a, b]| as Element of ( REAL 2) by EUCLID: 22;

      reconsider w1 = ((( - (2 * |((t - s), (s - |[a, b]|))|)) + ( sqrt ( delta (( Sum ( sqr (T - S))),(2 * |((t - s), (s - |[a, b]|))|),(( Sum ( sqr (S - X))) - (r ^2 )))))) / (2 * ( Sum ( sqr (T - S))))) as Real;

      consider e1 be Point of ( TOP-REAL 2) such that

       A4: ( {e1} = (( halfline (s,t)) /\ ( circle (a,b,r))) & e1 = (((1 - w1) * s) + (w1 * t))) by Th02, A1, A2, A3, TOPREAL9: 58;

      reconsider w2 = ((( - (2 * |((s - t), (t - |[a, b]|))|)) + ( sqrt ( delta (( Sum ( sqr (S - T))),(2 * |((s - t), (t - |[a, b]|))|),(( Sum ( sqr (T - X))) - (r ^2 )))))) / (2 * ( Sum ( sqr (S - T))))) as Real;

      consider e2 be Point of ( TOP-REAL 2) such that

       A5: ( {e2} = (( halfline (t,s)) /\ ( circle (a,b,r))) & e2 = (((1 - w2) * t) + (w2 * s))) by Th02, A1, A2, A3, TOPREAL9: 58;

      reconsider f = |[(e1 `1 ), (e1 `2 ), 1]| as Element of ( TOP-REAL 3);

      f is non zero by FINSEQ_1: 78, EUCLID_5: 4;

      then

      reconsider ee1 = f as non zero Element of ( TOP-REAL 3);

       |[(s `1 ), (s `2 )]| = |[(u . 1), (u . 2)]| & |[(t `1 ), (t `2 )]| = |[(v . 1), (v . 2)]| by EUCLID: 53;

      then (s `1 ) = (u . 1) & (s `2 ) = (u . 2) & (t `1 ) = (v . 1) & (t `2 ) = (v . 2) by FINSEQ_1: 77;

      then

       A6: (s . 1) = (u . 1) & (s . 2) = (u . 2) & (t . 1) = (v . 1) & (t . 2) = (v . 2) by EUCLID:def 9, EUCLID:def 10;

      reconsider P1 = ( Dir ee1) as Point of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

      (((1 * ee1) + (( - (1 - w1)) * u)) + (( - w1) * v)) = ( 0. ( TOP-REAL 3))

      proof

        

         A7: (1 * ee1) = |[(1 * (ee1 `1 )), (1 * (ee1 `2 )), (1 * (ee1 `3 ))]| by EUCLID_5: 7

        .= ee1 by EUCLID_5: 3;

        ee1 = (((1 - w1) * u) + (w1 * v))

        proof

          

           A8: (((1 - w1) * s) + (w1 * t)) = |[((((1 - w1) * s) + (w1 * t)) `1 ), ((((1 - w1) * s) + (w1 * t)) `2 )]| by EUCLID: 53;

          (((1 - w1) * s) + (w1 * t)) = |[((((1 - w1) * s) `1 ) + ((w1 * t) `1 )), ((((1 - w1) * s) `2 ) + ((w1 * t) `2 ))]| by EUCLID: 55

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) `1 )), ((((1 - w1) * s) `2 ) + ((w1 * t) `2 ))]| by EUCLID:def 9

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) . 1)), ((((1 - w1) * s) `2 ) + ((w1 * t) `2 ))]| by EUCLID:def 9

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) . 1)), ((((1 - w1) * s) . 2) + ((w1 * t) `2 ))]| by EUCLID:def 10

          .= |[((((1 - w1) * s) . 1) + ((w1 * t) . 1)), ((((1 - w1) * s) . 2) + ((w1 * t) . 2))]| by EUCLID:def 10

          .= |[(((1 - w1) * (s . 1)) + ((w1 * t) . 1)), ((((1 - w1) * s) . 2) + ((w1 * t) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w1) * (s . 1)) + (w1 * (t . 1))), ((((1 - w1) * s) . 2) + ((w1 * t) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w1) * (s . 1)) + (w1 * (t . 1))), (((1 - w1) * (s . 2)) + ((w1 * t) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w1) * (u . 1)) + (w1 * (v . 1))), (((1 - w1) * (u . 2)) + (w1 * (v . 2)))]| by A6, RVSUM_1: 44;

          then

           A9: (e1 `1 ) = (((1 - w1) * (u . 1)) + (w1 * (v . 1))) & (e1 `2 ) = (((1 - w1) * (u . 2)) + (w1 * (v . 2))) by A4, A8, FINSEQ_1: 77;

          (((1 - w1) * u) + (w1 * v)) = |[(((1 - w1) * (u . 1)) + (w1 * (v . 1))), (((1 - w1) * (u . 2)) + (w1 * (v . 2))), (((1 - w1) * (u . 3)) + (w1 * (v . 3)))]|

          proof

            (((1 - w1) * u) `1 ) = ((1 - w1) * (u `1 )) by EUCLID_5: 9

            .= ((1 - w1) * (u . 1)) by EUCLID_5:def 1;

            

            then

             A10: ((((1 - w1) * u) `1 ) + ((w1 * v) `1 )) = (((1 - w1) * (u . 1)) + ((w1 * v) . 1)) by EUCLID_5:def 1

            .= (((1 - w1) * (u . 1)) + (w1 * (v . 1))) by RVSUM_1: 44;

            (((1 - w1) * u) `2 ) = ((1 - w1) * (u `2 )) by EUCLID_5: 9

            .= ((1 - w1) * (u . 2)) by EUCLID_5:def 2;

            

            then

             A11: ((((1 - w1) * u) `2 ) + ((w1 * v) `2 )) = (((1 - w1) * (u . 2)) + ((w1 * v) . 2)) by EUCLID_5:def 2

            .= (((1 - w1) * (u . 2)) + (w1 * (v . 2))) by RVSUM_1: 44;

            (((1 - w1) * u) `3 ) = ((1 - w1) * (u `3 )) by EUCLID_5: 9

            .= ((1 - w1) * (u . 3)) by EUCLID_5:def 3;

            

            then ((((1 - w1) * u) `3 ) + ((w1 * v) `3 )) = (((1 - w1) * (u . 3)) + ((w1 * v) . 3)) by EUCLID_5:def 3

            .= (((1 - w1) * (u . 3)) + (w1 * (v . 3))) by RVSUM_1: 44;

            hence thesis by A10, A11, EUCLID_5: 5;

          end;

          hence thesis by A9, A2, A3;

        end;

        

        then ((ee1 + (( - (1 - w1)) * u)) + (( - w1) * v)) = ((((1 - w1) * u) + (w1 * v)) + ((( - (1 - w1)) * u) + (( - w1) * v))) by RVSUM_1: 15

        .= (((1 - w1) * u) + ((w1 * v) + ((( - (1 - w1)) * u) + (( - w1) * v)))) by RVSUM_1: 15

        .= (((1 - w1) * u) + ((( - (1 - w1)) * u) + ((w1 * v) + (( - w1) * v)))) by RVSUM_1: 15

        .= ((((1 - w1) * u) + (( - (1 - w1)) * u)) + ((w1 * v) + (( - w1) * v))) by RVSUM_1: 15

        .= (( 0. ( TOP-REAL 3)) + ((w1 * v) + (( - w1) * v))) by BKMODEL1: 4

        .= ( |[ 0 , 0 , 0 ]| + |[ 0 , 0 , 0 ]|) by BKMODEL1: 4, EUCLID_5: 4

        .= |[( 0 + 0 ), ( 0 + 0 ), ( 0 + 0 )]| by EUCLID_5: 6

        .= ( 0. ( TOP-REAL 3)) by EUCLID_5: 4;

        hence thesis by A7;

      end;

      then

       A12: (P1,P,Q) are_collinear by A2, A3, ANPROJ_8: 11;

      e1 in {e1} by TARSKI:def 1;

      then

       A13: e1 in ( circle ( 0 , 0 ,1)) by A4, XBOOLE_0:def 4;

      now

        

         A14: (ee1 `1 ) = (e1 `1 ) & (ee1 `2 ) = (e1 `2 ) & (ee1 `3 ) = 1 by EUCLID_5: 2;

        then (ee1 . 1) = (e1 `1 ) & (ee1 . 2) = (e1 `2 ) by EUCLID_5:def 1, EUCLID_5:def 2;

        hence |[(ee1 . 1), (ee1 . 2)]| in ( circle ( 0 , 0 ,1)) by A13, EUCLID: 53;

        thus (ee1 . 3) = 1 by A14, EUCLID_5:def 3;

      end;

      then

       A15: P1 is Element of absolute by BKMODEL1: 86;

      reconsider g = |[(e2 `1 ), (e2 `2 ), 1]| as Element of ( TOP-REAL 3);

      g is non zero by EUCLID_5: 4, FINSEQ_1: 78;

      then

      reconsider ee2 = g as non zero Element of ( TOP-REAL 3);

      reconsider P2 = ( Dir ee2) as Point of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

      (((1 * ee2) + (( - (1 - w2)) * v)) + (( - w2) * u)) = ( 0. ( TOP-REAL 3))

      proof

        

         A16: (1 * ee2) = |[(1 * (ee2 `1 )), (1 * (ee2 `2 )), (1 * (ee2 `3 ))]| by EUCLID_5: 7

        .= ee2 by EUCLID_5: 3;

        ee2 = (((1 - w2) * v) + (w2 * u))

        proof

          

           A17: (((1 - w2) * t) + (w2 * s)) = |[((((1 - w2) * t) + (w2 * s)) `1 ), ((((1 - w2) * t) + (w2 * s)) `2 )]| by EUCLID: 53;

          (((1 - w2) * t) + (w2 * s)) = |[((((1 - w2) * t) `1 ) + ((w2 * s) `1 )), ((((1 - w2) * t) `2 ) + ((w2 * s) `2 ))]| by EUCLID: 55

          .= |[((((1 - w2) * t) . 1) + ((w2 * s) `1 )), ((((1 - w2) * t) `2 ) + ((w2 * s) `2 ))]| by EUCLID:def 9

          .= |[((((1 - w2) * t) . 1) + ((w2 * s) . 1)), ((((1 - w2) * t) `2 ) + ((w2 * s) `2 ))]| by EUCLID:def 9

          .= |[((((1 - w2) * t) . 1) + ((w2 * s) . 1)), ((((1 - w2) * t) . 2) + ((w2 * s) `2 ))]| by EUCLID:def 10

          .= |[((((1 - w2) * t) . 1) + ((w2 * s) . 1)), ((((1 - w2) * t) . 2) + ((w2 * s) . 2))]| by EUCLID:def 10

          .= |[(((1 - w2) * (t . 1)) + ((w2 * s) . 1)), ((((1 - w2) * t) . 2) + ((w2 * s) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w2) * (t . 1)) + (w2 * (s . 1))), ((((1 - w2) * t) . 2) + ((w2 * s) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w2) * (t . 1)) + (w2 * (s . 1))), (((1 - w2) * (t . 2)) + ((w2 * s) . 2))]| by RVSUM_1: 44

          .= |[(((1 - w2) * (v . 1)) + (w2 * (u . 1))), (((1 - w2) * (v . 2)) + (w2 * (u . 2)))]| by A6, RVSUM_1: 44;

          then

           A18: (e2 `1 ) = (((1 - w2) * (v . 1)) + (w2 * (u . 1))) & (e2 `2 ) = (((1 - w2) * (v . 2)) + (w2 * (u . 2))) by A5, A17, FINSEQ_1: 77;

          (((1 - w2) * v) + (w2 * u)) = |[(((1 - w2) * (v . 1)) + (w2 * (u . 1))), (((1 - w2) * (v . 2)) + (w2 * (u . 2))), (((1 - w2) * (v . 3)) + (w2 * (u . 3)))]|

          proof

            (((1 - w2) * v) `1 ) = ((1 - w2) * (v `1 )) by EUCLID_5: 9

            .= ((1 - w2) * (v . 1)) by EUCLID_5:def 1;

            

            then

             A19: ((((1 - w2) * v) `1 ) + ((w2 * u) `1 )) = (((1 - w2) * (v . 1)) + ((w2 * u) . 1)) by EUCLID_5:def 1

            .= (((1 - w2) * (v . 1)) + (w2 * (u . 1))) by RVSUM_1: 44;

            (((1 - w2) * v) `2 ) = ((1 - w2) * (v `2 )) by EUCLID_5: 9

            .= ((1 - w2) * (v . 2)) by EUCLID_5:def 2;

            

            then

             A20: ((((1 - w2) * v) `2 ) + ((w2 * u) `2 )) = (((1 - w2) * (v . 2)) + ((w2 * u) . 2)) by EUCLID_5:def 2

            .= (((1 - w2) * (v . 2)) + (w2 * (u . 2))) by RVSUM_1: 44;

            (((1 - w2) * v) `3 ) = ((1 - w2) * (v `3 )) by EUCLID_5: 9

            .= ((1 - w2) * (v . 3)) by EUCLID_5:def 3;

            

            then ((((1 - w2) * v) `3 ) + ((w2 * u) `3 )) = (((1 - w2) * (v . 3)) + ((w2 * u) . 3)) by EUCLID_5:def 3

            .= (((1 - w2) * (v . 3)) + (w2 * (u . 3))) by RVSUM_1: 44;

            hence thesis by EUCLID_5: 5, A19, A20;

          end;

          hence thesis by A18, A2, A3;

        end;

        

        then ((ee2 + (( - (1 - w2)) * v)) + (( - w2) * u)) = ((((1 - w2) * v) + (w2 * u)) + ((( - (1 - w2)) * v) + (( - w2) * u))) by RVSUM_1: 15

        .= (((1 - w2) * v) + ((w2 * u) + ((( - (1 - w2)) * v) + (( - w2) * u)))) by RVSUM_1: 15

        .= (((1 - w2) * v) + ((( - (1 - w2)) * v) + ((w2 * u) + (( - w2) * u)))) by RVSUM_1: 15

        .= ((((1 - w2) * v) + (( - (1 - w2)) * v)) + ((w2 * u) + (( - w2) * u))) by RVSUM_1: 15

        .= (( 0. ( TOP-REAL 3)) + ((w2 * u) + (( - w2) * u))) by BKMODEL1: 4

        .= (( 0. ( TOP-REAL 3)) + ( 0. ( TOP-REAL 3))) by BKMODEL1: 4

        .= |[( 0 + 0 ), ( 0 + 0 ), ( 0 + 0 )]| by EUCLID_5: 4, EUCLID_5: 6

        .= ( 0. ( TOP-REAL 3)) by EUCLID_5: 4;

        hence thesis by A16;

      end;

      then

       A21: (P2,Q,P) are_collinear by A2, A3, ANPROJ_8: 11;

      e2 in (( halfline (t,s)) /\ ( circle (a,b,r))) by A5, TARSKI:def 1;

      then

       A22: e2 in ( circle ( 0 , 0 ,1)) by XBOOLE_0:def 4;

      now

        

         A23: (ee2 `1 ) = (e2 `1 ) & (ee2 `2 ) = (e2 `2 ) & (ee2 `3 ) = 1 by EUCLID_5: 2;

        then (ee2 . 1) = (e2 `1 ) & (ee2 . 2) = (e2 `2 ) by EUCLID_5:def 1, EUCLID_5:def 2;

        hence |[(ee2 . 1), (ee2 . 2)]| in ( circle ( 0 , 0 ,1)) by A22, EUCLID: 53;

        thus (ee2 . 3) = 1 by A23, EUCLID_5:def 3;

      end;

      then

       A24: P2 is Element of absolute by BKMODEL1: 86;

      

       A25: P1 <> P2

      proof

        assume P1 = P2;

        then are_Prop (ee1,ee2) by ANPROJ_1: 22;

        then

        consider l be Real such that l <> 0 and

         A26: ee1 = (l * ee2) by ANPROJ_1: 1;

         |[(e1 `1 ), (e1 `2 ), 1]| = |[(l * (e2 `1 )), (l * (e2 `2 )), (l * 1)]| by A26, EUCLID_5: 8;

        then

         A27: 1 = (l * 1) & (e1 `1 ) = (l * (e2 `1 )) & (e1 `2 ) = (l * (e2 `2 )) by FINSEQ_1: 78;

        

         A28: e1 = |[(e1 `1 ), (e1 `2 )]| by EUCLID: 53

        .= e2 by A27, EUCLID: 53;

        (1 - (w1 + w2)) <> 0

        proof

          assume

           A29: (1 - (w1 + w2)) = 0 ;

          

           A30: (2 * ( Sum ( sqr (S - T)))) = (2 * ( Sum ( sqr (T - S)))) by BKMODEL1: 6;

          ( Sum ( sqr (S - T))) is non zero

          proof

            assume

             A31: ( Sum ( sqr (S - T))) is zero;

            ( Sum ( sqr (S - T))) = ( |.(S - T).| ^2 ) by TOPREAL9: 5;

            then

             A32: |.(S - T).| = 0 by A31;

            reconsider n = 2 as Nat;

            S = T

            proof

              reconsider Sn = S, Tn = T as Element of (n -tuples_on REAL ) by EUCLID:def 1;

              Sn = ((Sn - Tn) + Tn) by RVSUM_1: 43

              .= (( 0* n) + Tn) by A32, EUCLID: 8

              .= Tn by EUCLID_4: 1;

              hence thesis;

            end;

            then

             A33: (u . 1) = (v . 1) & (u . 2) = (v . 2) & (u . 3) = (v . 3) by A2, A3, FINSEQ_1: 77;

            

             A34: |[(u . 1), (u . 2), (u . 3)]| = |[(u `1 ), (u . 2), (u . 3)]| by EUCLID_5:def 1

            .= |[(u `1 ), (u `2 ), (u . 3)]| by EUCLID_5:def 2

            .= |[(u `1 ), (u `2 ), (u `3 )]| by EUCLID_5:def 3

            .= u by EUCLID_5: 3;

             |[(v . 1), (v . 2), (v . 3)]| = |[(v `1 ), (v . 2), (v . 3)]| by EUCLID_5:def 1

            .= |[(v `1 ), (v `2 ), (v . 3)]| by EUCLID_5:def 2

            .= |[(v `1 ), (v `2 ), (v `3 )]| by EUCLID_5:def 3

            .= v by EUCLID_5: 3;

            hence contradiction by A1, A2, A3, A34, A33;

          end;

          then

          reconsider l = ( Sum ( sqr (S - T))) as non zero Real;

          

           A35: (s - |[a, b]|) = ( |[(s `1 ), (s `2 )]| - |[ 0 , 0 ]|) by EUCLID: 53

          .= |[((s `1 ) - 0 ), ((s `2 ) - 0 )]| by EUCLID: 62

          .= s by EUCLID: 53;

          

           A36: (t - |[a, b]|) = ( |[(t `1 ), (t `2 )]| - |[ 0 , 0 ]|) by EUCLID: 53

          .= |[((t `1 ) - 0 ), ((t `2 ) - 0 )]| by EUCLID: 62

          .= t by EUCLID: 53;

          

           A38: (w1 + w2) = (((( - (2 * |((t - s), s)|)) + ( sqrt ( delta (( Sum ( sqr (T - S))),(2 * |((t - s), s)|),(( Sum ( sqr (S - X))) - (r ^2 )))))) / (2 * l)) + ((( - (2 * |((s - t), t)|)) + ( sqrt ( delta (( Sum ( sqr (S - T))),(2 * |((s - t), t)|),(( Sum ( sqr (T - X))) - (r ^2 )))))) / (2 * l))) by A35, A36, BKMODEL1: 6

          .= (((( - (2 * |((t - s), s)|)) + ( sqrt ( delta (l,(2 * |((t - s), s)|),(( Sum ( sqr S)) - 1))))) / (2 * l)) + ((( - (2 * |((s - t), t)|)) + ( sqrt ( delta (l,(2 * |((s - t), t)|),(( Sum ( sqr T)) - 1))))) / (2 * l))) by A35, A36, BKMODEL1: 6;

          reconsider l2 = ( - (2 * |((t - s), s)|)), l3 = ( - (2 * |((s - t), t)|)), l4 = (( Sum ( sqr S)) - 1), l5 = (( Sum ( sqr T)) - 1) as Real;

          reconsider l6 = ( sqrt ( delta (l,( - l2),l4))), l7 = ( sqrt ( delta (l,( - l3),l5))), l8 = (2 * l) as Real;

           0 <= ( |.(S - T).| ^2 );

          then

           A39: 0 <= l by TOPREAL9: 5;

          ( |[(u . 1), (u . 2)]| - |[ 0 , 0 ]|) = |[((u . 1) - 0 ), ((u . 2) - 0 )]| by EUCLID: 62

          .= s;

          then

           A40: |.S.| < 1 by A2, TOPREAL9: 45;

          ( |[(v . 1), (v . 2)]| - |[ 0 , 0 ]|) = |[((v . 1) - 0 ), ((v . 2) - 0 )]| by EUCLID: 62

          .= t;

          then |.T.| < 1 by A3, TOPREAL9: 45;

          then

           A42: ( |.S.| ^2 ) <= 1 & ( |.T.| ^2 ) <= 1 by A40, XREAL_1: 160;

          then 0 <= ( delta (l,( - l2),l4)) & 0 <= ( delta (l,( - l3),l5)) by BKMODEL1: 18, A30;

          then

           A43: 0 <= l6 & 0 <= l7 by SQUARE_1:def 2;

          

           A44: (l2 + l3) = l8

          proof

            ( |((t - s), s)| + |((s - t), t)|) = (( |(t, s)| - |(s, s)|) + |((s - t), t)|) by EUCLID_2: 24

            .= ((( - |(s, s)|) + |(t, s)|) + ( |(s, t)| - |(t, t)|)) by EUCLID_2: 24

            .= ( - (( |(s, s)| - (2 * |(t, s)|)) + |(t, t)|))

            .= ( - |((s - t), (s - t))|) by EUCLID_2: 31

            .= ( - ( |.(S - T).| ^2 )) by EUCLID_2: 36

            .= ( - ( Sum ( sqr (S - T)))) by TOPREAL9: 5;

            hence thesis;

          end;

          (w1 + w2) = (((l2 / l8) + (l6 / l8)) + ((l3 + l7) / l8)) by A38, XCMPLX_1: 62

          .= (((l2 / l8) + (l6 / l8)) + ((l3 / l8) + (l7 / l8))) by XCMPLX_1: 62

          .= (((l2 / l8) + (l3 / l8)) + ((l6 / l8) + (l7 / l8)))

          .= ((l8 / l8) + ((l6 / l8) + (l7 / l8))) by A44, XCMPLX_1: 62

          .= (1 + ((l6 / l8) + (l7 / l8))) by XCMPLX_1: 60;

          then 0 = ((l6 + l7) / l8) by A29, XCMPLX_1: 62;

          then l6 = 0 & l7 = 0 by A43;

          then

           A45: ( delta (l,( - l2),l4)) = 0 & ( delta (l,( - l3),l5)) = 0 by A42, BKMODEL1: 18, A30, SQUARE_1: 24;

          l4 < 0

          proof

            ( |.S.| * |.S.|) < 1 by A40, XREAL_1: 162;

            then (( |.S.| ^2 ) - 1) < (1 - 1) by XREAL_1: 14;

            hence thesis by TOPREAL9: 5;

          end;

          hence contradiction by A45, A39, BKMODEL1: 5;

        end;

        then

        reconsider w1w2 = (1 - (w1 + w2)) as non zero Real;

        (w1w2 * s) = (w1w2 * t) by A28, A4, A5, BKMODEL1: 70;

        then s = t by EUCLID_4: 8;

        then

         A46: (u . 1) = (v . 1) & (u . 2) = (v . 2) & (u . 3) = (v . 3) by A2, A3, FINSEQ_1: 77;

        

         A47: |[(u . 1), (u . 2), (u . 3)]| = |[(u `1 ), (u . 2), (u . 3)]| by EUCLID_5:def 1

        .= |[(u `1 ), (u `2 ), (u . 3)]| by EUCLID_5:def 2

        .= |[(u `1 ), (u `2 ), (u `3 )]| by EUCLID_5:def 3

        .= u by EUCLID_5: 3;

         |[(v . 1), (v . 2), (v . 3)]| = |[(v `1 ), (v . 2), (v . 3)]| by EUCLID_5:def 1

        .= |[(v `1 ), (v `2 ), (v . 3)]| by EUCLID_5:def 2

        .= |[(v `1 ), (v `2 ), (v `3 )]| by EUCLID_5:def 3

        .= v by EUCLID_5: 3;

        hence contradiction by A1, A2, A3, A47, A46;

      end;

      

       A48: (P,Q,P1) are_collinear by COLLSP: 8, A12;

      (Q,P2,P) are_collinear by A21, COLLSP: 7;

      then (P2,P,Q) are_collinear by COLLSP: 8;

      then (P,Q,P2) are_collinear by COLLSP: 8;

      hence thesis by A15, A24, A25, A48;

    end;

    theorem :: BKMODEL2:21

    

     Th13: for P,Q,R be Element of real_projective_plane holds for u,v,w be non zero Element of ( TOP-REAL 3) holds for a,b,c,d be Real st P in BK_model & Q in absolute & P = ( Dir u) & Q = ( Dir v) & R = ( Dir w) & u = |[a, b, 1]| & v = |[c, d, 1]| & w = |[((a + c) / 2), ((b + d) / 2), 1]| holds R in BK_model & R <> P & (P,R,Q) are_collinear

    proof

      let P,Q,R be Element of real_projective_plane ;

      let u,v,w be non zero Element of ( TOP-REAL 3);

      let a,b,c,d be Real;

      assume that

       A1: P in BK_model and

       A2: Q in absolute and

       A3: P = ( Dir u) and

       A4: Q = ( Dir v) and

       A5: R = ( Dir w) and

       A6: u = |[a, b, 1]| and

       A7: v = |[c, d, 1]| and

       A8: w = |[((a + c) / 2), ((b + d) / 2), 1]|;

      reconsider PBK = P as Element of BK_model by A1;

      consider u2 be non zero Element of ( TOP-REAL 3) such that

       A9: ( Dir u2) = PBK & (u2 . 3) = 1 & ( BK_to_REAL2 PBK) = |[(u2 . 1), (u2 . 2)]| by Def01;

      

       A10: (u . 3) = (u `3 ) by EUCLID_5:def 3

      .= 1 by A6, EUCLID_5: 2;

      then

       A11: u = u2 by A3, A9, BKMODEL1: 43;

      reconsider S = |[(u . 1), (u . 2)]| as Element of ( TOP-REAL 2);

      

       A12: |.(S - |[ 0 , 0 ]|).| = |.( |[(S `1 ), (S `2 )]| - |[ 0 , 0 ]|).| by EUCLID: 53

      .= |. |[((S `1 ) - 0 ), ((S `2 ) - 0 )]|.| by EUCLID: 62

      .= |.S.| by EUCLID: 53;

      (1 ^2 ) = 1;

      then ( |.S.| ^2 ) < 1 by A9, A11, TOPREAL9: 45, A12, SQUARE_1: 16;

      then (((S `1 ) ^2 ) + ((S `2 ) ^2 )) < 1 by JGRAPH_3: 1;

      then (((u . 1) ^2 ) + ((S `2 ) ^2 )) < 1 by EUCLID: 52;

      then

       A13: (((u . 1) ^2 ) + ((u . 2) ^2 )) < 1 by EUCLID: 52;

      (u `1 ) = a & (u `2 ) = b & (v `1 ) = c & (v `2 ) = d by A6, A7, EUCLID_5: 2;

      then

       A14: (u . 1) = a & (u . 2) = b & (v . 1) = c & (v . 2) = d by EUCLID_5:def 1, EUCLID_5:def 2;

      (v `3 ) = 1 by A7, EUCLID_5: 2;

      then (v . 3) = 1 by EUCLID_5:def 3;

      then |[(v . 1), (v . 2)]| in ( circle ( 0 , 0 ,1)) by A2, A4, BKMODEL1: 84;

      then

      consider pp be Point of ( TOP-REAL 2) such that

       A15: |[(v . 1), (v . 2)]| = pp and

       A16: |.(pp - |[ 0 , 0 ]|).| = 1;

      1 = |. |[((v . 1) - 0 ), ((v . 2) - 0 )]|.| by A15, A16, EUCLID: 62

      .= |.pp.| by A15;

      

      then

       a17: (1 ^2 ) = (((pp `1 ) ^2 ) + ((pp `2 ) ^2 )) by JGRAPH_1: 29

      .= (((v . 1) ^2 ) + ((pp `2 ) ^2 )) by A15, EUCLID: 52

      .= (((v . 1) ^2 ) + ((v . 2) ^2 )) by A15, EUCLID: 52;

      (w `1 ) = ((a + c) / 2) & (w `2 ) = ((b + d) / 2) by A8, EUCLID_5: 2;

      then

       A18: (w . 1) = ((a + c) / 2) & (w . 2) = ((b + d) / 2) by EUCLID_5:def 1, EUCLID_5:def 2;

      reconsider R1 = |[(w . 1), (w . 2)]| as Element of ( TOP-REAL 2);

      ( |.(R1 - |[ 0 , 0 ]|).| ^2 ) < 1

      proof

        

         A19: (R1 `1 ) = (w . 1) & (R1 `2 ) = (w . 2) by EUCLID: 52;

        ( |.(R1 - |[ 0 , 0 ]|).| ^2 ) = ( |. |[((w . 1) - 0 ), ((w . 2) - 0 )]|.| ^2 ) by EUCLID: 62

        .= (((w . 1) ^2 ) + ((w . 2) ^2 )) by A19, JGRAPH_1: 29;

        hence thesis by A18, BKMODEL1: 17, A13, a17, A14;

      end;

      then |.(R1 - |[ 0 , 0 ]|).| < 1 by SQUARE_1: 52;

      then R1 in ( inside_of_circle ( 0 , 0 ,1));

      then

      reconsider R1 as Element of ( inside_of_circle ( 0 , 0 ,1));

      consider PR1 be Element of ( TOP-REAL 2) such that

       A20: PR1 = R1 and

       A21: ( REAL2_to_BK R1) = ( Dir |[(PR1 `1 ), (PR1 `2 ), 1]|) by Def02;

      

       A22: (w . 1) = (w `1 ) & (w . 2) = (w `2 ) & (w `3 ) = 1 by A8, EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5: 2;

      (PR1 `1 ) = (w . 1) & (PR1 `2 ) = (w . 2) by A20, EUCLID: 52;

      then

       A23: ( REAL2_to_BK R1) = ( Dir w) by A21, A22, EUCLID_5: 3;

      

       A24: P <> R

      proof

        assume

         A25: P = R;

        (w . 3) = (w `3 ) by EUCLID_5:def 3

        .= 1 by A8, EUCLID_5: 2;

        then

         A26: (u `1 ) = (w `1 ) & (u `2 ) = (w `2 ) by A25, A3, A5, A10, BKMODEL1: 43;

        (u `1 ) = a & (w `1 ) = ((a + c) / 2) & (u `2 ) = b & (w `2 ) = ((b + d) / 2) by A6, A8, EUCLID_5: 2;

        hence contradiction by A26, A6, A7, A3, A4, A1, A2, Th01, XBOOLE_0:def 4;

      end;

       0 = |{u, v, w}| by A6, A7, A8, BKMODEL1: 20

      .= ( - |{u, w, v}|) by ANPROJ_8: 29;

      hence thesis by A23, A24, A3, A4, A5, BKMODEL1: 1;

    end;

    theorem :: BKMODEL2:22

    

     Th14: for P,Q be Element of real_projective_plane st P in absolute & Q in BK_model holds ex R be Element of real_projective_plane st R in BK_model & Q <> R & (R,Q,P) are_collinear

    proof

      let P,Q be Element of real_projective_plane ;

      assume that

       A1: P in absolute and

       A2: Q in BK_model ;

      reconsider QBK = Q as Element of BK_model by A2;

      consider u be non zero Element of ( TOP-REAL 3) such that (((u . 1) ^2 ) + ((u . 2) ^2 )) = 1 and

       A3: (u . 3) = 1 and

       A4: P = ( Dir u) by A1, BKMODEL1: 89;

      consider v be non zero Element of ( TOP-REAL 3) such that

       A5: ( Dir v) = QBK & (v . 3) = 1 & ( BK_to_REAL2 QBK) = |[(v . 1), (v . 2)]| by Def01;

       |[(((v . 1) + (u . 1)) / 2), (((v . 2) + (u . 2)) / 2), 1]| is non zero by EUCLID_5: 4, FINSEQ_1: 78;

      then

      reconsider w = |[(((v . 1) + (u . 1)) / 2), (((v . 2) + (u . 2)) / 2), 1]| as non zero Element of ( TOP-REAL 3);

      reconsider R = ( Dir w) as Element of real_projective_plane by ANPROJ_1: 26;

      take R;

      now

        u = |[(u `1 ), (u `2 ), (u `3 )]| & v = |[(v `1 ), (v `2 ), (v `3 )]| by EUCLID_5: 3;

        then u = |[(u . 1), (u `2 ), (u `3 )]| & v = |[(v . 1), (v `2 ), (v `3 )]| by EUCLID_5:def 1;

        then u = |[(u . 1), (u . 2), (u `3 )]| & v = |[(v . 1), (v . 2), (v `3 )]| by EUCLID_5:def 2;

        hence u = |[(u . 1), (u . 2), 1]| & v = |[(v . 1), (v . 2), 1]| by EUCLID_5:def 3, A3, A5;

      end;

      then R in BK_model & R <> Q & (Q,R,P) are_collinear by A1, A4, A5, Th13;

      hence thesis by COLLSP: 4;

    end;

    theorem :: BKMODEL2:23

    

     Th15: for L be LINE of ( IncProjSp_of real_projective_plane ) holds for p,q be POINT of ( IncProjSp_of real_projective_plane ) holds for P,Q be Element of real_projective_plane st p = P & q = Q & P in BK_model & Q in absolute & q on L & p on L holds ex p1,p2 be POINT of ( IncProjSp_of real_projective_plane ), P1,P2 be Element of real_projective_plane st p1 = P1 & p2 = P2 & P1 <> P2 & P1 in absolute & P2 in absolute & p1 on L & p2 on L

    proof

      let L be LINE of ( IncProjSp_of real_projective_plane );

      let p,q be POINT of ( IncProjSp_of real_projective_plane );

      let P,Q be Element of real_projective_plane ;

      assume that

       A1: p = P and

       A2: q = Q and

       A3: P in BK_model and

       A4: Q in absolute and

       A5: q on L and

       A6: p on L;

      

       A7: P <> Q by Th01, A3, A4, XBOOLE_0:def 4;

      reconsider l = L as LINE of real_projective_plane by INCPROJ: 4;

      

       A8: P in l by A1, A6, INCPROJ: 5;

      reconsider PBK = P as Element of BK_model by A3;

      consider R be Element of real_projective_plane such that

       A9: R in BK_model and

       A10: P <> R and

       A11: (R,P,Q) are_collinear by A3, A4, Th14;

      reconsider r = R as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 3;

      consider LL be LINE of ( IncProjSp_of real_projective_plane ) such that

       A12: r on LL & p on LL & q on LL by A1, A2, A11, INCPROJ: 10;

      L = LL by A1, A2, A5, A6, A12, A7, INCPROJ: 8;

      then R in l by A12, INCPROJ: 5;

      then

       A13: l = ( Line (P,R)) by A8, A10, COLLSP: 19;

      reconsider RBK = R as Element of BK_model by A9;

      consider P1,P2 be Element of absolute such that

       A14: P1 <> P2 and

       A15: (PBK,RBK,P1) are_collinear and

       A16: (PBK,RBK,P2) are_collinear by A10, Th12;

      reconsider PP1 = P1, PP2 = P2 as Element of real_projective_plane ;

      

       A17: PP1 in ( Line (P,R)) & PP2 in ( Line (P,R)) by A15, A16, COLLSP: 11;

      reconsider p1 = P1, p2 = P2 as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 3;

      p1 on L & p2 on L by A13, A17, INCPROJ: 5;

      hence thesis by A14;

    end;

    theorem :: BKMODEL2:24

    

     Th16: for P be Element of BK_model holds for Q be Element of absolute holds ex R be Element of absolute st Q <> R & (Q,P,R) are_collinear

    proof

      let P be Element of BK_model ;

      let Q be Element of absolute ;

      

       A1: P <> Q by XBOOLE_0:def 4, Th01;

      reconsider p9 = P, q9 = Q as Element of real_projective_plane ;

      reconsider L9 = ( Line (p9,q9)) as LINE of real_projective_plane by A1, COLLSP:def 7;

      reconsider L = L9 as LINE of ( IncProjSp_of real_projective_plane ) by INCPROJ: 4;

      reconsider p = P, q = Q as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 3;

      p9 in L9 & q9 in L9 by COLLSP: 10;

      then p on L & q on L by INCPROJ: 5;

      then

      consider p1,p2 be POINT of ( IncProjSp_of real_projective_plane ), P1,P2 be Element of real_projective_plane such that

       A2: p1 = P1 & p2 = P2 & P1 <> P2 & P1 in absolute & P2 in absolute & p1 on L & p2 on L by Th15;

      reconsider p1, p2 as Element of real_projective_plane by INCPROJ: 3;

      

       A3: P1 in L9 & P2 in L9 by A2, INCPROJ: 5;

      then

       A4: (p9,q9,p1) are_collinear & (p9,q9,p2) are_collinear & P1 <> P2 & P1 in absolute & P2 in absolute by A2, COLLSP: 11;

      reconsider P1, P2 as Element of absolute by A2;

      per cases ;

        suppose

         A5: q9 = p1;

        take P2;

        now

          thus Q <> P2 by A5, A2;

          (P,Q,P2) are_collinear by A3, COLLSP: 11;

          hence (Q,P,P2) are_collinear by COLLSP: 4;

        end;

        hence thesis;

      end;

        suppose q9 <> p1;

        per cases ;

          suppose

           A6: Q <> P2;

          take P2;

          (P,Q,P2) are_collinear by A3, COLLSP: 11;

          hence thesis by A6, COLLSP: 4;

        end;

          suppose

           A7: Q = P2;

          take P1;

          thus thesis by A4, A7, A2, COLLSP: 4;

        end;

      end;

    end;

    theorem :: BKMODEL2:25

    

     Th17: for P be Element of BK_model holds for u be non zero Element of ( TOP-REAL 3) st P = ( Dir u) & (u . 3) = 1 holds (((u . 1) ^2 ) + ((u . 2) ^2 )) < 1

    proof

      let P be Element of BK_model ;

      let u be non zero Element of ( TOP-REAL 3);

      assume that

       A1: P = ( Dir u) and

       A2: (u . 3) = 1;

      consider u2 be non zero Element of ( TOP-REAL 3) such that

       A3: ( Dir u2) = P & (u2 . 3) = 1 & ( BK_to_REAL2 P) = |[(u2 . 1), (u2 . 2)]| by Def01;

      

       A4: u = u2 by A1, A2, A3, BKMODEL1: 43;

      reconsider S = |[(u . 1), (u . 2)]| as Element of ( TOP-REAL 2);

      

       A5: |.(S - |[ 0 , 0 ]|).| = |.( |[(S `1 ), (S `2 )]| - |[ 0 , 0 ]|).| by EUCLID: 53

      .= |. |[((S `1 ) - 0 ), ((S `2 ) - 0 )]|.| by EUCLID: 62

      .= |.S.| by EUCLID: 53;

      (1 ^2 ) = 1;

      then ( |.S.| ^2 ) < 1 by A4, A3, TOPREAL9: 45, A5, SQUARE_1: 16;

      then (((S `1 ) ^2 ) + ((S `2 ) ^2 )) < 1 by JGRAPH_3: 1;

      then (((u . 1) ^2 ) + ((S `2 ) ^2 )) < 1 by EUCLID: 52;

      hence thesis by EUCLID: 52;

    end;

    theorem :: BKMODEL2:26

    

     Th18: for P1,P2 be Element of absolute holds for Q be Element of BK_model holds for u,v,w be non zero Element of ( TOP-REAL 3) st ( Dir u) = P1 & ( Dir v) = P2 & ( Dir w) = Q & (u . 3) = 1 & (v . 3) = 1 & (w . 3) = 1 & (v . 1) = ( - (u . 1)) & (v . 2) = ( - (u . 2)) & (P1,Q,P2) are_collinear holds ex a be Real st ( - 1) < a < 1 & (w . 1) = (a * (u . 1)) & (w . 2) = (a * (u . 2))

    proof

      let P1,P2 be Element of absolute ;

      let Q be Element of BK_model ;

      let u,v,w be non zero Element of ( TOP-REAL 3);

      assume that

       A1: ( Dir u) = P1 & ( Dir v) = P2 & ( Dir w) = Q and

       A2: (u . 3) = 1 & (v . 3) = 1 & (w . 3) = 1 and

       A3: (v . 1) = ( - (u . 1)) & (v . 2) = ( - (u . 2)) and

       A4: (P1,Q,P2) are_collinear ;

      (u . 1) = (u `1 ) & (u . 2) = (u `2 ) by EUCLID_5:def 1, EUCLID_5:def 2;

      then

       A6: (u `3 ) = 1 & (v `3 ) = 1 & (w `3 ) = 1 & (v `1 ) = ( - (u `1 )) & (v `2 ) = ( - (u `2 )) by A2, A3, EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

      (P1,P2,Q) are_collinear by A4, COLLSP: 4;

      

      then

       A7: 0 = |{u, v, w}| by A1, BKMODEL1: 1

      .= ((((((((u `1 ) * ( - (u `2 ))) * 1) - ((1 * ( - (u `2 ))) * (w `1 ))) - (((u `1 ) * 1) * (w `2 ))) + (((u `2 ) * 1) * (w `1 ))) - (((u `2 ) * ( - (u `1 ))) * 1)) + ((1 * ( - (u `1 ))) * (w `2 ))) by A6, ANPROJ_8: 27

      .= (2 * (((u `2 ) * (w `1 )) - ((u `1 ) * (w `2 ))));

      consider u9 be non zero Element of ( TOP-REAL 3) such that

       A8: (((u9 . 1) ^2 ) + ((u9 . 2) ^2 )) = 1 and

       A9: (u9 . 3) = 1 and

       A10: P1 = ( Dir u9) by BKMODEL1: 89;

      

       A11: u = u9 by A9, A10, A1, A2, BKMODEL1: 43;

       not ((u `1 ) = 0 & (u `2 ) = 0 )

      proof

        assume (u `1 ) = 0 & (u `2 ) = 0 ;

        then (u . 1) = 0 & (u . 2) = 0 by EUCLID_5:def 1, EUCLID_5:def 2;

        hence contradiction by A11, A8;

      end;

      then

      consider e be Real such that

       A13: (w `1 ) = (e * (u `1 )) & (w `2 ) = (e * (u `2 )) by A7, BKMODEL1: 2;

      (w . 1) = (e * (u `1 )) & (w . 2) = (e * (u `2 )) by A13, EUCLID_5:def 1, EUCLID_5:def 2;

      then

       A14: (w . 1) = (e * (u . 1)) & (w . 2) = (e * (u . 2)) by EUCLID_5:def 1, EUCLID_5:def 2;

      per cases ;

        suppose e = 0 ;

        hence thesis by A14;

      end;

        suppose e <> 0 ;

        (((w . 1) ^2 ) + ((w . 2) ^2 )) = (((w `1 ) * (w . 1)) + ((w . 2) * (w . 2))) by EUCLID_5:def 1

        .= (((w `1 ) * (w `1 )) + ((w . 2) * (w . 2))) by EUCLID_5:def 1

        .= (((w `1 ) * (w `1 )) + ((w `2 ) * (w . 2))) by EUCLID_5:def 2

        .= (((w `1 ) * (w `1 )) + ((w `2 ) * (w `2 ))) by EUCLID_5:def 2

        .= ((e * e) * (((u `1 ) * (u `1 )) + ((u `2 ) * (u `2 )))) by A13

        .= ((e * e) * (((u . 1) * (u `1 )) + ((u `2 ) * (u `2 )))) by EUCLID_5:def 1

        .= ((e * e) * (((u . 1) * (u . 1)) + ((u `2 ) * (u `2 )))) by EUCLID_5:def 1

        .= ((e * e) * (((u . 1) * (u . 1)) + ((u . 2) * (u `2 )))) by EUCLID_5:def 2

        .= ((e * e) * (((u . 1) * (u . 1)) + ((u . 2) * (u . 2)))) by EUCLID_5:def 2

        .= (e * e) by A8, A11;

        then (e ^2 ) < 1 by A1, A2, Th17;

        then ( - 1) < e < 1 by SQUARE_1: 52;

        hence thesis by A14;

      end;

    end;

    begin

    definition

      let P be Element of absolute ;

      :: BKMODEL2:def4

      func pole_infty P -> Element of real_projective_plane means

      : Def03: ex u be non zero Element of ( TOP-REAL 3) st P = ( Dir u) & (u . 3) = 1 & (((u . 1) ^2 ) + ((u . 2) ^2 )) = 1 & it = ( Dir |[( - (u . 2)), (u . 1), 0 ]|);

      existence

      proof

        consider u be non zero Element of ( TOP-REAL 3) such that

         A1: (((u . 1) ^2 ) + ((u . 2) ^2 )) = 1 and

         A2: (u . 3) = 1 and

         A3: P = ( Dir u) by BKMODEL1: 89;

        ( Dir |[( - (u . 2)), (u . 1), 0 ]|) is Element of real_projective_plane by A1, BKMODEL1: 91, ANPROJ_1: 26;

        hence thesis by A1, A2, A3;

      end;

      uniqueness

      proof

        let P1,P2 be Element of real_projective_plane such that

         A4: ex u be non zero Element of ( TOP-REAL 3) st P = ( Dir u) & (u . 3) = 1 & (((u . 1) ^2 ) + ((u . 2) ^2 )) = 1 & P1 = ( Dir |[( - (u . 2)), (u . 1), 0 ]|) and

         A5: ex u be non zero Element of ( TOP-REAL 3) st P = ( Dir u) & (u . 3) = 1 & (((u . 1) ^2 ) + ((u . 2) ^2 )) = 1 & P2 = ( Dir |[( - (u . 2)), (u . 1), 0 ]|);

        consider u1 be non zero Element of ( TOP-REAL 3) such that

         A6: P = ( Dir u1) & (u1 . 3) = 1 & (((u1 . 1) ^2 ) + ((u1 . 2) ^2 )) = 1 & P1 = ( Dir |[( - (u1 . 2)), (u1 . 1), 0 ]|) by A4;

        consider u2 be non zero Element of ( TOP-REAL 3) such that

         A7: P = ( Dir u2) & (u2 . 3) = 1 & (((u2 . 1) ^2 ) + ((u2 . 2) ^2 )) = 1 & P2 = ( Dir |[( - (u2 . 2)), (u2 . 1), 0 ]|) by A5;

        

         A8: |[( - (u2 . 2)), (u2 . 1), 0 ]| is non zero by A7, BKMODEL1: 91;

        

         A9: |[( - (u1 . 2)), (u1 . 1), 0 ]| is non zero by A6, BKMODEL1: 91;

         are_Prop (u1,u2) by A6, A7, ANPROJ_1: 22;

        then

        consider a be Real such that

         A10: a <> 0 and

         A11: u2 = (a * u1) by ANPROJ_1: 1;

        

         A12: ( - (u2 . 2)) = ( - (a * (u1 . 2))) by A11, RVSUM_1: 44

        .= (a * ( - (u1 . 2)));

         |[( - (u2 . 2)), (u2 . 1), 0 ]| = |[(a * ( - (u1 . 2))), (a * (u1 . 1)), (a * 0 )]| by A11, RVSUM_1: 44, A12

        .= (a * |[( - (u1 . 2)), (u1 . 1), 0 ]|) by EUCLID_5: 8;

        then are_Prop ( |[( - (u2 . 2)), (u2 . 1), 0 ]|, |[( - (u1 . 2)), (u1 . 1), 0 ]|) by A10, ANPROJ_1: 1;

        hence thesis by A8, A9, A6, A7, ANPROJ_1: 22;

      end;

    end

    theorem :: BKMODEL2:27

    

     Th19: for P be Element of absolute holds P <> ( pole_infty P)

    proof

      let P be Element of absolute ;

      assume

       A1: P = ( pole_infty P);

      consider u be non zero Element of ( TOP-REAL 3) such that

       A2: P = ( Dir u) & (u . 3) = 1 & (((u . 1) ^2 ) + ((u . 2) ^2 )) = 1 & ( pole_infty P) = ( Dir |[( - (u . 2)), (u . 1), 0 ]|) by Def03;

      

       A3: |[( - (u . 2)), (u . 1), 0 ]| is non zero by A2, BKMODEL1: 91;

       are_Prop (u, |[( - (u . 2)), (u . 1), 0 ]|) by A1, A2, A3, ANPROJ_1: 22;

      then

      consider a be Real such that a <> 0 and

       A4: u = (a * |[( - (u . 2)), (u . 1), 0 ]|) by ANPROJ_1: 1;

      1 = (a * ( |[( - (u . 2)), (u . 1), 0 ]| . 3)) by A2, A4, RVSUM_1: 44

      .= (a * ( |[( - (u . 2)), (u . 1), 0 ]| `3 )) by EUCLID_5:def 3

      .= (a * 0 ) by EUCLID_5: 2

      .= 0 ;

      hence contradiction;

    end;

    theorem :: BKMODEL2:28

    

     Th20: for P1,P2 be Element of absolute st ( pole_infty P1) = ( pole_infty P2) holds P1 = P2 or (ex u be non zero Element of ( TOP-REAL 3) st P1 = ( Dir u) & P2 = ( Dir |[( - (u `1 )), ( - (u `2 )), 1]|) & (u `3 ) = 1)

    proof

      let P1,P2 be Element of absolute ;

      assume

       A1: ( pole_infty P1) = ( pole_infty P2);

      consider u1 be non zero Element of ( TOP-REAL 3) such that

       A2: P1 = ( Dir u1) & (u1 . 3) = 1 & (((u1 . 1) ^2 ) + ((u1 . 2) ^2 )) = 1 & ( pole_infty P1) = ( Dir |[( - (u1 . 2)), (u1 . 1), 0 ]|) by Def03;

      consider u2 be non zero Element of ( TOP-REAL 3) such that

       A3: P2 = ( Dir u2) & (u2 . 3) = 1 & (((u2 . 1) ^2 ) + ((u2 . 2) ^2 )) = 1 & ( pole_infty P2) = ( Dir |[( - (u2 . 2)), (u2 . 1), 0 ]|) by Def03;

      reconsider w1 = |[( - (u1 . 2)), (u1 . 1), 0 ]| as non zero Element of ( TOP-REAL 3) by A2, BKMODEL1: 91;

      reconsider w2 = |[( - (u2 . 2)), (u2 . 1), 0 ]| as non zero Element of ( TOP-REAL 3) by A3, BKMODEL1: 91;

       are_Prop (w1,w2) by A1, A2, A3, ANPROJ_1: 22;

      then

      consider a be Real such that a <> 0 and

       A5: w1 = (a * w2) by ANPROJ_1: 1;

      (a * w2) = |[(a * ( - (u2 . 2))), (a * (u2 . 1)), (a * 0 )]| by EUCLID_5: 8;

      then

       A6: ( - (u1 . 2)) = (a * ( - (u2 . 2))) & (u1 . 1) = (a * (u2 . 1)) by A5, FINSEQ_1: 78;

      

      then

       A7: 1 = (((a * (u2 . 1)) * (a * (u2 . 1))) + ((a * (u2 . 2)) ^2 )) by A2

      .= ((a * a) * (((u2 . 1) * (u2 . 1)) + ((u2 . 2) * (u2 . 2))))

      .= (a ^2 ) by A3;

      

       A8: a = 1 implies P1 = P2

      proof

        assume a = 1;

        then (u1 `1 ) = (u2 . 1) & (u1 `2 ) = (u2 . 2) & (u1 `3 ) = (u2 . 3) by A2, A3, A6, EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

        then

         A9: (u1 `1 ) = (u2 `1 ) & (u1 `2 ) = (u2 `2 ) & (u1 `3 ) = (u2 `3 ) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

        u1 = |[(u1 `1 ), (u1 `2 ), (u1 `3 )]| by EUCLID_5: 3

        .= u2 by A9, EUCLID_5: 3;

        hence thesis by A2, A3;

      end;

      a = ( - 1) implies ex u be non zero Element of ( TOP-REAL 3) st P1 = ( Dir u) & P2 = ( Dir |[( - (u `1 )), ( - (u `2 )), 1]|) & (u `3 ) = 1

      proof

        assume a = ( - 1);

        then (u1 `1 ) = ( - (u2 . 1)) & (u1 `2 ) = ( - (u2 . 2)) & (u2 `3 ) = 1 by A3, A6, EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

        then

         A10: (u1 `1 ) = ( - (u2 `1 )) & (u1 `2 ) = ( - (u2 `2 )) & (u2 `3 ) = 1 by EUCLID_5:def 1, EUCLID_5:def 2;

        take u1;

        thus thesis by A10, EUCLID_5: 3, A3, A2, EUCLID_5:def 3;

      end;

      hence thesis by A7, A8, SQUARE_1: 41;

    end;

    definition

      let P be Element of absolute ;

      :: BKMODEL2:def5

      func tangent P -> LINE of real_projective_plane means

      : Def04: ex p be Element of real_projective_plane st p = P & it = ( Line (p,( pole_infty P)));

      existence

      proof

        reconsider p = P as Element of real_projective_plane ;

        reconsider L = ( Line (p,( pole_infty P))) as LINE of real_projective_plane by Th19, COLLSP:def 7;

        take L;

        thus thesis;

      end;

      uniqueness ;

    end

    theorem :: BKMODEL2:29

    

     Th21: for P be Element of absolute holds P in ( tangent P)

    proof

      let P be Element of absolute ;

      ex p be Element of real_projective_plane st p = P & ( tangent P) = ( Line (p,( pole_infty P))) by Def04;

      hence thesis by COLLSP: 10;

    end;

    theorem :: BKMODEL2:30

    

     Th22: for P be 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 be 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, EUCLID_5:def 2, EUCLID_5: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, EUCLID_5:def 2, EUCLID_5: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 ;

              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;

                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, EUCLID_5:def 2, EUCLID_5: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;

                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, EUCLID_5:def 2, EUCLID_5: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, EUCLID_5:def 2, EUCLID_5:def 3;

                  then (w `1 ) = (v `1 ) & (w `2 ) = (v `2 ) & (w `3 ) = (v `3 ) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5: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, EUCLID_5:def 2, EUCLID_5:def 3;

                  then (w `1 ) = (v `1 ) & (w `2 ) = (v `2 ) & (w `3 ) = (v `3 ) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5: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;

    theorem :: BKMODEL2:31

    

     Th23: for P1,P2 be Element of absolute st ( tangent P1) = ( tangent P2) holds P1 = P2

    proof

      let P1,P2 be Element of absolute ;

      assume

       A1: ( tangent P1) = ( tangent P2);

      ( absolute /\ ( tangent P1)) = {P1} & ( absolute /\ ( tangent P2)) = {P2} by Th22;

      hence thesis by A1, ZFMISC_1: 3;

    end;

    theorem :: BKMODEL2:32

    

     Th24: for P,Q be Element of absolute holds ex R be Element of real_projective_plane st R in ( tangent P) & R in ( tangent Q)

    proof

      let P,Q be Element of absolute ;

      reconsider pP = ( tangent P), pQ = ( tangent Q) as LINE of ( IncProjSp_of real_projective_plane ) by INCPROJ: 4;

      consider R be POINT of ( IncProjSp_of real_projective_plane ) such that

       A1: R on pP and

       A2: R on pQ by BKMODEL1: 75, INCPROJ:def 9;

      reconsider S = R as Element of real_projective_plane by INCPROJ: 3;

      S in ( tangent P) & S in ( tangent Q) by A1, A2, INCPROJ: 5;

      hence thesis;

    end;

    theorem :: BKMODEL2:33

    

     Th25: for P1,P2 be Element of absolute st P1 <> P2 holds ex P be Element of real_projective_plane st (( tangent P1) /\ ( tangent P2)) = {P}

    proof

      let P1,P2 be Element of absolute ;

      assume P1 <> P2;

      then ( tangent P1) <> ( tangent P2) by Th23;

      hence thesis by COLLSP: 21, BKMODEL1: 76;

    end;

    theorem :: BKMODEL2:34

    

     Th26: for M be Matrix of 3, REAL holds for P be Element of absolute holds for Q be Element of real_projective_plane holds for u,v be non zero Element of ( TOP-REAL 3) holds for fp,fq be FinSequence of REAL st M = ( symmetric_3 (1,1,( - 1), 0 , 0 , 0 )) & P = ( Dir u) & Q = ( Dir v) & u = fp & v = fq & Q in ( tangent P) holds ( SumAll ( QuadraticForm (fq,M,fp))) = 0

    proof

      let M be Matrix of 3, REAL ;

      let P be Element of absolute ;

      let Q be Element of real_projective_plane ;

      let u,v be non zero Element of ( TOP-REAL 3);

      let fp,fq be FinSequence of REAL ;

      assume that

       A1: M = ( symmetric_3 (1,1,( - 1), 0 , 0 , 0 )) and

       A2: P = ( Dir u) and

       A3: Q = ( Dir v) and

       A4: u = fp and

       A5: v = fq and

       A6: Q in ( tangent P);

      consider p be Element of real_projective_plane such that

       A7: p = P and

       A8: ( tangent P) = ( Line (p,( pole_infty P))) by Def04;

      

       A9: (p,( pole_infty P),Q) are_collinear by A6, A8, COLLSP: 11;

      consider w be non zero Element of ( TOP-REAL 3) such that

       A10: P = ( Dir w) & (w . 3) = 1 & (((w . 1) ^2 ) + ((w . 2) ^2 )) = 1 & ( pole_infty P) = ( Dir |[( - (w . 2)), (w . 1), 0 ]|) by Def03;

       are_Prop (w,u) by A2, A10, ANPROJ_1: 22;

      then

      consider aa be Real such that

       A11: aa <> 0 and

       A12: w = (aa * u) by ANPROJ_1: 1;

      

       A13: (w . 1) = (aa * (u `1 )) & (w . 2) = (aa * (u `2 )) & (w . 3) = (aa * (u `3 ))

      proof

        

        thus (w . 1) = (aa * (u . 1)) by A12, RVSUM_1: 44

        .= (aa * (u `1 )) by EUCLID_5:def 1;

        

        thus (w . 2) = (aa * (u . 2)) by A12, RVSUM_1: 44

        .= (aa * (u `2 )) by EUCLID_5:def 2;

        

        thus (w . 3) = (aa * (u . 3)) by A12, RVSUM_1: 44

        .= (aa * (u `3 )) by EUCLID_5:def 3;

      end;

      then (1 * (w . 1)) = (aa * (u `1 )) & (1 * (w . 2)) = (aa * (u `2 )) & 1 = (aa * (u `3 )) by A10;

      then

       A14: ((aa * (1 / aa)) * (w . 1)) = (aa * (u `1 )) & ((aa * (1 / aa)) * (w . 2)) = (aa * (u `2 )) & ((aa * (1 / aa)) * 1) = (aa * (u `3 )) by A11, XCMPLX_1: 106;

      

       A16: 1 = ((aa ^2 ) * (((u `1 ) * (u `1 )) + ((u `2 ) * (u `2 )))) by A13, A10;

      

       A17: ( |[( - (w . 2)), (w . 1), 0 ]| `1 ) = ( - (aa * (u `2 ))) & ( |[( - (w . 2)), (w . 1), 0 ]| `2 ) = (aa * (u `1 )) & ( |[( - (w . 2)), (w . 1), 0 ]| `3 ) = 0 by A13, EUCLID_5: 2;

       |[( - (w . 2)), (w . 1), 0 ]| is non zero by BKMODEL1: 91, A10;

      

      then 0 = |{u, |[( - (w . 2)), (w . 1), 0 ]|, v}| by A7, A10, A2, A3, A9, BKMODEL1: 1

      .= ((((((((u `1 ) * ( |[( - (w . 2)), (w . 1), 0 ]| `2 )) * (v `3 )) - (((u `3 ) * ( |[( - (w . 2)), (w . 1), 0 ]| `2 )) * (v `1 ))) - (((u `1 ) * ( |[( - (w . 2)), (w . 1), 0 ]| `3 )) * (v `2 ))) + (((u `2 ) * ( |[( - (w . 2)), (w . 1), 0 ]| `3 )) * (v `1 ))) - (((u `2 ) * ( |[( - (w . 2)), (w . 1), 0 ]| `1 )) * (v `3 ))) + (((u `3 ) * ( |[( - (w . 2)), (w . 1), 0 ]| `1 )) * (v `2 ))) by ANPROJ_8: 27

      .= (aa * ((((((u `1 ) * (u `1 )) * (v `3 )) - (((u `1 ) * (u `3 )) * (v `1 ))) + (((u `2 ) * (u `2 )) * (v `3 ))) - (((u `2 ) * (u `3 )) * (v `2 )))) by A17;

      

      then 0 = (((v `3 ) * (((u `1 ) * (u `1 )) + ((u `2 ) * (u `2 )))) - ((u `3 ) * (((u `1 ) * (v `1 )) + ((u `2 ) * (v `2 ))))) by A11

      .= (((v `3 ) * (1 / (aa ^2 ))) - ((u `3 ) * (((u `1 ) * (v `1 )) + ((u `2 ) * (v `2 ))))) by A16, XCMPLX_1: 73

      .= (((v `3 ) * (1 / (aa ^2 ))) - ((1 / aa) * (((u `1 ) * (v `1 )) + ((u `2 ) * (v `2 ))))) by A14, A11, XCMPLX_1: 5

      .= (((v `3 ) * ((1 / aa) * (1 / aa))) - ((1 / aa) * (((u `1 ) * (v `1 )) + ((u `2 ) * (v `2 ))))) by XCMPLX_1: 102

      .= ((1 / aa) * (((v `3 ) * (1 / aa)) - (((u `1 ) * (v `1 )) + ((u `2 ) * (v `2 )))));

      then

       A18: (((v `3 ) * (1 / aa)) - (((u `1 ) * (v `1 )) + ((u `2 ) * (v `2 )))) = 0 by A11;

      

       A19: ( len fp) = ( width M) & ( len fq) = ( len M) & ( len fp) = ( len M) & ( len fq) = ( width M) & ( len fp) > 0 & ( len fq) > 0

      proof

        ( len M) = 3 & ( width M) = 3 by MATRIX_0: 24;

        hence thesis by A5, FINSEQ_3: 153, A4;

      end;

      

      then

       A20: ( SumAll ( QuadraticForm (fq,M,fp))) = |((fq * M), fp)| by MATRPROB: 46

      .= |(fq, (M * fp))| by A19, MATRPROB: 47;

      

       A21: (M * fp) = ( Col ((M * ( ColVec2Mx fp)),1)) by MATRIXR1:def 11;

      

       A22: fp is Element of ( REAL 3) by A4, EUCLID: 22;

      then

       A23: ( len fp) = 3 by EUCLID_8: 50;

      reconsider fa = 1, fb = ( - 1), z = 0 as Element of F_Real by XREAL_0:def 1;

      

       A24: M = <* <*fa, z, z*>, <*z, fa, z*>, <*z, z, fb*>*> by A1, PASCAL:def 3;

      reconsider fp1 = (fp . 1), fp2 = (fp . 2), fp3 = (fp . 3) as Element of F_Real by XREAL_0:def 1;

      

       A25: ( ColVec2Mx fp) = ( MXR2MXF ( ColVec2Mx fp)) by MATRIXR1:def 1

      .= ( <*fp*> @ ) by A22, ANPROJ_8: 72

      .= ( F2M fp) by A22, ANPROJ_8: 88, EUCLID_8: 50

      .= <* <*fp1*>, <*fp2*>, <*fp3*>*> by A23, ANPROJ_8:def 1;

      reconsider M1 = M as Matrix of 3, 3, F_Real ;

      reconsider M2 = <* <*fp1*>, <*fp2*>, <*fp3*>*> as Matrix of 3, 1, F_Real by ANPROJ_8: 4;

      

       A26: for n,k,m be Nat holds for A be Matrix of n, k, F_Real holds for B be Matrix of ( width A), m, F_Real holds (A * B) is Matrix of ( len A), ( width B), F_Real ;

      

       A27: ( len M1) = 3 & ( width M2) = 1 by MATRIX_0: 23;

      ( width M1) = 3 by MATRIX_0: 23;

      then

       A28: (M1 * M2) is Matrix of 3, 1, F_Real by A26, A27;

      

       A29: (M * ( ColVec2Mx fp)) = (M1 * M2) by A25, ANPROJ_8: 17;

      (M * ( ColVec2Mx fp)) = (M1 * M2) by A25, ANPROJ_8: 17

      .= <* <*(((fa * fp1) + (z * fp2)) + (z * fp3))*>, <*(((z * fp1) + (fa * fp2)) + (z * fp3))*>, <*(((z * fp1) + (z * fp2)) + (fb * fp3))*>*> by A24, ANPROJ_9: 7;

      

      then ( SumAll ( QuadraticForm (fq,M,fp))) = |(v, |[fp1, fp2, ( - fp3)]|)| by A5, A20, A21, A28, A29, ANPROJ_8: 5

      .= ((((v `1 ) * ( |[fp1, fp2, ( - fp3)]| `1 )) + ((v `2 ) * ( |[fp1, fp2, ( - fp3)]| `2 ))) + ((v `3 ) * ( |[fp1, fp2, ( - fp3)]| `3 ))) by EUCLID_5: 29

      .= ((((v `1 ) * fp1) + ((v `2 ) * ( |[fp1, fp2, ( - fp3)]| `2 ))) + ((v `3 ) * ( |[fp1, fp2, ( - fp3)]| `3 ))) by EUCLID_5: 2

      .= ((((v `1 ) * fp1) + ((v `2 ) * fp2)) + ((v `3 ) * ( |[fp1, fp2, ( - fp3)]| `3 ))) by EUCLID_5: 2

      .= ((((v `1 ) * fp1) + ((v `2 ) * fp2)) + ((v `3 ) * ( - fp3))) by EUCLID_5: 2

      .= ((((v `1 ) * (u . 1)) + ((v `2 ) * (u . 2))) - ((v `3 ) * (u . 3))) by A4

      .= ((((v `1 ) * (u `1 )) + ((v `2 ) * (u . 2))) - ((v `3 ) * (u . 3))) by EUCLID_5:def 1

      .= ((((v `1 ) * (u `1 )) + ((v `2 ) * (u `2 ))) - ((v `3 ) * (u . 3))) by EUCLID_5:def 2

      .= ((((v `1 ) * (u `1 )) + ((v `2 ) * (u `2 ))) - ((v `3 ) * (u `3 ))) by EUCLID_5:def 3

      .= 0 by A18, A14, A11, XCMPLX_1: 5;

      hence thesis;

    end;

    theorem :: BKMODEL2:35

    

     Th27: for P,Q,R be Element of absolute holds for P1,P2,P3,P4 be Point of real_projective_plane st (P,Q,R) are_mutually_distinct & P1 = P & P2 = Q & P3 = R & P4 in ( tangent P) & P4 in ( tangent Q) holds not (P1,P2,P3) are_collinear & not (P1,P2,P4) are_collinear & not (P1,P3,P4) are_collinear & not (P2,P3,P4) are_collinear

    proof

      let P,Q,R be Element of absolute ;

      let P1,P2,P3,P4 be Point of real_projective_plane ;

      assume that

       A1: (P,Q,R) are_mutually_distinct and

       A2: P1 = P & P2 = Q & P3 = R and

       A3: P4 in ( tangent P) and

       A4: P4 in ( tangent Q);

      

       A5: not P4 in absolute

      proof

        assume P4 in absolute ;

        then P4 in ( absolute /\ ( tangent P)) & P4 in ( absolute /\ ( tangent Q)) by A3, A4, XBOOLE_0:def 4;

        then P4 in {P} & P4 in {Q} by Th22;

        then P4 = P & P4 = Q by TARSKI:def 1;

        hence contradiction by A1;

      end;

      consider p be Element of real_projective_plane such that

       A6: p = P and

       A7: ( tangent P) = ( Line (p,( pole_infty P))) by Def04;

      

       A8: (p,( pole_infty P),P4) are_collinear by A3, A7, COLLSP: 11;

      

       A9: P4 <> p by A6, A5;

      consider q be Element of real_projective_plane such that

       A10: q = Q and

       A11: ( tangent Q) = ( Line (q,( pole_infty Q))) by Def04;

      

       A12: P4 <> q by A10, A5;

      

       A13: (q,( pole_infty Q),P4) are_collinear by A4, A11, COLLSP: 11;

      thus not (P1,P2,P3) are_collinear by A1, A2, BKMODEL1: 92;

      thus not (P1,P2,P4) are_collinear

      proof

        assume

         A14: (P1,P2,P4) are_collinear ;

        now

          thus P4 <> p by A6, A5;

          thus (P4,p,p) are_collinear by COLLSP: 2;

          (p,P4,( pole_infty P)) are_collinear by A8, COLLSP: 4;

          hence (P4,p,( pole_infty P)) are_collinear by COLLSP: 7;

          (p,P4,q) are_collinear by A14, A2, A6, A10, COLLSP: 4;

          hence (P4,p,q) are_collinear by COLLSP: 4;

        end;

        then Q in ( tangent P) by A10, A7, COLLSP: 3, COLLSP: 11;

        then Q in ( absolute /\ ( tangent P)) by XBOOLE_0:def 4;

        then Q in {P} by Th22;

        hence contradiction by A1, TARSKI:def 1;

      end;

      thus not (P1,P3,P4) are_collinear

      proof

        assume (P1,P3,P4) are_collinear ;

        then

         A15: (p,P4,P3) are_collinear by A2, A6, COLLSP: 4;

        (p,P4,( pole_infty P)) are_collinear by A8, COLLSP: 4;

        then P3 in ( tangent P) by A9, A15, A7, COLLSP: 6, COLLSP: 11;

        then P3 in ( absolute /\ ( tangent P)) by A2, XBOOLE_0:def 4;

        then P3 in {P} by Th22;

        hence contradiction by A1, A2, TARSKI:def 1;

      end;

      thus not (P2,P3,P4) are_collinear

      proof

        assume (P2,P3,P4) are_collinear ;

        then

         A16: (q,P4,P3) are_collinear by A2, A10, COLLSP: 4;

        (q,P4,( pole_infty Q)) are_collinear by A13, COLLSP: 4;

        then P3 in ( tangent Q) by A16, A12, A11, COLLSP: 6, COLLSP: 11;

        then P3 in ( absolute /\ ( tangent Q)) by A2, XBOOLE_0:def 4;

        then P3 in {Q} by Th22;

        hence contradiction by A1, A2, TARSKI:def 1;

      end;

    end;

    theorem :: BKMODEL2:36

    for P,Q be Element of absolute holds for R be Element of real_projective_plane holds for u,v,w be non zero Element of ( TOP-REAL 3) st P = ( Dir u) & Q = ( Dir v) & R = ( Dir w) & R in ( tangent P) & R in ( tangent Q) & (u . 3) = 1 & (v . 3) = 1 & (w . 3) = 0 holds P = Q or ((u . 1) = ( - (v . 1)) & (u . 2) = ( - (v . 2)))

    proof

      let P,Q be Element of absolute ;

      let R be Element of real_projective_plane ;

      let u,v,w be non zero Element of ( TOP-REAL 3);

      assume that

       A1: P = ( Dir u) & Q = ( Dir v) & R = ( Dir w) & R in ( tangent P) & R in ( tangent Q) & (u . 3) = 1 & (v . 3) = 1 & (w . 3) = 0 ;

      assume

       A2: P <> Q;

       |[(u . 1), (u . 2)]| in ( circle ( 0 , 0 ,1)) & |[(v . 1), (v . 2)]| in ( circle ( 0 , 0 ,1)) by A1, BKMODEL1: 84;

      then

       A3: (((u . 1) ^2 ) + ((u . 2) ^2 )) = 1 & (((v . 1) ^2 ) + ((v . 2) ^2 )) = 1 by BKMODEL1: 13;

      reconsider M = ( symmetric_3 (1,1,( - 1), 0 , 0 , 0 )) as Matrix of 3, REAL ;

      reconsider fp = u, fq = v, fr = w as FinSequence of REAL by EUCLID: 24;

      reconsider fr1 = (w `1 ), fr2 = (w `2 ), fr3 = (w `3 ) as Element of REAL by XREAL_0:def 1;

      

       A4: fr = <*fr1, fr2, fr3*> by EUCLID_5: 3;

      

       A5: ( SumAll ( QuadraticForm (fr,M,fp))) = 0 & ( SumAll ( QuadraticForm (fr,M,fq))) = 0 by A1, Th26;

      u is Element of ( REAL 3) & v is Element of ( REAL 3) & w is Element of ( REAL 3) by EUCLID: 22;

      then

       A6: ( len fp) = 3 & ( len fq) = 3 & ( len fr) = 3 by EUCLID_8: 50;

      ( len fr) = ( len M) & ( len fp) = ( width M) & ( len fp) > 0 & ( len fq) = ( width M) & ( len fq) > 0 by A6, MATRIX_0: 24;

      then

       A7: |(fr, (M * fp))| = 0 & |(fr, (M * fq))| = 0 by A5, MATRPROB: 44;

      reconsider m1 = 1, m2 = 0 , m3 = 0 , m4 = 0 , m5 = 1, m6 = 0 , m7 = 0 , m8 = 0 , m9 = ( - 1) as Element of REAL by XREAL_0:def 1;

      

       A8: M = <* <*m1, m2, m3*>, <*m4, m5, m6*>, <*m7, m8, m9*>*> by PASCAL:def 3;

      reconsider fp1 = (u `1 ), fp2 = (u `2 ), fp3 = (u `3 ), fq1 = (v `1 ), fq2 = (v `2 ), fq3 = (v `3 ) as Element of REAL by XREAL_0:def 1;

      

       A9: (u . 1) = fp1 & (u . 2) = fp2 & (v . 1) = fq1 & (v . 2) = fq2 & fp3 = 1 & fq3 = 1 & fr3 = 0 by A1, EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

      

       A10: fp = <*fp1, fp2, fp3*> & fq = <*fq1, fq2, fq3*> & fr = <*fr1, fr2, fr3*> by EUCLID_5: 3;

      

      then (M * fp) = <*(((1 * fp1) + ( 0 * fp2)) + ( 0 * fp3)), ((( 0 * fp1) + (1 * fp2)) + ( 0 * fp3)), ((( 0 * fp1) + ( 0 * fp2)) + (( - 1) * fp3))*> by A8, PASCAL: 9

      .= <*fp1, fp2, ( - fp3)*>;

      then

       A11: (((fr1 * fp1) + (fr2 * fp2)) + (fr3 * ( - fp3))) = 0 by A7, A4, EUCLID_5: 30;

      (M * fq) = <*(((1 * fq1) + ( 0 * fq2)) + ( 0 * fq3)), ((( 0 * fq1) + (1 * fq2)) + ( 0 * fq3)), ((( 0 * fq1) + ( 0 * fq2)) + (( - 1) * fq3))*> by A10, A8, PASCAL: 9

      .= <*fq1, fq2, ( - fq3)*>;

      then

       A12: (((fr1 * fq1) + (fr2 * fq2)) + (fr3 * ( - fq3))) = 0 by A7, A4, EUCLID_5: 30;

      

       A13: fr3 = 0 by A1, EUCLID_5:def 3;

      per cases ;

        suppose

         A14: fr2 = 0 ;

        then fr1 <> 0 by A10, EUCLID_5: 4, A1, EUCLID_5:def 3;

        then

         A15: fp1 = 0 & fq1 = 0 by A14, A13, A11, A12;

        then (fp2 = 1 or fp2 = ( - 1)) & (fq2 = 1 or fq2 = ( - 1)) by SQUARE_1: 41, A9, A3;

        hence thesis by A9, A1, A2, A15, A10;

      end;

        suppose

         A16: fr2 <> 0 ;

        (fq1 * ((fr1 * fp1) + (fr2 * fp2))) = 0 & (fp1 * ((fr1 * fq1) + (fr2 * fq2))) = 0 by A13, A11, A12;

        then (fr2 * ((fq1 * fp2) - (fp1 * fq2))) = 0 ;

        then

         A17: ((fq1 * fp2) - (fp1 * fq2)) = 0 by A16;

        per cases ;

          suppose

           A18: fp2 = 0 ;

          then fp1 = 0 or fq2 = 0 by A17;

          then (fq1 = 1 or fq1 = ( - 1)) & (fp1 = 1 or fp1 = ( - 1)) by SQUARE_1: 41, A3, A9, A18;

          hence thesis by A9, A1, A2, A10, A17;

        end;

          suppose

           A20: fp2 <> 0 ;

          per cases ;

            suppose fp1 = 0 ;

            hence thesis by A9, A11, A16, A20;

          end;

            suppose

             A21: fp1 <> 0 ;

            reconsider l = (fq1 / fp1) as Real;

            

             A22: l = (fq2 / fp2)

            proof

              fq1 = (fq1 * (fp2 / fp2)) by XCMPLX_1: 88, A20

              .= ((fp1 * fq2) / fp2) by A17, XCMPLX_1: 74

              .= (fp1 * (fq2 / fp2)) by XCMPLX_1: 74;

              hence thesis by A21, XCMPLX_1: 89;

            end;

            then

             A23: fq1 = (l * fp1) & fq2 = (l * fp2) by A21, A20, XCMPLX_1: 87;

            (v . 1) = (l * fp1) & (v . 2) = (l * fp2) by A22, A9, A21, A20, XCMPLX_1: 87;

            then l = 1 or l = ( - 1) by A9, A3, BKMODEL1: 3;

            hence thesis by A9, A1, A2, A10, A23;

          end;

        end;

      end;

    end;

    theorem :: BKMODEL2:37

    

     Th28: for P be Element of absolute holds for R be Element of real_projective_plane holds for u be non zero Element of ( TOP-REAL 3) st R in ( tangent P) & R = ( Dir u) & (u . 3) = 0 holds R = ( pole_infty P)

    proof

      let P be Element of absolute ;

      let R be Element of real_projective_plane ;

      let u be non zero Element of ( TOP-REAL 3);

      assume that

       A1: R in ( tangent P) and

       A2: R = ( Dir u) and

       A3: (u . 3) = 0 ;

      consider w be non zero Element of ( TOP-REAL 3) such that

       A4: P = ( Dir w) & (w . 3) = 1 & (((w . 1) ^2 ) + ((w . 2) ^2 )) = 1 & ( pole_infty P) = ( Dir |[( - (w . 2)), (w . 1), 0 ]|) by Def03;

      consider v be non zero Element of ( TOP-REAL 3) such that

       A5: (((v . 1) ^2 ) + ((v . 2) ^2 )) = 1 & (v . 3) = 1 & P = ( Dir v) by BKMODEL1: 89;

      reconsider M = ( symmetric_3 (1,1,( - 1), 0 , 0 , 0 )) as Matrix of 3, REAL ;

      reconsider fp = v, fr = u as FinSequence of REAL by EUCLID: 24;

      reconsider fr1 = (u `1 ), fr2 = (u `2 ), fr3 = (u `3 ) as Element of REAL by XREAL_0:def 1;

      

       A6: fr = <*fr1, fr2, fr3*> by EUCLID_5: 3;

      

       A7: ( SumAll ( QuadraticForm (fr,M,fp))) = 0 by A2, A1, A5, Th26;

      u is Element of ( REAL 3) & v is Element of ( REAL 3) by EUCLID: 22;

      then

       A8: ( len fp) = 3 & ( len fr) = 3 by EUCLID_8: 50;

      ( len M) = 3 & ( width M) = 3 by MATRIX_0: 24;

      then

       A9: |(fr, (M * fp))| = 0 by A7, A8, MATRPROB: 44;

      reconsider m1 = 1, m2 = 0 , m3 = 0 , m4 = 0 , m5 = 1, m6 = 0 , m7 = 0 , m8 = 0 , m9 = ( - 1) as Element of REAL by XREAL_0:def 1;

      

       A10: M = <* <*m1, m2, m3*>, <*m4, m5, m6*>, <*m7, m8, m9*>*> by PASCAL:def 3;

      reconsider fp1 = (v `1 ), fp2 = (v `2 ), fp3 = (v `3 ) as Element of REAL by XREAL_0:def 1;

      

       A11: (v . 1) = fp1 & (v . 2) = fp2 & fp3 = 1 & fr3 = 0 by A3, A5, EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

      

       A12: fp = <*fp1, fp2, fp3*> & fr = <*fr1, fr2, fr3*> by EUCLID_5: 3;

      

      then (M * fp) = <*(((1 * fp1) + ( 0 * fp2)) + ( 0 * fp3)), ((( 0 * fp1) + (1 * fp2)) + ( 0 * fp3)), ((( 0 * fp1) + ( 0 * fp2)) + (( - 1) * fp3))*> by A10, PASCAL: 9

      .= <*fp1, fp2, ( - fp3)*>;

      then

       A13: (((fr1 * fp1) + (fr2 * fp2)) + (fr3 * ( - fp3))) = 0 by A9, A6, EUCLID_5: 30;

      per cases ;

        suppose

         A14: fr1 = 0 ;

        then

         A15: fr2 <> 0 by A12, EUCLID_5: 4, A3, EUCLID_5:def 3;

        then

         A16: fp2 = 0 by A14, A11, A13;

        

         A17: (w . 1) = (v . 1) by A4, A5, BKMODEL1: 43

        .= fp1 by EUCLID_5:def 1;

        

         A18: (w . 2) = (v . 2) by A4, A5, BKMODEL1: 43

        .= 0 by A15, A14, A11, A13;

        now

          

           A19: fp1 <> 0 by A5, A16, A11;

          thus |[ 0 , fp1, 0 ]| is non zero by A5, A16, A11, FINSEQ_1: 78, EUCLID_5: 4;

          thus are_Prop (u, |[ 0 , fp1, 0 ]|)

          proof

            u = |[((fr2 / fp1) * 0 ), ((fr2 / fp1) * fp1), ((fr2 / fp1) * 0 )]| by A16, A11, A5, XCMPLX_1: 87, A12, A14

            .= ((fr2 / fp1) * |[ 0 , fp1, 0 ]|) by EUCLID_5: 8;

            hence thesis by A15, A19, ANPROJ_1: 1;

          end;

        end;

        hence thesis by A2, ANPROJ_1: 22, A4, A17, A18;

      end;

        suppose

         A20: fr1 <> 0 ;

        

         A21: fp2 <> 0

        proof

          assume

           A22: fp2 = 0 ;

          then fp1 = 0 by A13, A11, A20;

          hence contradiction by A22, A11, A5;

        end;

        

        then

         A23: fr2 = ((fp1 * ( - fr1)) / fp2) by A13, A11, XCMPLX_1: 89

        .= (fp1 * (( - fr1) / fp2)) by XCMPLX_1: 74;

        reconsider l = (( - fr1) / fp2) as non zero Real by A20, A21;

         A24:

        now

          thus |[( - fp2), fp1, 0 ]| is non zero by A21, FINSEQ_1: 78, EUCLID_5: 4;

          fr1 = ( - ( - fr1))

          .= ( - (l * fp2)) by A21, XCMPLX_1: 87;

          

          then fr = |[(l * ( - fp2)), (l * fp1), (l * 0 )]| by A23, A12, A3, EUCLID_5:def 3

          .= (l * |[( - fp2), fp1, 0 ]|) by EUCLID_5: 8;

          hence are_Prop (u, |[( - fp2), fp1, 0 ]|) by ANPROJ_1: 1;

        end;

        w = v by BKMODEL1: 43, A4, A5;

        hence thesis by A24, A2, ANPROJ_1: 22, A4, A11;

      end;

    end;

    theorem :: BKMODEL2:38

    

     Th29: for a be non zero Real holds for N be invertible Matrix of 3, F_Real st N = ( symmetric_3 (a,a,( - a), 0 , 0 , 0 )) holds (( homography N) .: absolute ) = absolute

    proof

      let a be non zero Real;

      let N be invertible Matrix of 3, F_Real ;

      assume

       A1: N = ( symmetric_3 (a,a,( - a), 0 , 0 , 0 ));

      

       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: (( homography N) . y) = x by FUNCT_1:def 6;

        

         A6: ( rng ( homography N)) c= the carrier of ( ProjectiveSpace ( TOP-REAL 3)) by RELAT_1:def 19;

        reconsider y9 = y as Element of ( ProjectiveSpace ( TOP-REAL 3)) by A3;

        consider u9 be non zero Element of ( TOP-REAL 3) such that

         A7: (((u9 . 1) ^2 ) + ((u9 . 2) ^2 )) = 1 & (u9 . 3) = 1 & y = ( Dir u9) by A4, BKMODEL1: 89;

        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

         A8: y9 = ( Dir u) & not u is zero & u = uf & p = (N * uf) & v = ( M2F p) & not v is zero & (( homography N) . y9) = ( Dir v) by ANPROJ_8:def 4;

        reconsider x9 = x as Element of ( ProjectiveSpace ( TOP-REAL 3)) by A5, A3, A6, FUNCT_1: 3;

        reconsider z1 = 0 , z2 = a, z3 = ( - a) as Element of F_Real by XREAL_0:def 1;

        

         A9: N = <* <*z2, z1, z1*>, <*z1, z2, z1*>, <*z1, z1, z3*>*> by A1, PASCAL:def 3;

        reconsider ux = (u `1 ), uy = (u `2 ), uz = (u `3 ) as Element of F_Real by XREAL_0:def 1;

         <*ux, uy, uz*> = uf by A8, EUCLID_5: 3;

        then

         A10: p = <* <*(((z2 * ux) + (z1 * uy)) + (z1 * uz))*>, <*(((z1 * ux) + (z2 * uy)) + (z1 * uz))*>, <*(((z1 * ux) + (z1 * uy)) + (z3 * uz))*>*> & v = <*(((z2 * ux) + (z1 * uy)) + (z1 * uz)), (((z1 * ux) + (z2 * uy)) + (z1 * uz)), (((z1 * ux) + (z1 * uy)) + (z3 * uz))*> by A8, A9, PASCAL: 8;

         are_Prop (u9,u) by A7, A8, ANPROJ_1: 22;

        then

        consider l be Real such that

         A11: l <> 0 and

         A12: u9 = (l * u) by ANPROJ_1: 1;

        

         A13: (u9 . 1) = (l * (u . 1)) & (u9 . 2) = (l * (u . 2)) & (u9 . 3) = (l * (u . 3)) by A12, RVSUM_1: 44;

        reconsider w = |[( - ((u . 1) * l)), ( - ((u . 2) * l)), ((u . 3) * l)]| as Element of ( TOP-REAL 3);

        

         A15: w is non zero

        proof

          assume w is zero;

          then (u . 1) = 0 & (u . 2) = 0 & (u . 3) = 0 by A11, FINSEQ_1: 78, EUCLID_5: 4;

          then (u `1 ) = 0 & (u `2 ) = 0 & (u `3 ) = 0 by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

          hence contradiction by A8, EUCLID_5: 3, EUCLID_5: 4;

        end;

        

         A16: (( - 1) * (w `1 )) = (( - 1) * ( - ((u . 1) * l))) & (( - 1) * (w `2 )) = (( - 1) * ( - ((u . 2) * l))) & (( - 1) * (w `3 )) = (( - 1) * ((u . 3) * l)) by EUCLID_5: 2;

        then (( - 1) * (w `1 )) = ((1 * (u . 1)) * l) & (( - 1) * (w `2 )) = ((1 * (u . 2)) * l) & (( - 1) * (w `3 )) = ( - ((1 * (u . 3)) * l));

        then

         A17: (( - 1) * (w `1 )) = ((u `1 ) * l) & (( - 1) * (w `2 )) = ((u `2 ) * l) & (( - 1) * (w `3 )) = ( - ((1 * (u `3 )) * l)) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

        now

          thus ( - (a / l)) <> 0 by A11;

          (( - (a / l)) * (w `1 )) = (v `1 ) & (( - (a / l)) * (w `2 )) = (v `2 ) & (( - (a / l)) * (w `3 )) = (v `3 )

          proof

            

            thus (( - (a / l)) * (w `1 )) = ((a / l) * (( - 1) * (w `1 )))

            .= ((a / l) * ((u `1 ) * l)) by A16, EUCLID_5:def 1

            .= (a * (u `1 )) by A11, XCMPLX_1: 90

            .= (v `1 ) by A10, EUCLID_5: 2;

            

            thus (( - (a / l)) * (w `2 )) = ((a / l) * (( - 1) * (w `2 )))

            .= ((a / l) * ((u `2 ) * l)) by A16, EUCLID_5:def 2

            .= (a * (u `2 )) by A11, XCMPLX_1: 90

            .= (v `2 ) by A10, EUCLID_5: 2;

            

            thus (( - (a / l)) * (w `3 )) = ( - ((a / l) * ((u `3 ) * l))) by A17

            .= ( - (a * (u `3 ))) by A11, XCMPLX_1: 90

            .= (v `3 ) by A10, EUCLID_5: 2;

          end;

          

          then |[(v `1 ), (v `2 ), (v `3 )]| = (( - (a / l)) * |[(w `1 ), (w `2 ), (w `3 )]|) by EUCLID_5: 8

          .= (( - (a / l)) * w) by EUCLID_5: 3;

          hence v = (( - (a / l)) * w) by EUCLID_5: 3;

        end;

        then are_Prop (w,v) by ANPROJ_1: 1;

        then

         A18: ( Dir w) = ( Dir v) by ANPROJ_1: 22, A15, A8;

        

         A19: (((w . 1) ^2 ) + ((w . 2) ^2 )) = (1 ^2 ) & (w . 3) = 1

        proof

          

          thus (1 ^2 ) = (((l * (u `1 )) * (l * (u . 1))) + ((l * (u . 2)) * (l * (u . 2)))) by EUCLID_5:def 1, A13, A7

          .= (((l * (u `1 )) * (l * (u `1 ))) + ((l * (u . 2)) * (l * (u . 2)))) by EUCLID_5:def 1

          .= (((l * (u `1 )) * (l * (u `1 ))) + ((l * (u `2 )) * (l * (u . 2)))) by EUCLID_5:def 2

          .= (((( - 1) * (w `1 )) * (( - 1) * (w `1 ))) + ((( - 1) * (w `2 )) * (( - 1) * (w `2 )))) by A17, EUCLID_5:def 2

          .= (((w `1 ) * (w `1 )) + ((w `2 ) * (w `2 )))

          .= (((w . 1) * (w `1 )) + ((w `2 ) * (w `2 ))) by EUCLID_5:def 1

          .= (((w . 1) * (w . 1)) + ((w `2 ) * (w `2 ))) by EUCLID_5:def 1

          .= (((w . 1) * (w . 1)) + ((w . 2) * (w `2 ))) by EUCLID_5:def 2

          .= (((w . 1) ^2 ) + ((w . 2) ^2 )) by EUCLID_5:def 2;

          (w `3 ) = ((u . 3) * l) by EUCLID_5: 2;

          hence (w . 3) = 1 by A13, A7, EUCLID_5:def 3;

        end;

        then |[(w . 1), (w . 2)]| in ( circle ( 0 , 0 ,1)) by BKMODEL1: 14;

        then x9 is Element of absolute by A15, A18, A8, A5, A19, BKMODEL1: 86;

        hence thesis;

      end;

       absolute c= (( homography N) .: absolute )

      proof

        let x be object;

        assume x in absolute ;

        then

        consider u be non zero Element of ( TOP-REAL 3) such that

         A20: (((u . 1) ^2 ) + ((u . 2) ^2 )) = 1 & (u . 3) = 1 & x = ( Dir u) by BKMODEL1: 89;

        reconsider w = |[((u . 1) / a), ((u . 2) / a), ( - ((u . 3) / a))]| as Element of ( TOP-REAL 3);

        

         A21: w is non zero

        proof

          assume w is zero;

          then (u . 1) = 0 & (u . 2) = 0 & (u . 3) = 0 by FINSEQ_1: 78, EUCLID_5: 4;

          then (u `1 ) = 0 & (u `2 ) = 0 & (u `3 ) = 0 by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

          hence contradiction by EUCLID_5: 3, EUCLID_5: 4;

        end;

        then

        reconsider P = ( Dir w) as Element of ( ProjectiveSpace ( TOP-REAL 3)) by ANPROJ_1: 26;

        reconsider v = |[( - (u . 1)), ( - (u . 2)), (u . 3)]| as Element of ( TOP-REAL 3);

        

         A22: (( - a) * ((u . 1) / a)) = ( - (a * ((u . 1) / a)))

        .= ( - (u . 1)) by XCMPLX_1: 87;

        

         A23: (( - a) * ((u . 2) / a)) = ( - (a * ((u . 2) / a)))

        .= ( - (u . 2)) by XCMPLX_1: 87;

        

         A24: (( - a) * ( - ((u . 3) / a))) = (a * ((u . 3) / a))

        .= (u . 3) by XCMPLX_1: 87;

        

         A25: v is non zero by A20, FINSEQ_1: 78, EUCLID_5: 4;

        v = (( - a) * w) by A22, A23, A24, EUCLID_5: 8;

        then are_Prop (v,w) by ANPROJ_1: 1;

        then

         A26: P = ( Dir v) by A21, A25, ANPROJ_1: 22;

        (v `3 ) = (u . 3) by EUCLID_5: 2;

        then

         A27: (v . 3) = 1 by A20, EUCLID_5:def 3;

         |[(v . 1), (v . 2)]| in ( circle ( 0 , 0 ,1))

        proof

          

           A28: (v `1 ) = ( - (u . 1)) & (v `2 ) = ( - (u . 2)) by EUCLID_5: 2;

          

           A29: ((v . 1) ^2 ) = ((v `1 ) * (v . 1)) by EUCLID_5:def 1

          .= (( - (u . 1)) * ( - (u . 1))) by A28, EUCLID_5:def 1

          .= ((u . 1) ^2 );

          

           A30: ((v . 2) ^2 ) = ((v `2 ) * (v . 2)) by EUCLID_5:def 2

          .= (( - (u . 2)) * ( - (u . 2))) by A28, EUCLID_5:def 2

          .= ((u . 2) ^2 );

          (((v . 1) ^2 ) + ((v . 2) ^2 )) = (1 ^2 ) by A20, A29, A30;

          hence thesis by BKMODEL1: 14;

        end;

        then

        reconsider P as Element of absolute by A26, A27, A25, BKMODEL1: 86;

        now

          ( dom ( homography N)) = the carrier of ( ProjectiveSpace ( TOP-REAL 3)) by FUNCT_2:def 1;

          hence P in ( dom ( homography N));

          consider u1,v1 be Element of ( TOP-REAL 3), uf be FinSequence of F_Real , p be FinSequence of (1 -tuples_on REAL ) such that

           A31: P = ( Dir u1) & not u1 is zero & u1 = uf & p = (N * uf) & v1 = ( M2F p) & not v1 is zero & (( homography N) . P) = ( Dir v1) by ANPROJ_8:def 4;

           are_Prop (u1,w) by A21, A31, ANPROJ_1: 22;

          then

          consider l be Real such that

           A32: l <> 0 and

           A33: u1 = (l * w) by ANPROJ_1: 1;

          u1 = |[(l * ((u . 1) / a)), (l * ((u . 2) / a)), (l * ( - ((u . 3) / a)))]| by A33, EUCLID_5: 8;

          then

           A34: (u1 `1 ) = (l * ((u . 1) / a)) & (u1 `2 ) = (l * ((u . 2) / a)) & (u1 `3 ) = (l * ( - ((u . 3) / a))) by EUCLID_5: 2;

          reconsider z1 = 0 , z2 = a, z3 = ( - a) as Element of F_Real by XREAL_0:def 1;

          

           A35: N = <* <*z2, z1, z1*>, <*z1, z2, z1*>, <*z1, z1, z3*>*> by A1, PASCAL:def 3;

          reconsider ux = (u1 `1 ), uy = (u1 `2 ), uz = (u1 `3 ) as Element of F_Real by XREAL_0:def 1;

          

           A36: (a * (l * ((u . 1) / a))) = (l * (a * ((u . 1) / a)))

          .= (l * (u . 1)) by XCMPLX_1: 87;

          

           A37: (a * (l * ((u . 2) / a))) = (l * (a * ((u . 2) / a)))

          .= (l * (u . 2)) by XCMPLX_1: 87;

          

           A38: (( - a) * (l * ( - ((u . 3) / a)))) = (l * (a * ((u . 3) / a)))

          .= (l * (u . 3)) by XCMPLX_1: 87;

           <*ux, uy, uz*> = uf by A31, EUCLID_5: 3;

          then p = <* <*(((z2 * ux) + (z1 * uy)) + (z1 * uz))*>, <*(((z1 * ux) + (z2 * uy)) + (z1 * uz))*>, <*(((z1 * ux) + (z1 * uy)) + (z3 * uz))*>*> & v1 = <*(((z2 * ux) + (z1 * uy)) + (z1 * uz)), (((z1 * ux) + (z2 * uy)) + (z1 * uz)), (((z1 * ux) + (z1 * uy)) + (z3 * uz))*> by A31, A35, PASCAL: 8;

          then (v1 `1 ) = (a * (u1 `1 )) & (v1 `2 ) = (a * (u1 `2 )) & (v1 `3 ) = (( - a) * (u1 `3 )) by EUCLID_5: 2;

          then (v1 `1 ) = (l * (u `1 )) & (v1 `2 ) = (l * (u `2 )) & (v1 `3 ) = (l * (u `3 )) by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3, A34, A36, A37, A38;

          

          then v1 = |[(l * (u `1 )), (l * (u `2 )), (l * (u `3 ))]| by EUCLID_5: 3

          .= (l * |[(u `1 ), (u `2 ), (u `3 )]|) by EUCLID_5: 8

          .= (l * u) by EUCLID_5: 3;

          then are_Prop (u,v1) by A32, ANPROJ_1: 1;

          hence x = (( homography N) . P) by A20, A31, ANPROJ_1: 22;

        end;

        hence thesis by FUNCT_1:def 6;

      end;

      hence thesis by A2;

    end;

    theorem :: BKMODEL2:39

    for ra be non zero Element of F_Real holds for M,O be invertible Matrix of 3, F_Real st O = ( symmetric_3 (1,1,( - 1), 0 , 0 , 0 )) & M = (ra * O) holds (( homography M) .: absolute ) = absolute

    proof

      let ra be non zero Element of F_Real ;

      let M,O be invertible Matrix of 3, F_Real ;

      assume that

       A1: O = ( symmetric_3 (1,1,( - 1), 0 , 0 , 0 )) and

       A2: M = (ra * O);

      reconsider z1 = 1, z2 = ( - 1), z3 = 0 as Element of F_Real by XREAL_0:def 1;

      O = <* <*z1, z3, z3*>, <*z3, z1, z3*>, <*z3, z3, z2*>*> by A1, PASCAL:def 3;

      

      then (ra * O) = <* <*(ra * z1), (ra * z3), (ra * z3)*>, <*(ra * z3), (ra * z1), (ra * z3)*>, <*(ra * z3), (ra * z3), (ra * z2)*>*> by BKMODEL1: 46

      .= <* <*ra, 0 , 0 *>, <* 0 , ra, 0 *>, <* 0 , 0 , ( - ra)*>*>;

      then

       A3: M = ( symmetric_3 (ra,ra,( - ra), 0 , 0 , 0 )) by A2, PASCAL:def 3;

      ra <> 0

      proof

        assume ra = 0 ;

        then ( Det M) = ( 0. F_Real );

        hence contradiction by LAPLACE: 34;

      end;

      hence thesis by A3, Th29;

    end;

    theorem :: BKMODEL2:40

    

     Th30: for P be Element of absolute holds ( tangent P) misses BK_model

    proof

      let P be Element of absolute ;

      assume not ( tangent P) misses BK_model ;

      then

      consider x be object such that

       A1: x in ( tangent P) and

       A2: x in BK_model by XBOOLE_0: 3;

      reconsider x as Element of real_projective_plane by A1;

      reconsider L = ( tangent P) as LINE of ( IncProjSp_of real_projective_plane ) by INCPROJ: 4;

      reconsider ip = P, iq = x as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 3;

      P in ( tangent P) & x in ( tangent P) by A1, Th21;

      then ip on L & iq on L by INCPROJ: 5;

      then

      consider p1,p2 be POINT of ( IncProjSp_of real_projective_plane ), P1,P2 be Element of real_projective_plane such that

       A3: p1 = P1 and

       A4: p2 = P2 and

       A5: P1 <> P2 and

       A6: P1 in absolute and

       A7: P2 in absolute and

       A8: p1 on L and

       A9: p2 on L by A2, Th15;

      P1 in L & P2 in L by A3, A4, A8, A9, INCPROJ: 5;

      then P1 in (( tangent P) /\ absolute ) & P2 in (( tangent P) /\ absolute ) by A6, A7, XBOOLE_0:def 4;

      then P1 in {P} & P2 in {P} by Th22;

      then P1 = P & P2 = P by TARSKI:def 1;

      hence contradiction by A5;

    end;

    theorem :: BKMODEL2:41

    

     Th31: for P,PP1,PP2 be Element of real_projective_plane holds for P1,P2 be Element of absolute holds for Q be Element of real_projective_plane st P1 <> P2 & PP1 = P1 & PP2 = P2 & P in BK_model & (P,PP1,PP2) are_collinear & Q in ( tangent P1) & Q in ( tangent P2) holds ex R be Element of real_projective_plane st R in absolute & (P,Q,R) are_collinear

    proof

      let P,PP1,PP2 be Element of real_projective_plane ;

      let P1,P2 be Element of absolute ;

      let Q be Element of real_projective_plane ;

      assume that

       A1: P1 <> P2 and

       A2: PP1 = P1 and

       A3: PP2 = P2 and

       A4: P in BK_model and

       A5: (P,PP1,PP2) are_collinear and

       A6: Q in ( tangent P1) and

       A7: Q in ( tangent P2);

      

       A8: P <> Q

      proof

        assume P = Q;

        then BK_model meets ( tangent P1) by A4, A6, XBOOLE_0:def 4;

        hence contradiction by Th30;

      end;

      consider u be Element of ( TOP-REAL 3) such that

       A9: u is non zero and

       A10: ( Dir u) = Q by ANPROJ_1: 26;

      per cases ;

        suppose

         A11: (u `3 ) <> 0 ;

        reconsider v = |[((u `1 ) / (u `3 )), ((u `2 ) / (u `3 )), 1]| as non zero Element of ( TOP-REAL 3) by BKMODEL1: 41;

        

         A12: (v . 3) = (v `3 ) by EUCLID_5:def 3

        .= 1 by EUCLID_5: 2;

        

         A13: ((u `3 ) * ((u `1 ) / (u `3 ))) = (u `1 ) & ((u `3 ) * ((u `2 ) / (u `3 ))) = (u `2 ) by A11, XCMPLX_1: 87;

        ((u `3 ) * v) = |[((u `3 ) * ((u `1 ) / (u `3 ))), ((u `3 ) * ((u `2 ) / (u `3 ))), ((u `3 ) * 1)]| by EUCLID_5: 8

        .= u by A13, EUCLID_5: 3;

        then are_Prop (v,u) by A11, ANPROJ_1: 1;

        then

         A14: ( Dir v) = Q & (v . 3) = 1 by A10, A12, A9, ANPROJ_1: 22;

        reconsider PP = P as Element of BK_model by A4;

        reconsider QQ = Q as Element of ( ProjectiveSpace ( TOP-REAL 3));

        consider R be Element of absolute such that

         A15: (PP,QQ,R) are_collinear by A8, A14, Th03;

        reconsider R as Element of real_projective_plane ;

        take R;

        thus thesis by A15;

      end;

        suppose (u `3 ) = 0 ;

        then

         A16: (u . 3) = 0 by EUCLID_5:def 3;

        then Q = ( pole_infty P1) & Q = ( pole_infty P2) by A6, A7, A9, A10, Th28;

        then

        consider up be non zero Element of ( TOP-REAL 3) such that

         A17: P1 = ( Dir up) and

         A18: P2 = ( Dir |[( - (up `1 )), ( - (up `2 )), 1]|) and

         A19: (up `3 ) = 1 by A1, Th20;

        consider up1 be non zero Element of ( TOP-REAL 3) such that

         A20: (((up1 . 1) ^2 ) + ((up1 . 2) ^2 )) = 1 and

         A21: (up1 . 3) = 1 and

         A22: P1 = ( Dir up1) by BKMODEL1: 89;

        (up . 3) = 1 by A19, EUCLID_5:def 3;

        then

         A23: up = up1 by A17, A21, A22, BKMODEL1: 43;

        reconsider PP = P as Element of BK_model by A4;

        consider w be non zero Element of ( TOP-REAL 3) such that

         A24: ( Dir w) = PP and

         A25: (w . 3) = 1 and ( BK_to_REAL2 PP) = |[(w . 1), (w . 2)]| by Def01;

        reconsider up2 = |[( - (up `1 )), ( - (up `2 )), 1]| as non zero Element of ( TOP-REAL 3) by BKMODEL1: 41;

        

         A26: (up2 . 1) = (up2 `1 ) by EUCLID_5:def 1

        .= ( - (up `1 )) by EUCLID_5: 2;

        

         A27: (up2 . 2) = (up2 `2 ) by EUCLID_5:def 2

        .= ( - (up `2 )) by EUCLID_5: 2;

        

         A28: (up2 . 3) = (up2 `3 ) by EUCLID_5:def 3

        .= 1 by EUCLID_5: 2;

        P1 is Element of absolute & P2 is Element of absolute & PP is Element of BK_model & up1 is non zero & up2 is non zero & w is non zero & P1 = ( Dir up1) & P2 = ( Dir up2) & PP = ( Dir w) & (up1 . 3) = 1 & (up2 . 3) = 1 & (w . 3) = 1 & (up2 . 1) = ( - (up1 . 1)) & (up2 . 2) = ( - (up1 . 2)) & (P1,PP,P2) are_collinear by A22, A18, A24, A21, A28, A25, A23, A26, A27, A2, A3, A5, COLLSP: 4, EUCLID_5:def 1, EUCLID_5:def 2;

        then

        consider a be Real such that

         A29: ( - 1) < a < 1 and

         A30: (w . 1) = (a * (up1 . 1)) & (w . 2) = (a * (up1 . 2)) by Th18;

        consider d,e,f be Real such that

         A31: e = (((d * a) * (up1 . 1)) + ((1 - d) * ( - (up1 . 2)))) and

         A32: f = (((d * a) * (up1 . 2)) + ((1 - d) * (up1 . 1))) and

         A33: ((e ^2 ) + (f ^2 )) = (d ^2 ) by A29, A20, BKMODEL1: 16;

        d <> 0 by A20, A31, A32, A33;

        then |[e, f, d]| is non zero by FINSEQ_1: 78, EUCLID_5: 4;

        then

        reconsider ur = |[e, f, d]| as non zero Element of ( TOP-REAL 3);

        reconsider S = ( Dir ur) as Element of real_projective_plane by ANPROJ_1: 26;

        take S;

        

         A35: ( qfconic (1,1,( - 1), 0 , 0 , 0 ,ur)) = 0

        proof

          

           A36: (ur . 1) = (ur `1 ) by EUCLID_5:def 1

          .= e by EUCLID_5: 2;

          

           A37: (ur . 2) = (ur `2 ) by EUCLID_5:def 2

          .= f by EUCLID_5: 2;

          

           A38: (ur . 3) = (ur `3 ) by EUCLID_5:def 3

          .= d by EUCLID_5: 2;

          ( qfconic (1,1,( - 1), 0 , 0 , 0 ,ur)) = (((((((1 * (ur . 1)) * (ur . 1)) + ((1 * (ur . 2)) * (ur . 2))) + ((( - 1) * (ur . 3)) * (ur . 3))) + (( 0 * (ur . 1)) * (ur . 2))) + (( 0 * (ur . 1)) * (ur . 3))) + (( 0 * (ur . 2)) * (ur . 3))) by PASCAL:def 1

          .= (((e ^2 ) + (f ^2 )) - (d ^2 )) by A36, A37, A38;

          hence thesis by A33;

        end;

         |{w, u, ur}| = 0

        proof

          consider u9 be non zero Element of ( TOP-REAL 3) such that

           A39: P1 = ( Dir u9) & (u9 . 3) = 1 & (((u9 . 1) ^2 ) + ((u9 . 2) ^2 )) = 1 & ( pole_infty P1) = ( Dir |[( - (u9 . 2)), (u9 . 1), 0 ]|) by Def03;

          (up . 3) = 1 by A19, EUCLID_5:def 3;

          then

           A40: u9 = up by A39, A17, BKMODEL1: 43;

          

           A41: Q = ( Dir |[( - (up . 2)), (up . 1), 0 ]|) by A39, A40, A16, A6, A9, A10, Th28;

           |[( - (up . 2)), (up . 1), 0 ]| is non zero

          proof

            assume |[( - (up . 2)), (up . 1), 0 ]| is zero;

            then (up1 . 1) = 0 & (up1 . 2) = 0 by A23, FINSEQ_1: 78, EUCLID_5: 4;

            hence contradiction by A20;

          end;

          then are_Prop (u, |[( - (up . 2)), (up . 1), 0 ]|) by A41, A10, A9, ANPROJ_1: 22;

          then

          consider g be Real such that g <> 0 and

           A42: u = (g * |[( - (up . 2)), (up . 1), 0 ]|) by ANPROJ_1: 1;

           |[(u `1 ), (u `2 ), (u `3 )]| = u by EUCLID_5: 3

          .= |[(g * ( - (up . 2))), (g * (up . 1)), (g * 0 )]| by A42, EUCLID_5: 8;

          then

           A43: (u `1 ) = (g * ( - (up . 2))) & (u `2 ) = (g * (up . 1)) & (u `3 ) = (g * 0 ) by FINSEQ_1: 78;

          

           A44: (w `3 ) = 1 by A25, EUCLID_5:def 3;

          

           A45: (w `1 ) = (a * (up1 . 1)) & (w `2 ) = (a * (up1 . 2)) by A30, EUCLID_5:def 1, EUCLID_5:def 2;

          (ur `1 ) = e & (ur `2 ) = f & (ur `3 ) = d by EUCLID_5: 2;

          

          then |{w, u, ur}| = ((((((((a * (up1 . 1)) * (g * (up . 1))) * d) - ((1 * (g * (up . 1))) * (((d * a) * (up1 . 1)) + ((1 - d) * ( - (up1 . 2)))))) - (((a * (up1 . 1)) * 0 ) * (((d * a) * (up1 . 2)) + ((1 - d) * (up1 . 1))))) + (((a * (up1 . 2)) * 0 ) * (((d * a) * (up1 . 1)) + ((1 - d) * ( - (up1 . 2)))))) - (((a * (up1 . 2)) * (g * ( - (up . 2)))) * d)) + ((1 * (g * ( - (up . 2)))) * (((d * a) * (up1 . 2)) + ((1 - d) * (up1 . 1))))) by A43, A44, A45, A31, A32, ANPROJ_8: 27

          .= 0 by A23;

          hence thesis;

        end;

        hence thesis by A35, PASCAL: 11, A9, A24, A10, BKMODEL1: 1;

      end;

    end;

    theorem :: BKMODEL2:42

    

     Th32: for P,R,S be Element of real_projective_plane holds for Q be Element of absolute st P in BK_model & R in ( tangent Q) & (P,S,R) are_collinear & R <> S holds Q <> S

    proof

      let P,R,S be Element of real_projective_plane ;

      let Q be Element of absolute ;

      assume that

       A1: P in BK_model and

       A2: R in ( tangent Q) and

       A3: (P,S,R) are_collinear and

       A4: R <> S;

      

       A5: (S,R,P) are_collinear by A3, COLLSP: 8;

      consider q be Element of real_projective_plane such that

       A6: q = Q & ( tangent Q) = ( Line (q,( pole_infty Q))) by Def04;

      assume Q = S;

      then (q,( pole_infty Q),S) are_collinear & (q,( pole_infty Q),R) are_collinear by A2, Th21, A6, COLLSP: 11;

      then

       A7: P in ( tangent Q) by A5, A4, COLLSP: 9, A6, COLLSP: 11;

      reconsider L = ( tangent Q) as LINE of ( IncProjSp_of real_projective_plane ) by INCPROJ: 4;

      reconsider ip = P, iq = Q as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 3;

      Q in ( tangent Q) by Th21;

      then ip on L & iq on L by A7, INCPROJ: 5;

      then

      consider p1,p2 be POINT of ( IncProjSp_of real_projective_plane ), P1,P2 be Element of real_projective_plane such that

       A8: p1 = P1 & p2 = P2 & P1 <> P2 & P1 in absolute & P2 in absolute & p1 on L & p2 on L by A1, Th15;

      P1 in L & P2 in L by A8, INCPROJ: 5;

      then P1 in (( tangent Q) /\ absolute ) & P2 in (( tangent Q) /\ absolute ) by A8, XBOOLE_0:def 4;

      then P1 in {Q} & P2 in {Q} by Th22;

      then P1 = Q & P2 = Q by TARSKI:def 1;

      hence contradiction by A8;

    end;

    begin

    definition

      let h be Element of EnsHomography3 ;

      :: BKMODEL2:def6

      pred h is_K-isometry means ex N be invertible Matrix of 3, F_Real st h = ( homography N) & (( homography N) .: absolute ) = absolute ;

    end

    theorem :: BKMODEL2:43

    

     Th33: for h be Element of EnsHomography3 st h = ( homography ( 1. ( F_Real ,3))) holds h is_K-isometry

    proof

      let h be Element of EnsHomography3 ;

      assume

       A1: h = ( homography ( 1. ( F_Real ,3)));

      reconsider N = ( 1. ( F_Real ,3)) as invertible Matrix of 3, F_Real ;

      h is_K-isometry

      proof

        

         A2: (( homography N) .: absolute ) c= absolute

        proof

          let x be object;

          assume x in (( homography N) .: absolute );

          then ex y be object st y in ( dom ( homography N)) & y in absolute & (( homography N) . y) = x by FUNCT_1:def 6;

          hence thesis by ANPROJ_9: 14;

        end;

         absolute c= (( homography N) .: absolute )

        proof

          let x be object;

          assume

           A3: x in absolute ;

          then

          reconsider y = x as Point of ( ProjectiveSpace ( TOP-REAL 3));

          

           A4: y = (( homography N) . y) by ANPROJ_9: 14;

          ( dom ( homography N)) = the carrier of ( ProjectiveSpace ( TOP-REAL 3)) by FUNCT_2:def 1;

          hence thesis by A4, A3, FUNCT_1: 108;

        end;

        then absolute = (( homography N) .: absolute ) by A2;

        hence thesis by A1;

      end;

      hence thesis;

    end;

    definition

      :: BKMODEL2:def7

      func EnsK-isometry -> non empty Subset of EnsHomography3 equals { h where h be Element of EnsHomography3 : h is_K-isometry };

      coherence

      proof

        set KI = { h where h be Element of EnsHomography3 : h is_K-isometry };

        KI c= EnsHomography3

        proof

          let x be object;

          assume x in KI;

          then ex h be Element of EnsHomography3 st x = h & h is_K-isometry ;

          hence thesis;

        end;

        then

        reconsider KI as Subset of EnsHomography3 ;

        reconsider N = ( 1. ( F_Real ,3)) as invertible Matrix of 3, F_Real ;

        ( homography N) in the set of all ( homography N) where N be invertible Matrix of 3, F_Real ;

        then

        reconsider h = ( homography N) as Element of EnsHomography3 by ANPROJ_9:def 1;

        h is_K-isometry by Th33;

        then h in KI;

        hence thesis;

      end;

    end

    definition

      :: BKMODEL2:def8

      func SubGroupK-isometry -> strict Subgroup of GroupHomography3 means

      : Def05: the carrier of it = EnsK-isometry ;

      existence

      proof

        set H = EnsK-isometry , G = GroupHomography3 ;

        reconsider N = ( 1. ( F_Real ,3)) as invertible Matrix of 3, F_Real ;

        ( homography N) in the set of all ( homography M) where M be invertible Matrix of 3, F_Real ;

        then

        reconsider idG = ( homography ( 1. ( F_Real ,3))) as Element of GroupHomography3 by ANPROJ_9:def 1, ANPROJ_9:def 4;

        

         A1: ( 1_ G) = idG

        proof

          for g be Element of G holds (idG * g) = g & (g * idG) = g

          proof

            let g be Element of G;

            g in EnsHomography3 by ANPROJ_9:def 4;

            then

            consider N be invertible Matrix of 3, F_Real such that

             A2: g = ( homography N) by ANPROJ_9:def 1;

            idG in EnsHomography3 & ( homography N) in EnsHomography3 by ANPROJ_9:def 1;

            then

            reconsider g1 = idG, g2 = ( homography N) as Element of EnsHomography3 ;

            thus (idG * g) = g

            proof

              (idG * g) = (g1 (*) g2) by A2, ANPROJ_9:def 3, ANPROJ_9:def 4

              .= ( homography (( 1. ( F_Real ,3)) * N)) by ANPROJ_9: 18

              .= g by A2, MATRIX_3: 18;

              hence thesis;

            end;

            thus (g * idG) = g

            proof

              (g * idG) = (g2 (*) g1) by A2, ANPROJ_9:def 3, ANPROJ_9:def 4

              .= ( homography (N * ( 1. ( F_Real ,3)))) by ANPROJ_9: 18

              .= g by A2, MATRIX_3: 19;

              hence thesis;

            end;

          end;

          hence thesis by GROUP_1:def 4;

        end;

        

         A3: for g1,g2 be Element of G st g1 in H & g2 in H holds (g1 * g2) in H

        proof

          let g1,g2 be Element of G;

          assume that

           A4: g1 in H and

           A5: g2 in H;

          consider h1 be Element of EnsHomography3 such that

           A5BIS: g1 = h1 & h1 is_K-isometry by A4;

          consider h2 be Element of EnsHomography3 such that

           A6: g2 = h2 & h2 is_K-isometry by A5;

          reconsider g3 = (g1 * g2) as Element of EnsHomography3 by ANPROJ_9:def 4;

          consider N1,N2 be invertible Matrix of 3, F_Real such that

           A7: h1 = ( homography N1) and

           A8: h2 = ( homography N2) and

           A9: (h1 (*) h2) = ( homography (N1 * N2)) by ANPROJ_9:def 2;

          

           A10: ( dom ( homography N1)) = the carrier of ( ProjectiveSpace ( TOP-REAL 3)) & ( dom ( homography N2)) = the carrier of ( ProjectiveSpace ( TOP-REAL 3)) & ( dom ( homography (N1 * N2))) = the carrier of ( ProjectiveSpace ( TOP-REAL 3)) by FUNCT_2:def 1;

          

           A11: (( homography (N1 * N2)) .: absolute ) c= absolute

          proof

            let x be object;

            assume x in (( homography (N1 * N2)) .: absolute );

            then

            consider y be object such that

             A12: y in ( dom ( homography (N1 * N2))) and

             A13: y in absolute and

             A14: (( homography (N1 * N2)) . y) = x by FUNCT_1:def 6;

            reconsider y as Point of ( ProjectiveSpace ( TOP-REAL 3)) by A12;

            ( dom ( homography N2)) = the carrier of ( ProjectiveSpace ( TOP-REAL 3)) by FUNCT_2:def 1;

            then

             A15: (( homography N2) . y) in absolute by A13, A6, A8, FUNCT_1: 108;

            ( dom ( homography N1)) = the carrier of ( ProjectiveSpace ( TOP-REAL 3)) by FUNCT_2:def 1;

            then (( homography N1) . (( homography N2) . y)) in (( homography N1) .: absolute ) by A15, FUNCT_1: 108;

            hence thesis by A5BIS, A7, A14, ANPROJ_9: 13;

          end;

           absolute c= (( homography (N1 * N2)) .: absolute )

          proof

            let x be object;

            assume

             A16: x in absolute ;

            then

            reconsider y = x as Point of ( ProjectiveSpace ( TOP-REAL 3));

            consider z be object such that

             A17: z in ( dom ( homography N1)) and

             A18: z in absolute and

             A19: y = (( homography N1) . z) by A16, A5BIS, A7, FUNCT_1:def 6;

            reconsider z as Point of ( ProjectiveSpace ( TOP-REAL 3)) by A17;

            consider t be object such that

             A20: t in ( dom ( homography N2)) and

             A21: t in absolute and

             A22: z = (( homography N2) . t) by A18, A6, A8, FUNCT_1:def 6;

            reconsider t as Point of ( ProjectiveSpace ( TOP-REAL 3)) by A20;

            y = (( homography (N1 * N2)) . t) by A22, A19, ANPROJ_9: 13;

            hence thesis by A10, A21, FUNCT_1:def 6;

          end;

          then (( homography (N1 * N2)) .: absolute ) = absolute by A11;

          then g3 is_K-isometry by A9, A5BIS, A6, ANPROJ_9:def 3, ANPROJ_9:def 4;

          hence thesis;

        end;

        for g be Element of G st g in H holds (g " ) in H

        proof

          let g be Element of G;

          assume g in H;

          then

          consider h be Element of EnsHomography3 such that

           A23: g = h & h is_K-isometry ;

          h in the set of all ( homography N) where N be invertible Matrix of 3, F_Real by ANPROJ_9:def 1;

          then

          consider N be invertible Matrix of 3, F_Real such that

           A24: h = ( homography N);

          reconsider h3 = (g " ) as Element of EnsHomography3 by ANPROJ_9:def 4;

          h3 in the set of all ( homography N) where N be invertible Matrix of 3, F_Real by ANPROJ_9:def 1;

          then

          consider N3 be invertible Matrix of 3, F_Real such that

           A25: h3 = ( homography N3);

          

           A26: (h (*) h3) = (g * (g " )) by A23, ANPROJ_9:def 3, ANPROJ_9:def 4

          .= ( 1_ G) by GROUP_1:def 5;

          

           A27: (h3 (*) h) = ((g " ) * g) by A23, ANPROJ_9:def 3, ANPROJ_9:def 4

          .= ( 1_ G) by GROUP_1:def 5;

          

           A28: ( homography (N * N3)) = ( homography ( 1. ( F_Real ,3))) by A26, A1, A25, A24, ANPROJ_9: 18;

          

           A29: ( homography (N3 * N)) = ( homography ( 1. ( F_Real ,3))) by A27, A1, A25, A24, ANPROJ_9: 18;

          

           A30: for P be Point of ( ProjectiveSpace ( TOP-REAL 3)) holds (( homography (N3 ~ )) . P) = (( homography N) . P)

          proof

            let P be Point of ( ProjectiveSpace ( TOP-REAL 3));

            (( homography N3) . (( homography N) . P)) = (( homography (N3 * N)) . P) by ANPROJ_9: 13

            .= P by A29, ANPROJ_9: 14;

            hence thesis by ANPROJ_9: 15;

          end;

          

           A31: for P be Point of ( ProjectiveSpace ( TOP-REAL 3)) holds (( homography N3) . P) = (( homography (N ~ )) . P)

          proof

            let P be Point of ( ProjectiveSpace ( TOP-REAL 3));

            (( homography N) . (( homography N3) . P)) = (( homography (N * N3)) . P) by ANPROJ_9: 13

            .= P by A28, ANPROJ_9: 14;

            hence thesis by ANPROJ_9: 15;

          end;

          

           A32: ( dom ( homography N)) = the carrier of ( ProjectiveSpace ( TOP-REAL 3)) & ( dom ( homography N3)) = the carrier of ( ProjectiveSpace ( TOP-REAL 3)) by FUNCT_2:def 1;

          (( homography N3) .: absolute ) = absolute

          proof

            

             A33: (( homography N3) .: absolute ) c= absolute

            proof

              let x be object;

              assume x in (( homography N3) .: absolute );

              then

              consider y be object such that

               A34: y in ( dom ( homography N3)) and

               A35: y in absolute and

               A36: (( homography N3) . y) = x by FUNCT_1:def 6;

              reconsider y as Point of ( ProjectiveSpace ( TOP-REAL 3)) by A34;

              

               A37: y = (( homography (N3 ~ )) . x) by A36, ANPROJ_9: 15;

              (( homography N3) . y) is Point of ( ProjectiveSpace ( TOP-REAL 3));

              then

              reconsider z = x as Point of ( ProjectiveSpace ( TOP-REAL 3)) by A36;

              (( homography N) . z) in absolute by A37, A30, A35;

              then

              consider t be object such that

               A38: t in ( dom ( homography N)) and

               A39: t in absolute and

               A40: (( homography N) . t) = (( homography N) . z) by A23, A24, FUNCT_1:def 6;

              reconsider t as Point of ( ProjectiveSpace ( TOP-REAL 3)) by A38;

              t = (( homography (N ~ )) . (( homography N) . t)) by ANPROJ_9: 15

              .= z by A40, ANPROJ_9: 15;

              hence thesis by A39;

            end;

             absolute c= (( homography N3) .: absolute )

            proof

              let x be object;

              assume

               A41: x in absolute ;

              then

              reconsider y = x as Point of ( ProjectiveSpace ( TOP-REAL 3));

              consider z be Point of ( ProjectiveSpace ( TOP-REAL 3)) such that

               A42: z in absolute and

               A43: (( homography N) . y) = z by A32, FUNCT_1: 108, A41, A23, A24;

              reconsider z as Point of ( ProjectiveSpace ( TOP-REAL 3));

              y = (( homography (N ~ )) . (( homography N) . y)) by ANPROJ_9: 15

              .= (( homography N3) . z) by A43, A31;

              hence thesis by A42, A32, FUNCT_1:def 6;

            end;

            hence thesis by A33;

          end;

          then h3 is_K-isometry by A25;

          hence (g " ) in H;

        end;

        hence thesis by ANPROJ_9:def 4, A3, GROUP_2: 52;

      end;

      uniqueness

      proof

        let H1,H2 be strict Subgroup of GroupHomography3 such that

         A44: the carrier of H1 = EnsK-isometry and

         A45: the carrier of H2 = EnsK-isometry ;

        for g be object holds g in H1 iff g in H2 by A44, A45;

        hence thesis;

      end;

    end

    theorem :: BKMODEL2:44

    for h be Element of EnsK-isometry holds for N be invertible Matrix of 3, F_Real st h = ( homography N) holds (( homography N) .: absolute ) = absolute

    proof

      let h be Element of EnsK-isometry ;

      let N be invertible Matrix of 3, F_Real ;

      assume

       A1: h = ( homography N);

      h in { h where h be Element of EnsHomography3 : h is_K-isometry };

      then

      consider g be Element of EnsHomography3 such that

       A2: h = g and

       A3: g is_K-isometry ;

      thus thesis by A1, A2, A3;

    end;

    theorem :: BKMODEL2:45

    

     Th34: ( homography ( 1. ( F_Real ,3))) = ( 1_ GroupHomography3 ) & ( homography ( 1. ( F_Real ,3))) = ( 1_ SubGroupK-isometry )

    proof

      set G = GroupHomography3 ;

      ( homography ( 1. ( F_Real ,3))) in G by ANPROJ_9:def 1, ANPROJ_9:def 4;

      then

      reconsider e = ( homography ( 1. ( F_Real ,3))) as Element of G;

      now

        let h be Element of GroupHomography3 ;

        h in EnsHomography3 by ANPROJ_9:def 4;

        then

        consider N be invertible Matrix of 3, F_Real such that

         A1: h = ( homography N) by ANPROJ_9:def 1;

        h in EnsHomography3 & e in EnsHomography3 by A1, ANPROJ_9:def 1;

        then

        reconsider h1 = h, h2 = e as Element of EnsHomography3 ;

        

        thus (h * e) = (h1 (*) h2) by ANPROJ_9:def 3, ANPROJ_9:def 4

        .= ( homography (N * ( 1. ( F_Real ,3)))) by A1, ANPROJ_9: 18

        .= h by A1, MATRIX_3: 19;

        

        thus (e * h) = (h2 (*) h1) by ANPROJ_9:def 3, ANPROJ_9:def 4

        .= ( homography (( 1. ( F_Real ,3)) * N)) by A1, ANPROJ_9: 18

        .= h by A1, MATRIX_3: 18;

      end;

      hence ( homography ( 1. ( F_Real ,3))) = ( 1_ GroupHomography3 ) by GROUP_1: 4;

      hence thesis by GROUP_2: 44;

    end;

    theorem :: BKMODEL2:46

    

     Th35: for N1,N2 be invertible Matrix of 3, F_Real holds for h1,h2 be Element of SubGroupK-isometry st h1 = ( homography N1) & h2 = ( homography N2) holds (h1 * h2) is Element of SubGroupK-isometry & (h1 * h2) = ( homography (N1 * N2))

    proof

      let N1,N2 be invertible Matrix of 3, F_Real ;

      let h1,h2 be Element of SubGroupK-isometry ;

      assume that

       A1: h1 = ( homography N1) and

       A2: h2 = ( homography N2);

      thus (h1 * h2) is Element of SubGroupK-isometry ;

      h1 in EnsHomography3 by A1, ANPROJ_9:def 1;

      then

      reconsider hh1 = h1 as Element of EnsHomography3 ;

      h2 in EnsHomography3 by A2, ANPROJ_9:def 1;

      then

      reconsider hh2 = h2 as Element of EnsHomography3 ;

      set G = GroupHomography3 ;

      reconsider h1g = hh1, h2g = hh2 as Element of G by ANPROJ_9:def 4;

      (h1g * h2g) = (hh1 (*) hh2) by ANPROJ_9:def 3, ANPROJ_9:def 4

      .= ( homography (N1 * N2)) by A1, A2, ANPROJ_9: 18;

      hence (h1 * h2) = ( homography (N1 * N2)) by GROUP_2: 43;

    end;

    theorem :: BKMODEL2:47

    

     Th36: for N be invertible Matrix of 3, F_Real holds for h be Element of SubGroupK-isometry st h = ( homography N) holds (h " ) = ( homography (N ~ )) & ( homography (N ~ )) is Element of SubGroupK-isometry

    proof

      let N be invertible Matrix of 3, F_Real ;

      let h be Element of SubGroupK-isometry ;

      assume

       A1: h = ( homography N);

      then h in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h1 = h as Element of EnsHomography3 ;

      ( homography (N ~ )) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h2 = ( homography (N ~ )) as Element of EnsHomography3 ;

      set G = GroupHomography3 ;

      reconsider h1g = h1, h2g = h2 as Element of G by ANPROJ_9:def 4;

      

       A2: N is_reverse_of (N ~ ) by MATRIX_6:def 4;

      

       A3: (h1g * h2g) = (h1 (*) h2) by ANPROJ_9:def 3, ANPROJ_9:def 4

      .= ( homography (N * (N ~ ))) by A1, ANPROJ_9: 18

      .= ( 1_ G) by A2, MATRIX_6:def 2, Th34;

      (h2g * h1g) = (h2 (*) h1) by ANPROJ_9:def 3, ANPROJ_9:def 4

      .= ( homography ((N ~ ) * N)) by A1, ANPROJ_9: 18

      .= ( 1_ G) by A2, MATRIX_6:def 2, Th34;

      then h2g = (h1g " ) by A3, GROUP_1: 5;

      hence (h " ) = ( homography (N ~ )) by GROUP_2: 48;

      hence thesis;

    end;

    theorem :: BKMODEL2:48

    

     Th37: for s be Element of ( ProjectiveSpace ( TOP-REAL 3)) holds for p,q,r be Element of absolute st (p,q,r) are_mutually_distinct & s in (( tangent p) /\ ( tangent q)) holds ex N be invertible Matrix of 3, F_Real st (( homography N) .: absolute ) = absolute & (( homography N) . Dir101 ) = p & (( homography N) . Dirm101 ) = q & (( homography N) . Dir011 ) = r & (( homography N) . Dir010 ) = s

    proof

      let s be Element of ( ProjectiveSpace ( TOP-REAL 3));

      let p,q,r be Element of absolute ;

      assume that

       A1: (p,q,r) are_mutually_distinct and

       A2: s in (( tangent p) /\ ( tangent q));

      reconsider P1 = p, P2 = q, P3 = r, P4 = s as Point of real_projective_plane ;

      P4 in ( tangent p) & P4 in ( tangent q) by A2, XBOOLE_0:def 4;

      then not (P1,P2,P3) are_collinear & not (P1,P2,P4) are_collinear & not (P1,P3,P4) are_collinear & not (P2,P3,P4) are_collinear by A1, Th27;

      then

      consider N be invertible Matrix of 3, F_Real such that

       A3: (( homography N) . Dir101 ) = P1 and

       A4: (( homography N) . Dirm101 ) = P2 and

       A5: (( homography N) . Dir011 ) = P3 and

       A6: (( homography N) . Dir010 ) = P4 by BKMODEL1: 44, ANPROJ_9: 31;

      consider na,nb,nc,nd,ne,nf,ng,nh,ni be Element of F_Real such that

       A7: N = <* <*na, nb, nc*>, <*nd, ne, nf*>, <*ng, nh, ni*>*> by PASCAL: 3;

      reconsider b = ( - 1) as Element of F_Real by XREAL_0:def 1;

      

       A8: b is non zero;

      reconsider a = 1 as Element of F_Real ;

      a is non zero;

      then

      reconsider a = 1, b = ( - 1) as non zero Element of F_Real by A8;

      reconsider N1 = <* <*a, 0 , 0 *>, <* 0 , a, 0 *>, <* 0 , 0 , b*>*> as invertible Matrix of 3, F_Real by ANPROJ_9: 9;

      reconsider M = (((N @ ) * N1) * N) as invertible Matrix of 3, F_Real ;

      

       A9: N1 = ( symmetric_3 (a,a,b, 0 , 0 , 0 )) by PASCAL:def 3;

      then

       A10: M is symmetric by PASCAL: 7, PASCAL: 12;

      consider va,vb,vc,vd,ve,vf,vg,vh,vi be Element of F_Real such that

       A11: M = <* <*va, vb, vc*>, <*vd, ve, vf*>, <*vg, vh, vi*>*> by PASCAL: 3;

      

       A12: vb = vd & vc = vg & vh = vf by A10, A11, PASCAL: 6;

      reconsider ra = va, rb = vb, rc = vc, re = ve, rf = vf, ri = vi as Real;

      

       A13: M = ( symmetric_3 (ra,re,ri,rb,rc,rf)) by A12, A11, PASCAL:def 3;

      

       A14: p in ( conic (1,1,( - 1), 0 , 0 , 0 )) & (N ~ ) is invertible;

      reconsider NR = ( MXF2MXR (N ~ )) as Matrix of 3, REAL by MATRIXR1:def 2;

      

       A15: N1 = ( symmetric_3 (1,1,( - 1),( 0 / 2),( 0 / 2),( 0 / 2))) by PASCAL:def 3;

      reconsider N2 = N1 as Matrix of 3, REAL ;

      

       A16: M = ((( MXF2MXR (( MXR2MXF (NR @ )) ~ )) * N2) * ( MXF2MXR (( MXR2MXF NR) ~ ))) by A15, BKMODEL1: 53;

      

       A17: (( homography (N ~ )) . p) = Dir101 by A3, ANPROJ_9: 15;

      

       A18: not (ra = 0 & re = 0 & ri = 0 & rb = 0 & rc = 0 & rf = 0 ) & (( homography (N ~ )) . p) in ( conic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf))) by A13, A14, A15, A16, PASCAL: 16;

       Dir101 in { P where P be Point of ( ProjectiveSpace ( TOP-REAL 3)) : for u be Element of ( TOP-REAL 3) st u is non zero & P = ( Dir u) holds ( qfconic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf),u)) = 0 } by A18, A17, PASCAL:def 2;

      then

      consider Q be Point of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A19: Dir101 = Q and

       A20: for u be Element of ( TOP-REAL 3) st u is non zero & Q = ( Dir u) holds ( qfconic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf),u)) = 0 ;

      ( |[1, 0 , 1]| `1 ) = 1 & ( |[1, 0 , 1]| `2 ) = 0 & ( |[1, 0 , 1]| `3 ) = 1 by EUCLID_5: 2;

      then

       A21: ( |[1, 0 , 1]| . 1) = 1 & ( |[1, 0 , 1]| . 2) = 0 & ( |[1, 0 , 1]| . 3) = 1 by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

      ( qfconic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf), |[1, 0 , 1]|)) = 0 by A19, A20, BKMODEL1: 41;

      

      then

       A22: 0 = (((((((ra * 1) * 1) + ((re * 0 ) * 0 )) + ((ri * 1) * 1)) + (((2 * rb) * 1) * 0 )) + (((2 * rc) * 1) * 1)) + (((2 * rf) * 0 ) * 1)) by A21, PASCAL:def 1

      .= ((ra + ri) + (2 * rc));

      

       A23: (( homography (N ~ )) . q) = Dirm101 by A4, ANPROJ_9: 15;

      q in ( conic (1,1,( - 1), 0 , 0 , 0 )) & (N ~ ) is invertible;

      then

       A25: not (ra = 0 & re = 0 & ri = 0 & rb = 0 & rc = 0 & rf = 0 ) & (( homography (N ~ )) . q) in ( conic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf))) by A13, PASCAL: 16, A15, A16;

       Dirm101 in { P where P be Point of ( ProjectiveSpace ( TOP-REAL 3)) : for u be Element of ( TOP-REAL 3) st u is non zero & P = ( Dir u) holds ( qfconic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf),u)) = 0 } by A23, A25, PASCAL:def 2;

      then

      consider Q be Point of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A26: Dirm101 = Q and

       A27: for u be Element of ( TOP-REAL 3) st u is non zero & Q = ( Dir u) holds ( qfconic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf),u)) = 0 ;

      ( |[( - 1), 0 , 1]| `1 ) = ( - 1) & ( |[( - 1), 0 , 1]| `2 ) = 0 & ( |[( - 1), 0 , 1]| `3 ) = 1 by EUCLID_5: 2;

      then

       A28: ( |[( - 1), 0 , 1]| . 1) = ( - 1) & ( |[( - 1), 0 , 1]| . 2) = 0 & ( |[( - 1), 0 , 1]| . 3) = 1 by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

      ( qfconic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf), |[( - 1), 0 , 1]|)) = 0 by A26, A27, BKMODEL1: 41;

      

      then

       A29: 0 = (((((((ra * ( - 1)) * ( - 1)) + ((re * 0 ) * 0 )) + ((ri * 1) * 1)) + (((2 * rb) * ( - 1)) * 0 )) + (((2 * rc) * ( - 1)) * 1)) + (((2 * rf) * 0 ) * 1)) by A28, PASCAL:def 1

      .= ((ra + ri) - (2 * rc));

      

       A30: (( homography (N ~ )) . r) = Dir011 by A5, ANPROJ_9: 15;

      r in ( conic (1,1,( - 1), 0 , 0 , 0 )) & (N ~ ) is invertible;

      then

       A31: for fa,fb,fc,fe,fi,ff be Real holds for N1,M be Matrix of 3, REAL holds for NR be Matrix of 3, REAL st N1 = ( symmetric_3 (1,1,( - 1),( 0 / 2),( 0 / 2),( 0 / 2))) & NR = ( MXF2MXR (N ~ )) & M = ((( MXF2MXR (( MXR2MXF (NR @ )) ~ )) * N1) * ( MXF2MXR (( MXR2MXF NR) ~ ))) & M = ( symmetric_3 (fa,fe,fi,fb,fc,ff)) holds not (fa = 0 & fe = 0 & fi = 0 & fb = 0 & ff = 0 & fc = 0 ) & (( homography (N ~ )) . r) in ( conic (fa,fe,fi,(2 * fb),(2 * fc),(2 * ff))) by PASCAL: 16;

       not (ra = 0 & re = 0 & ri = 0 & rb = 0 & rc = 0 & rf = 0 ) & (( homography (N ~ )) . r) in ( conic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf))) by A9, A13, A31, A16;

      then Dir011 in { P where P be Point of ( ProjectiveSpace ( TOP-REAL 3)) : for u be Element of ( TOP-REAL 3) st u is non zero & P = ( Dir u) holds ( qfconic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf),u)) = 0 } by A30, PASCAL:def 2;

      then

      consider Q be Point of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A32: Dir011 = Q and

       A33: for u be Element of ( TOP-REAL 3) st u is non zero & Q = ( Dir u) holds ( qfconic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf),u)) = 0 ;

      ( |[ 0 , 1, 1]| `1 ) = 0 & ( |[ 0 , 1, 1]| `2 ) = 1 & ( |[ 0 , 1, 1]| `3 ) = 1 by EUCLID_5: 2;

      then

       A34: ( |[ 0 , 1, 1]| . 1) = 0 & ( |[ 0 , 1, 1]| . 2) = 1 & ( |[ 0 , 1, 1]| . 3) = 1 by EUCLID_5:def 1, EUCLID_5:def 2, EUCLID_5:def 3;

      ( qfconic (ra,re,ri,(2 * rb),(2 * rc),(2 * rf), |[ 0 , 1, 1]|)) = 0 by A32, A33, BKMODEL1: 41;

      

      then

       A35: 0 = (((((((ra * 0 ) * 0 ) + ((re * 1) * 1)) + ((ri * 1) * 1)) + (((2 * rb) * 0 ) * 1)) + (((2 * rc) * 0 ) * 1)) + (((2 * rf) * 1) * 1)) by A34, PASCAL:def 1

      .= ((re + ri) + (2 * rf));

      rc = 0 & ra = ( - ri) & rb = 0 & rf = 0 & ra = re

      proof

        thus rc = 0 by A22, A29;

        thus ra = ( - ri) by A22, A29;

        consider k1 be Element of ( TOP-REAL 3) such that

         A36: k1 is non zero and

         A37: P1 = ( Dir k1) by ANPROJ_1: 26;

        consider k1b be Element of ( TOP-REAL 3) such that

         A38: k1b is non zero and

         A39: P2 = ( Dir k1b) by ANPROJ_1: 26;

        consider k2 be Element of ( TOP-REAL 3) such that

         A40: k2 is non zero and

         A41: P4 = ( Dir k2) by ANPROJ_1: 26;

        reconsider kf1 = k1, kf1b = k1b, kf2 = k2 as FinSequence of REAL by EUCLID: 24;

        

         A42: P4 in ( tangent p) & N2 is Matrix of 3, REAL & p is Element of absolute & Q is Element of real_projective_plane & k1 is non zero Element of ( TOP-REAL 3) & k2 is non zero Element of ( TOP-REAL 3) & kf1 is FinSequence of REAL & kf2 is FinSequence of REAL & N2 = ( symmetric_3 (1,1,( - 1), 0 , 0 , 0 )) & p = ( Dir k1) & P4 = ( Dir k2) & k1 = kf1 & k2 = kf2 by A2, XBOOLE_0:def 4, PASCAL:def 3, A36, A37, A40, A41;

        

         A43: P4 in ( tangent q) & N2 is Matrix of 3, REAL & p is Element of absolute & k1b is non zero Element of ( TOP-REAL 3) & k2 is non zero Element of ( TOP-REAL 3) & kf1b is FinSequence of REAL & kf2 is FinSequence of REAL & N2 = ( symmetric_3 (1,1,( - 1), 0 , 0 , 0 )) & q = ( Dir k1b) & P4 = ( Dir k2) & k1b = kf1b & k2 = kf2 by A2, A38, A40, PASCAL:def 3, A39, A41, XBOOLE_0:def 4;

        consider ua,va be Element of ( TOP-REAL 3), ufa be FinSequence of F_Real , pa be FinSequence of (1 -tuples_on REAL ) such that

         A44: Dir101 = ( Dir ua) & not ua is zero & ua = ufa & pa = (N * ufa) & va = ( M2F pa) & not va is zero & (( homography N) . Dir101 ) = ( Dir va) by ANPROJ_8:def 4;

         are_Prop (k1,va) by A3, A37, A44, ANPROJ_1: 22, A36;

        then

        consider li be Real such that

         A45: li <> 0 and

         A46: k1 = (li * va) by ANPROJ_1: 1;

        

         A47: ( len (N * ( <*ufa*> @ ))) = ( len N)

        proof

          ufa in ( TOP-REAL 3) by A44;

          then

           A48: ufa in ( REAL 3) by EUCLID: 22;

          then ( len ufa) = 3 by EUCLID_8: 50;

          then ( width <*ufa*>) = 3 by ANPROJ_8: 75;

          

          then

           A50: ( len ( <*ufa*> @ )) = ( width <*ufa*>) by MATRIX_0: 29

          .= ( len ufa) by MATRIX_0: 23;

          ( width N) = 3 by MATRIX_0: 24

          .= ( len ( <*ufa*> @ )) by A50, A48, EUCLID_8: 50;

          hence thesis by MATRIX_3:def 4;

        end;

        

         A51: ( len pa) = ( len N) by A47, A44, LAPLACE:def 9

        .= 3 by MATRIX_0: 23;

        then

         A52: kf1 = ( M2F (li * pa)) by A44, A46, ANPROJ_8: 83;

        consider ub,vb be Element of ( TOP-REAL 3), ufb be FinSequence of F_Real , pb be FinSequence of (1 -tuples_on REAL ) such that

         A53: Dir010 = ( Dir ub) & not ub is zero & ub = ufb & pb = (N * ufb) & vb = ( M2F pb) & not vb is zero & (( homography N) . Dir010 ) = ( Dir vb) by ANPROJ_8:def 4;

         are_Prop (ub, |[ 0 , 1, 0 ]|) by A53, ANPROJ_1: 22, ANPROJ_9:def 6, ANPROJ_9: 10;

        then

        consider lub be Real such that

         A54: lub <> 0 and

         A55: ub = (lub * |[ 0 , 1, 0 ]|) by ANPROJ_1: 1;

        

         A56: ufb = |[(lub * 0 ), (lub * 1), (lub * 0 )]| by A53, A55, EUCLID_5: 8

        .= |[ 0 , lub, 0 ]|;

        lub in REAL by XREAL_0:def 1;

        then

        reconsider MUFB = <*ufb*> as Matrix of 1, 3, F_Real by A56, BKMODEL1: 27;

         A57:

        now

          ( len ufb) = 3 by A56, FINSEQ_1: 45;

          then ( dom ufb) = {1, 2, 3} by FINSEQ_1:def 3, FINSEQ_3: 1;

          then 1 in ( dom ufb) & 2 in ( dom ufb) & 3 in ( dom ufb) by ENUMSET1:def 1;

          then (MUFB * (1,1)) = ( |[ 0 , lub, 0 ]| . 1) & (MUFB * (1,2)) = ( |[ 0 , lub, 0 ]| . 2) & (MUFB * (1,3)) = ( |[ 0 , lub, 0 ]| . 3) by A56, ANPROJ_8: 70;

          hence (MUFB * (1,1)) = 0 & (MUFB * (1,2)) = lub & (MUFB * (1,3)) = 0 by FINSEQ_1: 45;

        end;

         are_Prop (k2,vb) by A41, A6, A53, ANPROJ_1: 22, A40;

        then

        consider lj be Real such that

         A58: lj <> 0 and

         A59: k2 = (lj * vb) by ANPROJ_1: 1;

        

         A60: ( len (N * ( <*ufb*> @ ))) = ( len N)

        proof

          ufb in ( TOP-REAL 3) by A53;

          then

           A61: ufb in ( REAL 3) by EUCLID: 22;

          then ( len ufb) = 3 by EUCLID_8: 50;

          then ( width <*ufb*>) = 3 by ANPROJ_8: 75;

          

          then

           A61bis: ( len ( <*ufb*> @ )) = ( width <*ufb*>) by MATRIX_0: 29

          .= ( len ufb) by MATRIX_0: 23;

          ( width N) = 3 by MATRIX_0: 24

          .= ( len ( <*ufb*> @ )) by A61, A61bis, EUCLID_8: 50;

          hence thesis by MATRIX_3:def 4;

        end;

        

         A62: ( len pb) = ( len N) by A60, A53, LAPLACE:def 9

        .= 3 by MATRIX_0: 23;

        then

         A63: ( M2F pb) is Element of ( TOP-REAL 3) by ANPROJ_8: 82;

        then

         A64: ( M2F pb) is Element of ( REAL 3) by EUCLID: 22;

        then

         A65: ( len ( M2F pb)) = 3 by EUCLID_8: 50;

        ( M2F pa) is Element of ( TOP-REAL 3) by A51, ANPROJ_8: 82;

        then

         A66: ( M2F pa) is Element of ( REAL 3) by EUCLID: 22;

        then

         A67: ( len ( M2F pa)) = 3 by EUCLID_8: 50;

        

         A68: (li * (N2 * ( M2F pa))) = (N2 * (li * ( M2F pa)))

        proof

          ( width N2) = 3 by MATRIX_0: 23;

          hence thesis by A67, MATRIXR1: 59;

        end;

        

         A69: ( len (li * (N2 * ( M2F pa)))) = 3 & ( len (N2 * ( M2F pa))) = 3

        proof

          ( ColVec2Mx ( M2F pa)) = pa by A51, BKMODEL1: 33;

          then

          reconsider Mpa = pa as Matrix of REAL ;

          

           A70: ( len (N2 * ( M2F pa))) = ( len ( Col ((N2 * ( ColVec2Mx ( M2F pa))),1))) by MATRIXR1:def 11

          .= ( len (N2 * ( ColVec2Mx ( M2F pa)))) by MATRIX_0:def 8

          .= ( len (N2 * Mpa)) by A51, BKMODEL1: 33;

          reconsider N2F = N2, MpaF = Mpa as Matrix of F_Real ;

          

           A71: ( width N2F) = ( len Mpa) by A51, MATRIX_0: 23;

          ( len (N2 * Mpa)) = ( len (N2F * MpaF)) by ANPROJ_8: 17

          .= ( len N2F) by A71, MATRIX_3:def 4

          .= 3 by MATRIX_0: 23;

          hence thesis by A70, RVSUM_1: 117;

        end;

        then

         A72: ( len ( M2F pb)) = ( len (N2 * ( M2F (li * pa)))) by A51, ANPROJ_8: 83, A68, A65;

        

         A73: ( len ( M2F pb)) = ( len (N2 * ( M2F pa))) by A69, A64, EUCLID_8: 50;

        

         A74: kf2 = ( M2F (lj * pb)) by A59, A53, A62, ANPROJ_8: 83;

        

         A75: ( len ( M2F (lj * pb))) = ( len N2) & ( len ( M2F (li * pa))) = ( width N2) & ( len ( M2F (li * pa))) > 0

        proof

          

           A76: ( len N2) = 3 & ( width N2) = 3 by MATRIX_0: 23;

          consider p1,p2,p3 be Real such that

           A77: p1 = ((pb . 1) . 1) & p2 = ((pb . 2) . 1) & p3 = ((pb . 3) . 1) & (lj * pb) = <* <*(lj * p1)*>, <*(lj * p2)*>, <*(lj * p3)*>*> by A62, ANPROJ_8:def 3;

          ( len (lj * pb)) = 3 by A77, FINSEQ_1: 45;

          then

           A78: ( M2F (lj * pb)) = <*(((lj * pb) . 1) . 1), (((lj * pb) . 2) . 1), (((lj * pb) . 3) . 1)*> by ANPROJ_8:def 2;

          consider p1,p2,p3 be Real such that

           A79: p1 = ((pa . 1) . 1) & p2 = ((pa . 2) . 1) & p3 = ((pa . 3) . 1) & (li * pa) = <* <*(li * p1)*>, <*(li * p2)*>, <*(li * p3)*>*> by A51, ANPROJ_8:def 3;

          ( len (li * pa)) = 3 by A79, FINSEQ_1: 45;

          then ( M2F (li * pa)) = <*(((li * pa) . 1) . 1), (((li * pa) . 2) . 1), (((li * pa) . 3) . 1)*> by ANPROJ_8:def 2;

          hence thesis by A76, A78, FINSEQ_1: 45;

        end;

        

         A80: 0 = ( SumAll ( QuadraticForm (( M2F (lj * pb)),N2,( M2F (li * pa))))) by A42, Th26, A52, A74

        .= |(( M2F (lj * pb)), (N2 * ( M2F (li * pa))))| by A75, MATRPROB: 44

        .= |((lj * ( M2F pb)), (N2 * ( M2F (li * pa))))| by A62, ANPROJ_8: 83

        .= (lj * |(( M2F pb), (N2 * ( M2F (li * pa))))|) by A72, RVSUM_1: 121

        .= (lj * |(( M2F pb), (li * (N2 * ( M2F pa))))|) by A68, A51, ANPROJ_8: 83

        .= (lj * (li * |(( M2F pb), (N2 * ( M2F pa)))|)) by A73, RVSUM_1: 121

        .= ((lj * li) * |(( M2F pb), (N2 * ( M2F pa)))|);

        

         A81: ((nb * (( - na) + nc)) + (ne * (( - nd) + nf))) = (nh * (( - ng) + ni))

        proof

          consider ua2,va2 be Element of ( TOP-REAL 3), ufa2 be FinSequence of F_Real , pa2 be FinSequence of (1 -tuples_on REAL ) such that

           A82: Dirm101 = ( Dir ua2) & not ua2 is zero & ua2 = ufa2 & pa2 = (N * ufa2) & va2 = ( M2F pa2) & not va2 is zero & (( homography N) . Dirm101 ) = ( Dir va2) by ANPROJ_8:def 4;

           are_Prop (k1b,va2) by A39, A4, A82, ANPROJ_1: 22, A38;

          then

          consider li2 be Real such that

           A83: li2 <> 0 and

           A84: k1b = (li2 * va2) by ANPROJ_1: 1;

          

           A85: ( len (N * ( <*ufa2*> @ ))) = ( len N)

          proof

            ufa2 in ( TOP-REAL 3) by A82;

            then

             A86: ufa2 in ( REAL 3) by EUCLID: 22;

            

             A87: ( len ufa2) = 3 by A86, EUCLID_8: 50;

            ( width <*ufa2*>) = 3 by A87, ANPROJ_8: 75;

            

            then

             A88: ( len ( <*ufa2*> @ )) = ( width <*ufa2*>) by MATRIX_0: 29

            .= ( len ufa2) by MATRIX_0: 23;

            ( width N) = 3 by MATRIX_0: 24

            .= ( len ( <*ufa2*> @ )) by A88, A86, EUCLID_8: 50;

            hence thesis by MATRIX_3:def 4;

          end;

          

           A89: ( len pa2) = ( len N) by A85, A82, LAPLACE:def 9

          .= 3 by MATRIX_0: 23;

          

           A90: kf1b = ( M2F (li2 * pa2)) by A82, A84, A89, ANPROJ_8: 83;

          ( M2F pa2) is Element of ( TOP-REAL 3) by A89, ANPROJ_8: 82;

          then

           A91: ( M2F pa2) is Element of ( REAL 3) by EUCLID: 22;

          then

           A92: ( len ( M2F pa2)) = 3 by EUCLID_8: 50;

          

           A93: (li2 * (N2 * ( M2F pa2))) = (N2 * (li2 * ( M2F pa2)))

          proof

            ( width N2) = 3 by MATRIX_0: 23;

            hence thesis by MATRIXR1: 59, A92;

          end;

          

           A94: ( len (li2 * (N2 * ( M2F pa2)))) = 3 & ( len (N2 * ( M2F pa2))) = 3

          proof

            ( ColVec2Mx ( M2F pa2)) = pa2 by A89, BKMODEL1: 33;

            then

            reconsider Mpa2 = pa2 as Matrix of REAL ;

            

             A95: ( len (N2 * ( M2F pa2))) = ( len ( Col ((N2 * ( ColVec2Mx ( M2F pa2))),1))) by MATRIXR1:def 11

            .= ( len (N2 * ( ColVec2Mx ( M2F pa2)))) by MATRIX_0:def 8

            .= ( len (N2 * Mpa2)) by A89, BKMODEL1: 33;

            reconsider N2F2 = N2, MpaF2 = Mpa2 as Matrix of F_Real ;

            

             A96: ( width N2F2) = ( len Mpa2) by A89, MATRIX_0: 23;

            ( len (N2 * Mpa2)) = ( len (N2F2 * MpaF2)) by ANPROJ_8: 17

            .= ( len N2F2) by A96, MATRIX_3:def 4

            .= 3 by MATRIX_0: 23;

            hence thesis by A95, RVSUM_1: 117;

          end;

          then

           A97: ( len ( M2F pb)) = ( len (N2 * ( M2F (li2 * pa2)))) by A89, ANPROJ_8: 83, A93, A65;

          

           A98: ( len ( M2F pb)) = ( len (N2 * ( M2F pa2))) by A94, A64, EUCLID_8: 50;

          

           A99: kf2 = ( M2F (lj * pb)) by A59, A53, A62, ANPROJ_8: 83;

          

           A100: ( len ( M2F (lj * pb))) = ( len N2) & ( len ( M2F (li2 * pa2))) = ( width N2) & ( len ( M2F (li2 * pa2))) > 0

          proof

            

             A101: ( len N2) = 3 & ( width N2) = 3 by MATRIX_0: 23;

            consider p1,p2,p3 be Real such that

             A102: p1 = ((pb . 1) . 1) & p2 = ((pb . 2) . 1) & p3 = ((pb . 3) . 1) & (lj * pb) = <* <*(lj * p1)*>, <*(lj * p2)*>, <*(lj * p3)*>*> by A62, ANPROJ_8:def 3;

            ( len (lj * pb)) = 3 by A102, FINSEQ_1: 45;

            then

             A103: ( M2F (lj * pb)) = <*(((lj * pb) . 1) . 1), (((lj * pb) . 2) . 1), (((lj * pb) . 3) . 1)*> by ANPROJ_8:def 2;

            consider p1b,p2b,p3b be Real such that

             A104: p1b = ((pa2 . 1) . 1) & p2b = ((pa2 . 2) . 1) & p3b = ((pa2 . 3) . 1) & (li2 * pa2) = <* <*(li2 * p1b)*>, <*(li2 * p2b)*>, <*(li2 * p3b)*>*> by A89, ANPROJ_8:def 3;

            ( len (li2 * pa2)) = 3 by A104, FINSEQ_1: 45;

            then ( M2F (li2 * pa2)) = <*(((li2 * pa2) . 1) . 1), (((li2 * pa2) . 2) . 1), (((li2 * pa2) . 3) . 1)*> by ANPROJ_8:def 2;

            hence thesis by A101, A103, FINSEQ_1: 45;

          end;

          

           A105: 0 = ( SumAll ( QuadraticForm (( M2F (lj * pb)),N2,( M2F (li2 * pa2))))) by A43, Th26, A90, A99

          .= |(( M2F (lj * pb)), (N2 * ( M2F (li2 * pa2))))| by A100, MATRPROB: 44

          .= |((lj * ( M2F pb)), (N2 * ( M2F (li2 * pa2))))| by A62, ANPROJ_8: 83

          .= (lj * |(( M2F pb), (N2 * ( M2F (li2 * pa2))))|) by A97, RVSUM_1: 121

          .= (lj * |(( M2F pb), (li2 * (N2 * ( M2F pa2))))|) by A93, A89, ANPROJ_8: 83

          .= (lj * (li2 * |(( M2F pb), (N2 * ( M2F pa2)))|)) by A98, RVSUM_1: 121

          .= ((lj * li2) * |(( M2F pb), (N2 * ( M2F pa2)))|);

          

           A106: ( M2F pa2) = <*((pa2 . 1) . 1), ((pa2 . 2) . 1), ((pa2 . 3) . 1)*> by A89, ANPROJ_8:def 2;

          ( dom ( M2F pa2)) = ( Seg 3) by A91, EUCLID_8: 50;

          then (( M2F pa2) . 1) in REAL & (( M2F pa2) . 2) in REAL & (( M2F pa2) . 3) in REAL by FINSEQ_1: 1, FINSEQ_2: 11;

          then

          reconsider s1 = ((pa2 . 1) . 1), s2 = ((pa2 . 2) . 1), s3 = ((pa2 . 3) . 1) as Element of REAL by A106, FINSEQ_1: 45;

          

           A107: ( M2F pb) = <*((pb . 1) . 1), ((pb . 2) . 1), ((pb . 3) . 1)*> by A62, ANPROJ_8:def 2;

          ( dom ( M2F pb)) = ( Seg 3) by A64, EUCLID_8: 50;

          then (( M2F pb) . 1) in REAL & (( M2F pb) . 2) in REAL & (( M2F pb) . 3) in REAL by FINSEQ_1: 1, FINSEQ_2: 11;

          then

          reconsider t1 = ((pb . 1) . 1), t2 = ((pb . 2) . 1), t3 = ((pb . 3) . 1) as Element of F_Real by A107, FINSEQ_1: 45;

          reconsider r1 = 1, r2 = 0 , r3 = ( - 1) as Element of F_Real by XREAL_0:def 1;

          ((nb * (( - na) + nc)) + (ne * (( - nd) + nf))) = (nh * (( - ng) + ni))

          proof

            reconsider r1 = 1, r2 = 0 , r3 = ( - 1) as Element of F_Real by XREAL_0:def 1;

            

             A108: ( M2F pa2) = <*((pa2 . 1) . 1), ((pa2 . 2) . 1), ((pa2 . 3) . 1)*> by A89, ANPROJ_8:def 2;

            ( dom ( M2F pa2)) = ( Seg 3) by A91, EUCLID_8: 50;

            then (( M2F pa2) . 1) in REAL & (( M2F pa2) . 2) in REAL & (( M2F pa2) . 3) in REAL by FINSEQ_1: 1, FINSEQ_2: 11;

            then

            reconsider s1 = ((pa2 . 1) . 1), s2 = ((pa2 . 2) . 1), s3 = ((pa2 . 3) . 1) as Element of REAL by A108, FINSEQ_1: 45;

            

             A109: ( M2F pb) = <*((pb . 1) . 1), ((pb . 2) . 1), ((pb . 3) . 1)*> by A62, ANPROJ_8:def 2;

            ( dom ( M2F pb)) = ( Seg 3) by A64, EUCLID_8: 50;

            then (( M2F pb) . 1) in REAL & (( M2F pb) . 2) in REAL & (( M2F pb) . 3) in REAL by FINSEQ_1: 1, FINSEQ_2: 11;

            then

            reconsider t1 = ((pb . 1) . 1), t2 = ((pb . 2) . 1), t3 = ((pb . 3) . 1) as Element of F_Real by A109, FINSEQ_1: 45;

            ( M2F pa2) = <*s1, s2, s3*> by A89, ANPROJ_8:def 2;

            

            then

             A110: (N2 * ( M2F pa2)) = <*(((1 * s1) + ( 0 * s2)) + ( 0 * s3)), ((( 0 * s1) + (1 * s2)) + ( 0 * s3)), ((( 0 * s1) + ( 0 * s2)) + (( - 1) * s3))*> by PASCAL: 9

            .= <*s1, s2, ( - s3)*>;

            ( M2F pb) = <*t1, t2, t3*> by A62, ANPROJ_8:def 2;

            then

             A111: (( M2F pb) . 1) = t1 & (( M2F pb) . 2) = t2 & (( M2F pb) . 3) = t3 & ( <*s1, s2, ( - s3)*> . 1) = s1 & ( <*s1, s2, ( - s3)*> . 2) = s2 & ( <*s1, s2, ( - s3)*> . 3) = ( - s3) by FINSEQ_1: 45;

            

             A112: ( M2F pb) is Element of ( REAL 3) by A63, EUCLID: 22;

            

             A113: |[s1, s2, ( - s3)]| is Element of ( REAL 3) by EUCLID: 22;

             0 = |(( M2F pb), <*s1, s2, ( - s3)*>)| by A110, A105, A83, A58

            .= (((t1 * s1) + (t2 * s2)) + (t3 * ( - s3))) by A112, A113, EUCLID_8: 63, A111;

            then

             A114: ((t1 * s1) + (t2 * s2)) = (t3 * s3);

             |[( - 1), 0 , 1]| is non zero by EUCLID_5: 4, FINSEQ_1: 78;

            then are_Prop (ua2, |[( - 1), 0 , 1]|) by A82, ANPROJ_1: 22;

            then

            consider lua2 be Real such that

             A115: lua2 <> 0 and

             A116: ua2 = (lua2 * |[( - 1), 0 , 1]|) by ANPROJ_1: 1;

            

             A117: ua2 = |[(lua2 * ( - 1)), (lua2 * 0 ), (lua2 * 1)]| by A116, EUCLID_5: 8

            .= |[( - lua2), 0 , lua2]|;

            reconsider za1 = ( - lua2), za2 = 0 , za3 = lua2 as Element of F_Real by XREAL_0:def 1;

            lua2 in REAL & ( - lua2) in REAL by XREAL_0:def 1;

            then

            reconsider MUFA = <*ufa2*> as Matrix of 1, 3, F_Real by A117, A82, BKMODEL1: 27;

            now

              ( len ufa2) = 3 by A117, A82, FINSEQ_1: 45;

              then ( dom ufa2) = {1, 2, 3} by FINSEQ_1:def 3, FINSEQ_3: 1;

              then 1 in ( dom ufa2) & 2 in ( dom ufa2) & 3 in ( dom ufa2) by ENUMSET1:def 1;

              then (MUFA * (1,1)) = ( |[( - lua2), 0 , lua2]| . 1) & (MUFA * (1,2)) = ( |[( - lua2), 0 , lua2]| . 2) & (MUFA * (1,3)) = ( |[( - lua2), 0 , lua2]| . 3) by A117, A82, ANPROJ_8: 70;

              hence (MUFA * (1,1)) = ( - lua2) & (MUFA * (1,2)) = 0 & (MUFA * (1,3)) = lua2 by FINSEQ_1: 45;

            end;

            then

             A119: ( <*ufa2*> @ ) = <* <*( - lua2)*>, <* 0 *>, <*lua2*>*> by BKMODEL1: 31;

            reconsider nlua2 = ( - lua2) as Element of F_Real by XREAL_0:def 1;

             0 is Element of F_Real & lua2 is Element of F_Real by XREAL_0:def 1;

            then

            reconsider MUFAT = <* <*nlua2*>, <* 0 *>, <*lua2*>*> as Matrix of 3, 1, F_Real by BKMODEL1: 28;

            

             A120: (N * MUFAT) is Matrix of 3, 1, F_Real by BKMODEL1: 24;

            

             A121: (N * ufa2) = (N * MUFAT) by A119, LAPLACE:def 9;

            then (N * ufa2) = <* <*((N * ufa2) * (1,1))*>, <*((N * ufa2) * (2,1))*>, <*((N * ufa2) * (3,1))*>*> by A120, BKMODEL1: 30;

            then

             A122: (pa2 . 1) = <*((N * ufa2) * (1,1))*> & (pa2 . 2) = <*((N * ufa2) * (2,1))*> & (pa2 . 3) = <*((N * ufa2) * (3,1))*> by A82, FINSEQ_1: 45;

            (N * MUFAT) is Matrix of 3, 1, F_Real by BKMODEL1: 24;

            then

             A123: ( Indices (N * MUFAT)) = [:( Seg 3), ( Seg 1):] by MATRIX_0: 23;

            ( width N) = 3 by MATRIX_0: 24;

            then

             A124: ( width N) = ( len MUFAT) by MATRIX_0: 23;

            

             A125: ( Col (MUFAT,1)) = <*za1, za2, za3*> by ANPROJ_8: 5;

            

             A126: ( Line (N,1)) = <*na, nb, nc*> & ( Line (N,2)) = <*nd, ne, nf*> & ( Line (N,3)) = <*ng, nh, ni*> by A7, ANPROJ_9: 4;

            ((N * MUFAT) * (1,1)) = (( Line (N,1)) "*" ( Col (MUFAT,1))) & ((N * MUFAT) * (2,1)) = (( Line (N,2)) "*" ( Col (MUFAT,1))) & ((N * MUFAT) * (3,1)) = (( Line (N,3)) "*" ( Col (MUFAT,1))) by A123, A124, MATRIX_3:def 4, ANPROJ_8: 2;

            then ((N * MUFAT) * (1,1)) = (((na * za1) + (nb * za2)) + (nc * za3)) & ((N * MUFAT) * (2,1)) = (((nd * za1) + (ne * za2)) + (nf * za3)) & ((N * MUFAT) * (3,1)) = (((ng * za1) + (nh * za2)) + (ni * za3)) by A125, A126, ANPROJ_8: 7;

            then

             A127: ((pa2 . 1) . 1) = ((na * nlua2) + (nc * lua2)) & ((pa2 . 2) . 1) = ((nd * nlua2) + (nf * lua2)) & ((pa2 . 3) . 1) = ((ng * nlua2) + (ni * lua2)) by A121, A122, FINSEQ_1: 40;

            reconsider z1 = 0 , z2 = lub, z3 = 0 as Element of F_Real by XREAL_0:def 1;

             0 is Element of F_Real & lub is Element of F_Real by XREAL_0:def 1;

            then

            reconsider MUFBT = <* <* 0 *>, <*lub*>, <* 0 *>*> as Matrix of 3, 1, F_Real by BKMODEL1: 28;

            

             A128: (N * MUFBT) is Matrix of 3, 1, F_Real by BKMODEL1: 24;

            

             A129: (N * ufb) = (N * ( <*ufb*> @ )) by LAPLACE:def 9

            .= (N * MUFBT) by A57, BKMODEL1: 31;

            then (N * ufb) = <* <*((N * ufb) * (1,1))*>, <*((N * ufb) * (2,1))*>, <*((N * ufb) * (3,1))*>*> by A128, BKMODEL1: 30;

            then

             A130: (pb . 1) = <*((N * ufb) * (1,1))*> & (pb . 2) = <*((N * ufb) * (2,1))*> & (pb . 3) = <*((N * ufb) * (3,1))*> by A53, FINSEQ_1: 45;

            (N * MUFBT) is Matrix of 3, 1, F_Real by BKMODEL1: 24;

            then

             A131: ( Indices (N * MUFBT)) = [:( Seg 3), ( Seg 1):] by MATRIX_0: 23;

            ( width N) = 3 by MATRIX_0: 24;

            then

             A132: ( width N) = ( len MUFBT) by MATRIX_0: 23;

            reconsider z1 = 0 , z2 = lub, z3 = 0 as Element of F_Real by XREAL_0:def 1;

            

             A133: ( Col (MUFBT,1)) = <*z1, z2, z3*> by ANPROJ_8: 5;

            

             A134: ( Line (N,1)) = <*na, nb, nc*> & ( Line (N,2)) = <*nd, ne, nf*> & ( Line (N,3)) = <*ng, nh, ni*> by ANPROJ_9: 4, A7;

            ((N * MUFBT) * (1,1)) = (( Line (N,1)) "*" ( Col (MUFBT,1))) & ((N * MUFBT) * (2,1)) = (( Line (N,2)) "*" ( Col (MUFBT,1))) & ((N * MUFBT) * (3,1)) = (( Line (N,3)) "*" ( Col (MUFBT,1))) by A132, MATRIX_3:def 4, A131, ANPROJ_8: 2;

            then ((N * MUFBT) * (1,1)) = (((na * z1) + (nb * z2)) + (nc * z3)) & ((N * MUFBT) * (2,1)) = (((nd * z1) + (ne * z2)) + (nf * z3)) & ((N * MUFBT) * (3,1)) = (((ng * z1) + (nh * z2)) + (ni * z3)) by A133, A134, ANPROJ_8: 7;

            then ((pb . 1) . 1) = (nb * lub) & ((pb . 2) . 1) = (ne * lub) & ((pb . 3) . 1) = (nh * lub) by A129, A130, FINSEQ_1: 40;

            then (((lua2 * lub) * (nb * (( - na) + nc))) + ((lua2 * lub) * (ne * (( - nd) + nf)))) = ((lua2 * lub) * (nh * (( - ng) + ni))) by A127, A114;

            then ((((lua2 * lub) * (nb * (( - na) + nc))) + ((lua2 * lub) * (ne * (( - nd) + nf)))) - ((lua2 * lub) * (nh * (( - ng) + ni)))) = 0 ;

            then ((lua2 * lub) * (((nb * (( - na) + nc)) + (ne * (( - nd) + nf))) - (nh * (( - ng) + ni)))) = 0 ;

            then (((nb * (( - na) + nc)) + (ne * (( - nd) + nf))) - (nh * (( - ng) + ni))) = 0 by A54, A115;

            hence thesis;

          end;

          hence thesis;

        end;

        

         A136: ((nb * (na + nc)) + (ne * (nd + nf))) = (nh * (ng + ni))

        proof

          reconsider r1 = 1, r2 = 0 , r3 = ( - 1) as Element of F_Real by XREAL_0:def 1;

          

           A137: ( M2F pa) = <*((pa . 1) . 1), ((pa . 2) . 1), ((pa . 3) . 1)*> by A51, ANPROJ_8:def 2;

          ( dom ( M2F pa)) = ( Seg 3) by A66, EUCLID_8: 50;

          then (( M2F pa) . 1) in REAL & (( M2F pa) . 2) in REAL & (( M2F pa) . 3) in REAL by FINSEQ_1: 1, FINSEQ_2: 11;

          then

          reconsider s1 = ((pa . 1) . 1), s2 = ((pa . 2) . 1), s3 = ((pa . 3) . 1) as Element of REAL by A137, FINSEQ_1: 45;

          

           A138: ( M2F pb) = <*((pb . 1) . 1), ((pb . 2) . 1), ((pb . 3) . 1)*> by A62, ANPROJ_8:def 2;

          ( dom ( M2F pb)) = ( Seg 3) by A64, EUCLID_8: 50;

          then (( M2F pb) . 1) in REAL & (( M2F pb) . 2) in REAL & (( M2F pb) . 3) in REAL by FINSEQ_1: 1, FINSEQ_2: 11;

          then

          reconsider t1 = ((pb . 1) . 1), t2 = ((pb . 2) . 1), t3 = ((pb . 3) . 1) as Element of F_Real by A138, FINSEQ_1: 45;

          ( M2F pa) = <*s1, s2, s3*> by A51, ANPROJ_8:def 2;

          

          then

           A139: (N2 * ( M2F pa)) = <*(((1 * s1) + ( 0 * s2)) + ( 0 * s3)), ((( 0 * s1) + (1 * s2)) + ( 0 * s3)), ((( 0 * s1) + ( 0 * s2)) + (( - 1) * s3))*> by PASCAL: 9

          .= <*s1, s2, ( - s3)*>;

          ( M2F pb) = <*t1, t2, t3*> by A62, ANPROJ_8:def 2;

          then

           A140: (( M2F pb) . 1) = t1 & (( M2F pb) . 2) = t2 & (( M2F pb) . 3) = t3 & ( <*s1, s2, ( - s3)*> . 1) = s1 & ( <*s1, s2, ( - s3)*> . 2) = s2 & ( <*s1, s2, ( - s3)*> . 3) = ( - s3) by FINSEQ_1: 45;

          

           A141: ( M2F pb) is Element of ( REAL 3) by A63, EUCLID: 22;

          

           A142: |[s1, s2, ( - s3)]| is Element of ( REAL 3) by EUCLID: 22;

           0 = |(( M2F pb), <*s1, s2, ( - s3)*>)| by A139, A80, A45, A58

          .= (((t1 * s1) + (t2 * s2)) + (t3 * ( - s3))) by A141, A142, EUCLID_8: 63, A140;

          then

           A143: ((t1 * s1) + (t2 * s2)) = (t3 * s3);

           |[1, 0 , 1]| is non zero by EUCLID_5: 4, FINSEQ_1: 78;

          then are_Prop (ua, |[1, 0 , 1]|) by A44, ANPROJ_1: 22;

          then

          consider lua be Real such that

           A145: lua <> 0 and

           A146: ua = (lua * |[1, 0 , 1]|) by ANPROJ_1: 1;

          

           A147: ua = |[(lua * 1), (lua * 0 ), (lua * 1)]| by A146, EUCLID_5: 8

          .= |[lua, 0 , lua]|;

          reconsider za1 = lua, za2 = 0 , za3 = lua as Element of F_Real by XREAL_0:def 1;

          lua in REAL by XREAL_0:def 1;

          then

          reconsider MUFA = <*ufa*> as Matrix of 1, 3, F_Real by A147, A44, BKMODEL1: 27;

          now

            ( len ufa) = 3 by A147, A44, FINSEQ_1: 45;

            then ( dom ufa) = {1, 2, 3} by FINSEQ_1:def 3, FINSEQ_3: 1;

            then 1 in ( dom ufa) & 2 in ( dom ufa) & 3 in ( dom ufa) by ENUMSET1:def 1;

            then (MUFA * (1,1)) = ( |[lua, 0 , lua]| . 1) & (MUFA * (1,2)) = ( |[lua, 0 , lua]| . 2) & (MUFA * (1,3)) = ( |[lua, 0 , lua]| . 3) by A147, A44, ANPROJ_8: 70;

            hence (MUFA * (1,1)) = lua & (MUFA * (1,2)) = 0 & (MUFA * (1,3)) = lua by FINSEQ_1: 45;

          end;

          then

           A148: ( <*ufa*> @ ) = <* <*lua*>, <* 0 *>, <*lua*>*> by BKMODEL1: 31;

           0 is Element of F_Real & lua is Element of F_Real by XREAL_0:def 1;

          then

          reconsider MUFAT = <* <*lua*>, <* 0 *>, <*lua*>*> as Matrix of 3, 1, F_Real by BKMODEL1: 28;

          

           A149: (N * MUFAT) is Matrix of 3, 1, F_Real by BKMODEL1: 24;

          

           A150: (N * ufa) = (N * MUFAT) by A148, LAPLACE:def 9;

          (N * ufa) = <* <*((N * ufa) * (1,1))*>, <*((N * ufa) * (2,1))*>, <*((N * ufa) * (3,1))*>*> by A149, A150, BKMODEL1: 30;

          then (pa . 1) = <*((N * ufa) * (1,1))*> & (pa . 2) = <*((N * ufa) * (2,1))*> & (pa . 3) = <*((N * ufa) * (3,1))*> by A44, FINSEQ_1: 45;

          then

           A152: ((pa . 1) . 1) = ((N * MUFAT) * (1,1)) & ((pa . 2) . 1) = ((N * MUFAT) * (2,1)) & ((pa . 3) . 1) = ((N * MUFAT) * (3,1)) by A150, FINSEQ_1: 40;

          (N * MUFAT) is Matrix of 3, 1, F_Real by BKMODEL1: 24;

          then

           A153: ( Indices (N * MUFAT)) = [:( Seg 3), ( Seg 1):] by MATRIX_0: 23;

          ( width N) = 3 by MATRIX_0: 24;

          then

           A154: ( width N) = ( len MUFAT) by MATRIX_0: 23;

          

           A155: ( Col (MUFAT,1)) = <*za1, za2, za3*> by ANPROJ_8: 5;

          

           A156: ( Line (N,1)) = <*na, nb, nc*> & ( Line (N,2)) = <*nd, ne, nf*> & ( Line (N,3)) = <*ng, nh, ni*> by ANPROJ_9: 4, A7;

          ((N * MUFAT) * (1,1)) = (( Line (N,1)) "*" ( Col (MUFAT,1))) & ((N * MUFAT) * (2,1)) = (( Line (N,2)) "*" ( Col (MUFAT,1))) & ((N * MUFAT) * (3,1)) = (( Line (N,3)) "*" ( Col (MUFAT,1))) by A154, MATRIX_3:def 4, A153, ANPROJ_8: 2;

          then

           A157: ((pa . 1) . 1) = (((na * za1) + (nb * za2)) + (nc * za3)) & ((pa . 2) . 1) = (((nd * za1) + (ne * za2)) + (nf * za3)) & ((pa . 3) . 1) = (((ng * za1) + (nh * za2)) + (ni * za3)) by A152, A155, A156, ANPROJ_8: 7;

          reconsider z1 = 0 , z2 = lub, z3 = 0 as Element of F_Real by XREAL_0:def 1;

           0 is Element of F_Real & lub is Element of F_Real by XREAL_0:def 1;

          then

          reconsider MUFBT = <* <* 0 *>, <*lub*>, <* 0 *>*> as Matrix of 3, 1, F_Real by BKMODEL1: 28;

          

           A158: (N * MUFBT) is Matrix of 3, 1, F_Real by BKMODEL1: 24;

          

           A159: (N * ufb) = (N * ( <*ufb*> @ )) by LAPLACE:def 9

          .= (N * MUFBT) by A57, BKMODEL1: 31;

          (N * ufb) = <* <*((N * ufb) * (1,1))*>, <*((N * ufb) * (2,1))*>, <*((N * ufb) * (3,1))*>*> by A158, A159, BKMODEL1: 30;

          then

           A160: (pb . 1) = <*((N * ufb) * (1,1))*> & (pb . 2) = <*((N * ufb) * (2,1))*> & (pb . 3) = <*((N * ufb) * (3,1))*> by A53, FINSEQ_1: 45;

          (N * MUFBT) is Matrix of 3, 1, F_Real by BKMODEL1: 24;

          then

           A161: ( Indices (N * MUFBT)) = [:( Seg 3), ( Seg 1):] by MATRIX_0: 23;

          ( width N) = 3 by MATRIX_0: 24;

          then

           A162: ( width N) = ( len MUFBT) by MATRIX_0: 23;

          reconsider z1 = 0 , z2 = lub, z3 = 0 as Element of F_Real by XREAL_0:def 1;

          

           A163: ( Col (MUFBT,1)) = <*z1, z2, z3*> by ANPROJ_8: 5;

          

           A164: ( Line (N,1)) = <*na, nb, nc*> & ( Line (N,2)) = <*nd, ne, nf*> & ( Line (N,3)) = <*ng, nh, ni*> by ANPROJ_9: 4, A7;

          ((N * MUFBT) * (1,1)) = (( Line (N,1)) "*" ( Col (MUFBT,1))) & ((N * MUFBT) * (2,1)) = (( Line (N,2)) "*" ( Col (MUFBT,1))) & ((N * MUFBT) * (3,1)) = (( Line (N,3)) "*" ( Col (MUFBT,1))) by A162, MATRIX_3:def 4, A161, ANPROJ_8: 2;

          then ((N * MUFBT) * (1,1)) = (((na * z1) + (nb * z2)) + (nc * z3)) & ((N * MUFBT) * (2,1)) = (((nd * z1) + (ne * z2)) + (nf * z3)) & ((N * MUFBT) * (3,1)) = (((ng * z1) + (nh * z2)) + (ni * z3)) by A163, A164, ANPROJ_8: 7;

          then ((pb . 1) . 1) = (nb * lub) & ((pb . 2) . 1) = (ne * lub) & ((pb . 3) . 1) = (nh * lub) by A160, FINSEQ_1: 40, A159;

          then (((lua * lub) * (nb * (na + nc))) + ((lua * lub) * (ne * (nd + nf)))) = ((lua * lub) * (nh * (ng + ni))) by A157, A143;

          then ((((lua * lub) * (nb * (na + nc))) + ((lua * lub) * (ne * (nd + nf)))) - ((lua * lub) * (nh * (ng + ni)))) = 0 ;

          then ((lua * lub) * (((nb * (na + nc)) + (ne * (nd + nf))) - (nh * (ng + ni)))) = 0 ;

          then (((nb * (na + nc)) + (ne * (nd + nf))) - (nh * (ng + ni))) = 0 by A54, A145;

          hence thesis;

        end;

         <* <*na, nb, nc*>, <*nd, ne, nf*>, <*ng, nh, ni*>*> = <* <*(N * (1,1)), (N * (1,2)), (N * (1,3))*>, <*(N * (2,1)), (N * (2,2)), (N * (2,3))*>, <*(N * (3,1)), (N * (3,2)), (N * (3,3))*>*> by A7, MATRIXR2: 37;

        then

         A166: na = (N * (1,1)) & nb = (N * (1,2)) & nc = (N * (1,3)) & nd = (N * (2,1)) & ne = (N * (2,2)) & nf = (N * (2,3)) & ng = (N * (3,1)) & nh = (N * (3,2)) & ni = (N * (3,3)) by PASCAL: 2;

        ( width N) > 0 by MATRIX_0: 23;

        then ( len N) = 3 & ( len N1) = 3 & ( width N1) = 3 & ( width (N @ )) = ( len N) by MATRIX_0: 29, MATRIX_0: 23;

        

        then

         A167: <* <*ra, rb, rc*>, <*rb, re, rf*>, <*rc, rf, ri*>*> = ((N @ ) * (N1 * N)) by A12, A11, MATRIX_3: 33

        .= <* <*(((N @ ) * (N1 * N)) * (1,1)), (((N @ ) * (N1 * N)) * (1,2)), (((N @ ) * (N1 * N)) * (1,3))*>, <*(((N @ ) * (N1 * N)) * (2,1)), (((N @ ) * (N1 * N)) * (2,2)), (((N @ ) * (N1 * N)) * (2,3))*>, <*(((N @ ) * (N1 * N)) * (3,1)), (((N @ ) * (N1 * N)) * (3,2)), (((N @ ) * (N1 * N)) * (3,3))*>*> by MATRIXR2: 37;

        

         A168: (((N @ ) * (N1 * N)) * (1,1)) = ((((a * (N * (1,1))) * (N * (1,1))) + ((a * (N * (2,1))) * (N * (2,1)))) + ((b * (N * (3,1))) * (N * (3,1)))) & (((N @ ) * (N1 * N)) * (1,2)) = ((((a * (N * (1,1))) * (N * (1,2))) + ((a * (N * (2,1))) * (N * (2,2)))) + ((b * (N * (3,1))) * (N * (3,2)))) & (((N @ ) * (N1 * N)) * (1,3)) = ((((a * (N * (1,1))) * (N * (1,3))) + ((a * (N * (2,1))) * (N * (2,3)))) + ((b * (N * (3,1))) * (N * (3,3)))) & (((N @ ) * (N1 * N)) * (2,1)) = ((((a * (N * (1,2))) * (N * (1,1))) + ((a * (N * (2,2))) * (N * (2,1)))) + ((b * (N * (3,2))) * (N * (3,1)))) & (((N @ ) * (N1 * N)) * (2,2)) = ((((a * (N * (1,2))) * (N * (1,2))) + ((a * (N * (2,2))) * (N * (2,2)))) + ((b * (N * (3,2))) * (N * (3,2)))) & (((N @ ) * (N1 * N)) * (2,3)) = ((((a * (N * (1,2))) * (N * (1,3))) + ((a * (N * (2,2))) * (N * (2,3)))) + ((b * (N * (3,2))) * (N * (3,3)))) & (((N @ ) * (N1 * N)) * (3,1)) = ((((a * (N * (1,3))) * (N * (1,1))) + ((a * (N * (2,3))) * (N * (2,1)))) + ((b * (N * (3,3))) * (N * (3,1)))) & (((N @ ) * (N1 * N)) * (3,2)) = ((((a * (N * (1,3))) * (N * (1,2))) + ((a * (N * (2,3))) * (N * (2,2)))) + ((b * (N * (3,3))) * (N * (3,2)))) & (((N @ ) * (N1 * N)) * (3,3)) = ((((a * (N * (1,3))) * (N * (1,3))) + ((a * (N * (2,3))) * (N * (2,3)))) + ((b * (N * (3,3))) * (N * (3,3)))) by BKMODEL1: 23;

        

         A169: ra = (((na * na) + (nd * nd)) - (ng * ng)) & rb = (((na * nb) + (nd * ne)) - (ng * nh)) & rc = (((na * nc) + (nd * nf)) - (ng * ni)) & rb = (((na * nb) + (nd * ne)) - (ng * nh)) & re = (((nb * nb) + (ne * ne)) - (nh * nh)) & rf = (((nb * nc) + (ne * nf)) - (nh * ni)) & rc = (((na * nc) + (nd * nf)) - (ng * ni)) & rf = (((nb * nc) + (ne * nf)) - (nh * ni)) & ri = (((nc * nc) + (nf * nf)) - (ni * ni)) by A166, A167, PASCAL: 2, A168;

        

         A170: ((((nb * na) + (nb * nc)) + (ne * nd)) + (ne * nf)) = ((nh * ng) + (nh * ni)) by A136;

        (((( - (nb * na)) + (nb * nc)) + ( - (ne * nd))) + (ne * nf)) = (( - (nh * ng)) + (nh * ni)) by A81;

        hence thesis by A170, A169, A22, A29, A35;

      end;

      then

       A170: M = ( symmetric_3 (ra,ra,( - ra), 0 , 0 , 0 )) by A12, A11, PASCAL:def 3;

      

       A171: ra <> 0

      proof

        assume ra = 0 ;

        then ( Det M) = ( 0. F_Real ) by A170, BKMODEL1: 22;

        hence contradiction by LAPLACE: 34;

      end;

      then

       A172: (( homography M) .: absolute ) = absolute by A170, Th29;

      take N;

      thus thesis by A9, A171, A172, A170, A3, A4, A5, A6, BKMODEL1: 93;

    end;

    theorem :: BKMODEL2:49

    

     Th38: for p1,q1,r1,p2,q2,r2 be Element of absolute holds for s1,s2 be Element of real_projective_plane st (p1,q1,r1) are_mutually_distinct & (p2,q2,r2) are_mutually_distinct & s1 in (( tangent p1) /\ ( tangent q1)) & s2 in (( tangent p2) /\ ( tangent q2)) holds ex N be invertible Matrix of 3, F_Real st (( homography N) .: absolute ) = absolute & (( homography N) . p1) = p2 & (( homography N) . q1) = q2 & (( homography N) . r1) = r2 & (( homography N) . s1) = s2

    proof

      let p1,q1,r1,p2,q2,r2 be Element of absolute ;

      let s1,s2 be Element of real_projective_plane ;

      assume that

       A1: (p1,q1,r1) are_mutually_distinct and

       A2: (p2,q2,r2) are_mutually_distinct and

       A3: s1 in (( tangent p1) /\ ( tangent q1)) and

       A4: s2 in (( tangent p2) /\ ( tangent q2));

      consider N1 be invertible Matrix of 3, F_Real such that

       A5: (( homography N1) .: absolute ) = absolute & (( homography N1) . Dir101 ) = p1 & (( homography N1) . Dirm101 ) = q1 & (( homography N1) . Dir011 ) = r1 & (( homography N1) . Dir010 ) = s1 by A1, A3, Th37;

      consider N2 be invertible Matrix of 3, F_Real such that

       A7: (( homography N2) .: absolute ) = absolute & (( homography N2) . Dir101 ) = p2 & (( homography N2) . Dirm101 ) = q2 & (( homography N2) . Dir011 ) = r2 & (( homography N2) . Dir010 ) = s2 by A2, A4, Th37;

      reconsider N = (N2 * (N1 ~ )) as invertible Matrix of 3, F_Real ;

      

       A20: (( homography N) . p1) = (( homography N2) . (( homography (N1 ~ )) . p1)) by ANPROJ_9: 13

      .= p2 by A5, A7, ANPROJ_9: 15;

      

       A21: (( homography N) . q1) = (( homography N2) . (( homography (N1 ~ )) . q1)) by ANPROJ_9: 13

      .= q2 by A5, A7, ANPROJ_9: 15;

      

       A22: (( homography N) . r1) = (( homography N2) . (( homography (N1 ~ )) . r1)) by ANPROJ_9: 13

      .= r2 by A5, A7, ANPROJ_9: 15;

      

       A23: (( homography N) . s1) = (( homography N2) . (( homography (N1 ~ )) . s1)) by ANPROJ_9: 13

      .= s2 by A5, A7, ANPROJ_9: 15;

      ( homography N1) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h1 = ( homography N1) as Element of EnsHomography3 ;

      h1 is_K-isometry by A5;

      then h1 in EnsK-isometry ;

      then

      reconsider hsg1 = h1 as Element of SubGroupK-isometry by Def05;

      ( homography N2) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h2 = ( homography N2) as Element of EnsHomography3 ;

      h2 is_K-isometry by A7;

      then h2 in EnsK-isometry ;

      then

      reconsider hsg2 = h2 as Element of SubGroupK-isometry by Def05;

      ( homography (N1 ~ )) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h3 = ( homography (N1 ~ )) as Element of EnsHomography3 ;

      

       A24: (hsg1 " ) = h3 by Th36;

      set H = EnsK-isometry , G = GroupHomography3 ;

      reconsider hg1 = hsg1, hg2 = hsg2, hg3 = (hsg1 " ) as Element of G by A24, ANPROJ_9:def 4;

      reconsider hsg3 = h3 as Element of SubGroupK-isometry by A24;

      reconsider h4 = (hsg2 * hsg3) as Element of SubGroupK-isometry ;

      

       A25: h4 = (hg2 * hg3) by A24, GROUP_2: 43

      .= (h2 (*) h3) by A24, ANPROJ_9:def 3, ANPROJ_9:def 4

      .= ( homography N) by ANPROJ_9: 18;

      h4 in the carrier of SubGroupK-isometry ;

      then h4 in EnsK-isometry by Def05;

      then ex h be Element of EnsHomography3 st h4 = h & h is_K-isometry ;

      hence thesis by A20, A21, A22, A23, A25;

    end;

    theorem :: BKMODEL2:50

    for p1,q1,r1,p2,q2,r2 be Element of absolute st (p1,q1,r1) are_mutually_distinct & (p2,q2,r2) are_mutually_distinct holds ex N be invertible Matrix of 3, F_Real st (( homography N) .: absolute ) = absolute & (( homography N) . p1) = p2 & (( homography N) . q1) = q2 & (( homography N) . r1) = r2

    proof

      let p1,q1,r1,p2,q2,r2 be Element of absolute ;

      assume that

       A1: (p1,q1,r1) are_mutually_distinct and

       A2: (p2,q2,r2) are_mutually_distinct ;

      consider t be Element of real_projective_plane such that

       A3: (( tangent p1) /\ ( tangent q1)) = {t} by A1, Th25;

      t in (( tangent p1) /\ ( tangent q1)) by A3, TARSKI:def 1;

      then

      consider N1 be invertible Matrix of 3, F_Real such that

       A5: (( homography N1) .: absolute ) = absolute & (( homography N1) . Dir101 ) = p1 & (( homography N1) . Dirm101 ) = q1 & (( homography N1) . Dir011 ) = r1 & (( homography N1) . Dir010 ) = t by A1, Th37;

      consider u be Element of real_projective_plane such that

       A6: (( tangent q2) /\ ( tangent p2)) = {u} by A2, Th25;

      u in (( tangent p2) /\ ( tangent q2)) by A6, TARSKI:def 1;

      then

      consider N2 be invertible Matrix of 3, F_Real such that

       A7: (( homography N2) .: absolute ) = absolute & (( homography N2) . Dir101 ) = p2 & (( homography N2) . Dirm101 ) = q2 & (( homography N2) . Dir011 ) = r2 & (( homography N2) . Dir010 ) = u by A2, Th37;

      reconsider N = (N2 * (N1 ~ )) as invertible Matrix of 3, F_Real ;

      

       A20: (( homography N) . p1) = (( homography N2) . (( homography (N1 ~ )) . p1)) by ANPROJ_9: 13

      .= p2 by A5, A7, ANPROJ_9: 15;

      

       A21: (( homography N) . q1) = (( homography N2) . (( homography (N1 ~ )) . q1)) by ANPROJ_9: 13

      .= q2 by A5, A7, ANPROJ_9: 15;

      

       A22: (( homography N) . r1) = (( homography N2) . (( homography (N1 ~ )) . r1)) by ANPROJ_9: 13

      .= r2 by A5, A7, ANPROJ_9: 15;

      ( homography N1) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h1 = ( homography N1) as Element of EnsHomography3 ;

      h1 is_K-isometry by A5;

      then h1 in EnsK-isometry ;

      then

      reconsider hsg1 = h1 as Element of SubGroupK-isometry by Def05;

      ( homography N2) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h2 = ( homography N2) as Element of EnsHomography3 ;

      h2 is_K-isometry by A7;

      then h2 in EnsK-isometry ;

      then

      reconsider hsg2 = h2 as Element of SubGroupK-isometry by Def05;

      ( homography (N1 ~ )) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h3 = ( homography (N1 ~ )) as Element of EnsHomography3 ;

      

       A24: (hsg1 " ) = h3 by Th36;

      set H = EnsK-isometry , G = GroupHomography3 ;

      reconsider hg1 = hsg1, hg2 = hsg2, hg3 = (hsg1 " ) as Element of G by A24, ANPROJ_9:def 4;

      reconsider hsg3 = h3 as Element of SubGroupK-isometry by A24;

      reconsider h4 = (hsg2 * hsg3) as Element of SubGroupK-isometry ;

      

       A25: h4 = (hg2 * hg3) by A24, GROUP_2: 43

      .= (h2 (*) h3) by A24, ANPROJ_9:def 3, ANPROJ_9:def 4

      .= ( homography N) by ANPROJ_9: 18;

      h4 in the carrier of SubGroupK-isometry ;

      then h4 in EnsK-isometry by Def05;

      then ex h be Element of EnsHomography3 st h4 = h & h is_K-isometry ;

      hence thesis by A20, A21, A22, A25;

    end;

    theorem :: BKMODEL2:51

    

     Th39: for CLSP be CollSp holds for p,q,r,s be Element of CLSP st ( Line (p,q)) = ( Line (r,s)) holds (r,s,p) are_collinear by COLLSP: 10, COLLSP: 11;

    theorem :: BKMODEL2:52

    

     Th40: for CLSP be CollSp holds for p,q be Element of CLSP holds ( Line (p,q)) = ( Line (q,p))

    proof

      let CLSP be CollSp;

      let p,q be Element of CLSP;

      

       A1: ( Line (p,q)) c= ( Line (q,p))

      proof

        let x be object;

        assume x in ( Line (p,q));

        then x in { y where y be Element of CLSP : (p,q,y) are_collinear } by COLLSP:def 5;

        then

        consider y be Element of CLSP such that

         A2: y = x and

         A3: (p,q,y) are_collinear ;

        (q,p,y) are_collinear by A3, COLLSP: 4;

        then y in { y where y be Element of CLSP : (q,p,y) are_collinear };

        hence thesis by A2, COLLSP:def 5;

      end;

      ( Line (q,p)) c= ( Line (p,q))

      proof

        let x be object;

        assume x in ( Line (q,p));

        then x in { y where y be Element of CLSP : (q,p,y) are_collinear } by COLLSP:def 5;

        then

        consider y be Element of CLSP such that

         A4: y = x and

         A5: (q,p,y) are_collinear ;

        (p,q,y) are_collinear by A5, COLLSP: 4;

        then y in { y where y be Element of CLSP : (p,q,y) are_collinear };

        hence thesis by A4, COLLSP:def 5;

      end;

      hence thesis by A1;

    end;

    theorem :: BKMODEL2:53

    

     Th41: for N be invertible Matrix of 3, F_Real holds for p,q,r,s be Element of ( ProjectiveSpace ( TOP-REAL 3)) st ( Line ((( homography N) . p),(( homography N) . q))) = ( Line ((( homography N) . r),(( homography N) . s))) holds (p,q,r) are_collinear & (p,q,s) are_collinear & (r,s,p) are_collinear & (r,s,q) are_collinear

    proof

      let N be invertible Matrix of 3, F_Real ;

      let p,q,r,s be Element of ( ProjectiveSpace ( TOP-REAL 3));

      assume

       A1: ( Line ((( homography N) . p),(( homography N) . q))) = ( Line ((( homography N) . r),(( homography N) . s)));

      hence (p,q,r) are_collinear by ANPROJ_8: 102, Th39;

      ( Line ((( homography N) . p),(( homography N) . q))) = ( Line ((( homography N) . s),(( homography N) . r))) by A1, Th40;

      hence (p,q,s) are_collinear by ANPROJ_8: 102, Th39;

      thus (r,s,p) are_collinear by A1, ANPROJ_8: 102, Th39;

      ( Line ((( homography N) . q),(( homography N) . p))) = ( Line ((( homography N) . r),(( homography N) . s))) by A1, Th40;

      hence (r,s,q) are_collinear by ANPROJ_8: 102, Th39;

    end;

    theorem :: BKMODEL2:54

    

     Th42: for N be invertible Matrix of 3, F_Real holds for p,q,r,s,t,u,np,nq,nr,ns be Element of real_projective_plane st p <> q & r <> s & np <> nq & nr <> ns & (p,q,t) are_collinear & (r,s,t) are_collinear & np = (( homography N) . p) & nq = (( homography N) . q) & nr = (( homography N) . r) & ns = (( homography N) . s) & (np,nq,u) are_collinear & (nr,ns,u) are_collinear holds u = (( homography N) . t) or ( Line (np,nq)) = ( Line (nr,ns))

    proof

      let N be invertible Matrix of 3, F_Real ;

      let p,q,r,s,t,u,np,nq,nr,ns be Element of real_projective_plane ;

      assume that

       A0: p <> q & r <> s & np <> nq & nr <> ns and

       A1: (p,q,t) are_collinear and

       A2: (r,s,t) are_collinear and

       A3: np = (( homography N) . p) and

       A4: nq = (( homography N) . q) and

       A5: nr = (( homography N) . r) and

       A6: ns = (( homography N) . s) and

       A7: (np,nq,u) are_collinear and

       A8: (nr,ns,u) are_collinear ;

      

       A9: t in ( Line (p,q)) & t in ( Line (r,s)) & u in ( Line (np,nq)) & u in ( Line (nr,ns)) by A1, A2, A7, A8, COLLSP: 11;

      reconsider L1 = ( Line (p,q)), L2 = ( Line (r,s)), L3 = ( Line (np,nq)), L4 = ( Line (nr,ns)) as LINE of real_projective_plane by A0, COLLSP:def 7;

      reconsider LL1 = L1, LL2 = L2, LL3 = L3, LL4 = L4 as LINE of ( IncProjSp_of real_projective_plane ) by INCPROJ: 4;

      reconsider t9 = t, u9 = u as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 3;

      

       A10: t9 on LL1 & t9 on LL2 & u9 on LL3 & u9 on LL4 by A9, INCPROJ: 5;

      reconsider nt = (( homography N) . t) as Element of real_projective_plane by FUNCT_2: 5;

      

       A11: nt in ( Line (np,nq)) & nt in ( Line (nr,ns)) by A1, A2, A3, A4, A5, A6, ANPROJ_8: 102, COLLSP: 11;

      reconsider nt9 = nt as POINT of ( IncProjSp_of real_projective_plane ) by INCPROJ: 3;

      nt9 on LL3 & nt9 on LL4 by A11, INCPROJ: 5;

      hence thesis by A10, INCPROJ: 8;

    end;

    theorem :: BKMODEL2:55

    

     Th43: for N be invertible Matrix of 3, F_Real holds for p,q,r,s,t,u,np,nq,nr,ns be Element of real_projective_plane st p <> q & r <> s & np <> nq & nr <> ns & (p,q,t) are_collinear & (r,s,t) are_collinear & np = (( homography N) . p) & nq = (( homography N) . q) & nr = (( homography N) . r) & ns = (( homography N) . s) & (np,nq,u) are_collinear & (nr,ns,u) are_collinear & not (p,q,r) are_collinear holds u = (( homography N) . t)

    proof

      let N be invertible Matrix of 3, F_Real ;

      let p,q,r,s,t,u,np,nq,nr,ns be Element of real_projective_plane ;

      assume that

       A1: p <> q & r <> s & np <> nq & nr <> ns & (p,q,t) are_collinear & (r,s,t) are_collinear & np = (( homography N) . p) & nq = (( homography N) . q) & nr = (( homography N) . r) & ns = (( homography N) . s) & (np,nq,u) are_collinear & (nr,ns,u) are_collinear and

       A2: not (p,q,r) are_collinear ;

      u = (( homography N) . t) or ( Line (np,nq)) = ( Line (nr,ns)) by A1, Th42;

      hence thesis by A1, A2, Th41;

    end;

    theorem :: BKMODEL2:56

    for p,q be Element of absolute holds for a,b be Element of BK_model holds ex N be invertible Matrix of 3, F_Real st (( homography N) .: absolute ) = absolute & (( homography N) . a) = b & (( homography N) . p) = q

    proof

      let p,q be Element of absolute ;

      let a,b be Element of BK_model ;

      consider p9 be Element of absolute such that

       A1: p <> p9 and

       A2: (p,a,p9) are_collinear by Th16;

      consider q9 be Element of absolute such that

       A3: q <> q9 and

       A4: (q,b,q9) are_collinear by Th16;

      consider t be Element of real_projective_plane such that

       A5: (( tangent p) /\ ( tangent p9)) = {t} by A1, Th25;

      

       A6: t in (( tangent p) /\ ( tangent p9)) by A5, TARSKI:def 1;

      consider u be Element of real_projective_plane such that

       A7: (( tangent q) /\ ( tangent q9)) = {u} by A3, Th25;

      

       A8: u in (( tangent q) /\ ( tangent q9)) by A7, TARSKI:def 1;

      reconsider a9 = a as Element of real_projective_plane ;

      p <> p9 & a in BK_model & (a,p,p9) are_collinear & t in ( tangent p) & t in ( tangent p9) by A1, A2, A6, XBOOLE_0:def 4, COLLSP: 4;

      then

      consider Ra be Element of real_projective_plane such that

       A9: Ra in absolute and

       A10: (a9,t,Ra) are_collinear by Th31;

      reconsider RRa = Ra as Element of absolute by A9;

      reconsider b9 = b as Element of real_projective_plane ;

      q <> q9 & b in BK_model & (b,q,q9) are_collinear & u in ( tangent q) & u in ( tangent q9) by A3, A4, A8, XBOOLE_0:def 4, COLLSP: 4;

      then

      consider Rb be Element of real_projective_plane such that

       A11: Rb in absolute and

       A12: (b9,u,Rb) are_collinear by Th31;

      reconsider RRb = Rb as Element of absolute by A11;

      

       A13: (p,p9,Ra) are_mutually_distinct

      proof

        now

          consider ra be Element of real_projective_plane such that

           A14: ra = Ra & ( tangent RRa) = ( Line (ra,( pole_infty RRa))) by Def04;

          thus p <> Ra

          proof

            assume p = Ra;

            then t in ( Line (ra,( pole_infty RRa))) by A14, A6, XBOOLE_0:def 4;

            then (ra,( pole_infty RRa),t) are_collinear by COLLSP: 11;

            then

             A15: (ra,t,( pole_infty RRa)) are_collinear by COLLSP: 4;

            

             A16: (ra,t,a9) are_collinear by A14, A10, HESSENBE: 1;

            ra <> t

            proof

              assume ra = t;

              then t in absolute & t in ( tangent p) & t in ( tangent p9) by A14, A6, XBOOLE_0:def 4;

              then t in (( tangent p) /\ absolute ) & t in (( tangent p9) /\ absolute ) by XBOOLE_0:def 4;

              then t in {p} & t in {p9} by Th22;

              then t = p & t = p9 by TARSKI:def 1;

              hence contradiction by A1;

            end;

            then a in ( tangent RRa) & a in BK_model by A16, A15, A14, COLLSP: 6, COLLSP: 11;

            then ( tangent RRa) meets BK_model by XBOOLE_0:def 4;

            hence contradiction by Th30;

          end;

          thus p9 <> Ra

          proof

            assume p9 = Ra;

            then t in ( Line (ra,( pole_infty RRa))) by A14, A6, XBOOLE_0:def 4;

            then (ra,( pole_infty RRa),t) are_collinear by COLLSP: 11;

            then

             A17: (ra,t,( pole_infty RRa)) are_collinear by COLLSP: 4;

            

             A18: (ra,t,a9) are_collinear by A14, A10, HESSENBE: 1;

            ra <> t

            proof

              assume ra = t;

              then t in absolute & t in ( tangent p) & t in ( tangent p9) by A14, A6, XBOOLE_0:def 4;

              then t in (( tangent p) /\ absolute ) & t in (( tangent p9) /\ absolute ) by XBOOLE_0:def 4;

              then t in {p} & t in {p9} by Th22;

              then t = p & t = p9 by TARSKI:def 1;

              hence contradiction by A1;

            end;

            then a in ( tangent RRa) & a in BK_model by A18, A17, A14, COLLSP: 6, COLLSP: 11;

            then ( tangent RRa) meets BK_model by XBOOLE_0:def 4;

            hence contradiction by Th30;

          end;

        end;

        hence thesis by A1;

      end;

      now

        now

          consider rb be Element of real_projective_plane such that

           A19: rb = Rb & ( tangent RRb) = ( Line (rb,( pole_infty RRb))) by Def04;

          thus q <> Rb

          proof

            assume q = Rb;

            then u in ( Line (rb,( pole_infty RRb))) by A19, A8, XBOOLE_0:def 4;

            then (rb,( pole_infty RRb),u) are_collinear by COLLSP: 11;

            then

             A20: (rb,u,( pole_infty RRb)) are_collinear by COLLSP: 4;

            

             A21: (rb,u,b9) are_collinear by A19, A12, HESSENBE: 1;

            rb <> u

            proof

              assume rb = u;

              then u in absolute & u in ( tangent q) & u in ( tangent q9) by A19, A8, XBOOLE_0:def 4;

              then u in (( tangent q) /\ absolute ) & u in (( tangent q9) /\ absolute ) by XBOOLE_0:def 4;

              then u in {q} & u in {q9} by Th22;

              then u = q & u = q9 by TARSKI:def 1;

              hence contradiction by A3;

            end;

            then b in ( tangent RRb) & b in BK_model by A21, A20, A19, COLLSP: 6, COLLSP: 11;

            then ( tangent RRb) meets BK_model by XBOOLE_0:def 4;

            hence contradiction by Th30;

          end;

          thus q9 <> Rb

          proof

            assume q9 = Rb;

            then u in ( Line (rb,( pole_infty RRb))) by A19, A8, XBOOLE_0:def 4;

            then (rb,( pole_infty RRb),u) are_collinear by COLLSP: 11;

            then

             A22: (rb,u,( pole_infty RRb)) are_collinear by COLLSP: 4;

            

             A23: (rb,u,b9) are_collinear by A19, A12, HESSENBE: 1;

            rb <> u

            proof

              assume rb = u;

              then u in absolute & u in ( tangent q) & u in ( tangent q9) by A19, A8, XBOOLE_0:def 4;

              then u in (( tangent q) /\ absolute ) & u in (( tangent q9) /\ absolute ) by XBOOLE_0:def 4;

              then u in {q} & u in {q9} by Th22;

              then u = q & u = q9 by TARSKI:def 1;

              hence contradiction by A3;

            end;

            then b in ( tangent RRb) & b in BK_model by A23, A22, A19, COLLSP: 6, COLLSP: 11;

            then ( tangent RRb) meets BK_model by XBOOLE_0:def 4;

            hence contradiction by Th30;

          end;

        end;

        hence (q,q9,Rb) are_mutually_distinct by A3;

      end;

      then

      consider N be invertible Matrix of 3, F_Real such that

       A24: (( homography N) .: absolute ) = absolute and

       A25: (( homography N) . p) = q and

       A26: (( homography N) . p9) = q9 and

       A27: (( homography N) . Ra) = Rb and

       A28: (( homography N) . t) = u by A9, A11, A6, A8, A13, Th38;

      reconsider plp = p, plq = p9, plr = Ra, pls = t, plt = a, np = q, nq = q9, nr = Rb, ns = u, nu = b as Element of real_projective_plane ;

      now

        thus plp <> plq by A1;

        thus np <> nq by A3;

        thus nr <> ns

        proof

          consider rb be Element of real_projective_plane such that

           A29: rb = Rb & ( tangent RRb) = ( Line (rb,( pole_infty RRb))) by Def04;

          rb <> u

          proof

            assume rb = u;

            then u in absolute & u in ( tangent q) & u in ( tangent q9) by A29, A8, XBOOLE_0:def 4;

            then u in (( tangent q) /\ absolute ) & u in (( tangent q9) /\ absolute ) by XBOOLE_0:def 4;

            then u in {q} & u in {q9} by Th22;

            then u = q & u = q9 by TARSKI:def 1;

            hence contradiction by A3;

          end;

          hence thesis by A29;

        end;

        thus plr <> pls

        proof

          consider ra be Element of real_projective_plane such that

           A30: ra = Ra & ( tangent RRa) = ( Line (ra,( pole_infty RRa))) by Def04;

          ra <> t

          proof

            assume ra = t;

            then t in absolute & t in ( tangent p) & t in ( tangent p9) by A30, A6, XBOOLE_0:def 4;

            then t in (( tangent p) /\ absolute ) & t in (( tangent p9) /\ absolute ) by XBOOLE_0:def 4;

            then t in {p} & t in {p9} by Th22;

            then t = p & t = p9 by TARSKI:def 1;

            hence contradiction by A1;

          end;

          hence thesis by A30;

        end;

        thus (plp,plq,plt) are_collinear by A2, COLLSP: 4;

        thus (plr,pls,plt) are_collinear by A10, HESSENBE: 1;

        thus np = (( homography N) . plp) & nq = (( homography N) . plq) & nr = (( homography N) . plr) & ns = (( homography N) . pls) by A25, A26, A27, A28;

        thus (np,nq,nu) are_collinear by A4, HESSENBE: 1;

        thus (nr,ns,nu) are_collinear by A12, HESSENBE: 1;

        thus not (plp,plq,plr) are_collinear

        proof

          assume (plp,plq,plr) are_collinear ;

          then (p,p9,RRa) are_collinear ;

          hence contradiction by A13, BKMODEL1: 92;

        end;

      end;

      then nu = (( homography N) . plt) by Th43;

      hence thesis by A24, A25;

    end;

    theorem :: BKMODEL2:57

    for p,q,r,s be Element of absolute st (p,q,r) are_mutually_distinct & (q,p,s) are_mutually_distinct holds ex N be invertible Matrix of 3, F_Real st (( homography N) .: absolute ) = absolute & (( homography N) . p) = q & (( homography N) . q) = p & (( homography N) . r) = s & (for t be Element of real_projective_plane st t in (( tangent p) /\ ( tangent q)) holds (( homography N) . t) = t)

    proof

      let p,q,r,s be Element of absolute ;

      assume that

       A1: (p,q,r) are_mutually_distinct and

       A2: (q,p,s) are_mutually_distinct ;

      consider t be Element of real_projective_plane such that

       A3: (( tangent p) /\ ( tangent q)) = {t} by A1, Th25;

      

       A4: t in (( tangent p) /\ ( tangent q)) by A3, TARSKI:def 1;

      then

      consider N1 be invertible Matrix of 3, F_Real such that

       A5: (( homography N1) .: absolute ) = absolute & (( homography N1) . Dir101 ) = p & (( homography N1) . Dirm101 ) = q & (( homography N1) . Dir011 ) = r & (( homography N1) . Dir010 ) = t by A1, Th37;

      consider N2 be invertible Matrix of 3, F_Real such that

       A7: (( homography N2) .: absolute ) = absolute & (( homography N2) . Dir101 ) = q & (( homography N2) . Dirm101 ) = p & (( homography N2) . Dir011 ) = s & (( homography N2) . Dir010 ) = t by A2, A4, Th37;

      reconsider N = (N2 * (N1 ~ )) as invertible Matrix of 3, F_Real ;

      

       A20: (( homography N) . p) = (( homography N2) . (( homography (N1 ~ )) . p)) by ANPROJ_9: 13

      .= q by A5, A7, ANPROJ_9: 15;

      

       A21: (( homography N) . q) = (( homography N2) . (( homography (N1 ~ )) . q)) by ANPROJ_9: 13

      .= p by A5, A7, ANPROJ_9: 15;

      

       A22: (( homography N) . r) = (( homography N2) . (( homography (N1 ~ )) . r)) by ANPROJ_9: 13

      .= s by A5, A7, ANPROJ_9: 15;

      

       A23: (( homography N) . t) = (( homography N2) . (( homography (N1 ~ )) . t)) by ANPROJ_9: 13

      .= t by A5, A7, ANPROJ_9: 15;

      ( homography N1) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h1 = ( homography N1) as Element of EnsHomography3 ;

      h1 is_K-isometry by A5;

      then h1 in EnsK-isometry ;

      then

      reconsider hsg1 = h1 as Element of SubGroupK-isometry by Def05;

      ( homography N2) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h2 = ( homography N2) as Element of EnsHomography3 ;

      h2 is_K-isometry by A7;

      then h2 in EnsK-isometry ;

      then

      reconsider hsg2 = h2 as Element of SubGroupK-isometry by Def05;

      ( homography (N1 ~ )) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h3 = ( homography (N1 ~ )) as Element of EnsHomography3 ;

      

       A24: (hsg1 " ) = h3 by Th36;

      set H = EnsK-isometry , G = GroupHomography3 ;

      reconsider hg1 = hsg1, hg2 = hsg2, hg3 = (hsg1 " ) as Element of G by A24, ANPROJ_9:def 4;

      reconsider hsg3 = h3 as Element of SubGroupK-isometry by A24;

      reconsider h4 = (hsg2 * hsg3) as Element of SubGroupK-isometry ;

      

       A25: h4 = (hg2 * hg3) by A24, GROUP_2: 43

      .= (h2 (*) h3) by A24, ANPROJ_9:def 3, ANPROJ_9:def 4

      .= ( homography N) by ANPROJ_9: 18;

      h4 in the carrier of SubGroupK-isometry ;

      then h4 in EnsK-isometry by Def05;

      then

      consider h be Element of EnsHomography3 such that

       A26: h4 = h and

       A27: h is_K-isometry ;

      take N;

      thus (( homography N) .: absolute ) = absolute by A25, A26, A27;

      thus (( homography N) . p) = q & (( homography N) . q) = p & (( homography N) . r) = s by A20, A21, A22;

      thus for t be Element of real_projective_plane st t in (( tangent p) /\ ( tangent q)) holds (( homography N) . t) = t

      proof

        let v be Element of real_projective_plane ;

        assume v in (( tangent p) /\ ( tangent q));

        then v = t by A3, TARSKI:def 1;

        hence thesis by A23;

      end;

    end;

    theorem :: BKMODEL2:58

    

     Th44: for P,Q be Element of BK_model st P <> Q holds ex P1,P2,P3,P4 be Element of absolute , P5 be Element of ( ProjectiveSpace ( TOP-REAL 3)) st P1 <> P2 & (P,Q,P1) are_collinear & (P,Q,P2) are_collinear & (P,P5,P3) are_collinear & (Q,P5,P4) are_collinear & (P1,P2,P3) are_mutually_distinct & (P1,P2,P4) are_mutually_distinct & P5 in (( tangent P1) /\ ( tangent P2))

    proof

      let P,Q be Element of BK_model ;

      assume

       A1: P <> Q;

      then

      consider P1,P2 be Element of absolute such that

       A2: P1 <> P2 and

       A3: (P,Q,P1) are_collinear and

       A4: (P,Q,P2) are_collinear by Th12;

      consider R be Element of real_projective_plane such that

       A5: R in ( tangent P1) & R in ( tangent P2) by Th24;

      consider u be Element of ( TOP-REAL 3) such that

       A6: u is non zero and

       A7: R = ( Dir u) by ANPROJ_1: 26;

      per cases ;

        suppose (u . 3) = 0 ;

        reconsider RR = R as Element of ( ProjectiveSpace ( TOP-REAL 3));

        (P,P1,P2) are_collinear by A1, A3, A4, COLLSP: 6;

        then

        consider PT1 be Element of ( ProjectiveSpace ( TOP-REAL 3)) such that

         A8: PT1 in absolute and

         A9: (P,RR,PT1) are_collinear by A2, A5, Th31;

        (Q,P,P1) are_collinear & (Q,P,P2) are_collinear by A3, A4, COLLSP: 4;

        then (Q,P1,P2) are_collinear by A1, COLLSP: 6;

        then

        consider PT2 be Element of ( ProjectiveSpace ( TOP-REAL 3)) such that

         A10: PT2 in absolute and

         A11: (Q,RR,PT2) are_collinear by A2, A5, Th31;

        now

          thus (P,Q,P1) are_collinear by A3;

          thus (P,Q,P2) are_collinear by A4;

          

           A12: PT1 <> RR

          proof

            assume PT1 = RR;

            then PT1 in ( absolute /\ ( tangent P1)) & PT1 in ( absolute /\ ( tangent P2)) by A5, A8, XBOOLE_0:def 4;

            then PT1 in {P1} & PT1 in {P2} by Th22;

            then PT1 = P1 & PT1 = P2 by TARSKI:def 1;

            hence contradiction by A2;

          end;

          

           A13: PT2 <> RR

          proof

            assume PT2 = RR;

            then PT2 in ( absolute /\ ( tangent P1)) & PT2 in ( absolute /\ ( tangent P2)) by A5, A10, XBOOLE_0:def 4;

            then PT2 in {P1} & PT2 in {P2} by Th22;

            then PT2 = P1 & PT2 = P2 by TARSKI:def 1;

            hence contradiction by A2;

          end;

          

           A14: P2 <> PT1

          proof

            (P,PT1,RR) are_collinear by A9, COLLSP: 4;

            hence thesis by A5, A12, Th32;

          end;

          P1 <> PT1

          proof

            assume

             A15: P1 = PT1;

            consider p1 be Element of real_projective_plane such that

             A16: p1 = P1 & ( tangent P1) = ( Line (p1,( pole_infty P1))) by Def04;

            reconsider pt1 = PT1, rr = RR, p = P as Element of real_projective_plane ;

            

             A17: (p1,( pole_infty P1),pt1) are_collinear & (p1,( pole_infty P1),rr) are_collinear by A15, A5, Th21, A16, COLLSP: 11;

            (rr,pt1,p) are_collinear by A9, COLLSP: 8;

            then P in ( tangent P1) & P in BK_model by A12, A17, A16, COLLSP: 9, COLLSP: 11;

            then ( tangent P1) meets BK_model by XBOOLE_0:def 4;

            hence contradiction by Th30;

          end;

          hence (P1,P2,PT1) are_mutually_distinct by A14, A2;

          

           A18: P1 <> PT2

          proof

            (Q,PT2,RR) are_collinear by A11, COLLSP: 4;

            hence thesis by A5, A13, Th32;

          end;

          P2 <> PT2

          proof

            assume

             A19: P2 = PT2;

            consider p2 be Element of real_projective_plane such that

             A20: p2 = P2 & ( tangent P2) = ( Line (p2,( pole_infty P2))) by Def04;

            reconsider pt2 = PT2, rr = RR, q = Q as Element of real_projective_plane ;

            

             A21: (p2,( pole_infty P2),pt2) are_collinear & (p2,( pole_infty P2),rr) are_collinear by A20, A19, A5, Th21, COLLSP: 11;

            (rr,pt2,q) are_collinear by A11, COLLSP: 8;

            then Q in ( tangent P2) & Q in BK_model by A20, A13, A21, COLLSP: 9, COLLSP: 11;

            then ( tangent P2) meets BK_model by XBOOLE_0:def 4;

            hence contradiction by Th30;

          end;

          hence (P1,P2,PT2) are_mutually_distinct by A18, A2;

          thus R in (( tangent P1) /\ ( tangent P2)) by A5, XBOOLE_0:def 4;

          thus (P,RR,PT1) are_collinear by A9;

          thus (Q,RR,PT2) are_collinear by A11;

        end;

        hence thesis by A8, A10;

      end;

        suppose

         A22: (u . 3) <> 0 ;

        reconsider v = |[((u . 1) / (u . 3)), ((u . 2) / (u . 3)), 1]| as non zero Element of ( TOP-REAL 3) by BKMODEL1: 41;

        

         A23: (v . 3) = (v `3 ) by EUCLID_5:def 3

        .= 1 by EUCLID_5: 2;

        

         A24: ((u . 3) * ((u . 1) / (u . 3))) = (u . 1) & ((u . 3) * ((u . 2) / (u . 3))) = (u . 2) by A22, XCMPLX_1: 87;

        ((u . 3) * v) = |[((u . 3) * ((u . 1) / (u . 3))), ((u . 3) * ((u . 2) / (u . 3))), ((u . 3) * 1)]| by EUCLID_5: 8

        .= |[(u `1 ), (u . 2), (u . 3)]| by A24, EUCLID_5:def 1

        .= |[(u `1 ), (u `2 ), (u . 3)]| by EUCLID_5:def 2

        .= |[(u `1 ), (u `2 ), (u `3 )]| by EUCLID_5:def 3

        .= u by EUCLID_5: 3;

        then are_Prop (v,u) by A22, ANPROJ_1: 1;

        then

         A25: R = ( Dir v) & (v . 3) = 1 by A6, A7, A23, ANPROJ_1: 22;

        reconsider RR = R as Element of ( ProjectiveSpace ( TOP-REAL 3));

        P <> RR

        proof

          assume P = RR;

          then BK_model meets ( tangent P1) by A5, XBOOLE_0:def 4;

          hence contradiction by Th30;

        end;

        then

        consider PT1 be Element of absolute such that

         A26: (P,RR,PT1) are_collinear by A25, Th03;

        Q <> RR

        proof

          assume Q = RR;

          then BK_model meets ( tangent P2) by A5, XBOOLE_0:def 4;

          hence contradiction by Th30;

        end;

        then

        consider PT2 be Element of absolute such that

         A27: (Q,RR,PT2) are_collinear by A25, Th03;

        now

          thus (P,Q,P1) are_collinear by A3;

          thus (P,Q,P2) are_collinear by A4;

          

           A28: PT1 <> RR

          proof

            assume PT1 = RR;

            then PT1 in ( absolute /\ ( tangent P1)) & PT1 in ( absolute /\ ( tangent P2)) by A5, XBOOLE_0:def 4;

            then PT1 in {P1} & PT1 in {P2} by Th22;

            then PT1 = P1 & PT1 = P2 by TARSKI:def 1;

            hence contradiction by A2;

          end;

          

           A29: PT2 <> RR

          proof

            assume PT2 = RR;

            then PT2 in ( absolute /\ ( tangent P1)) & PT2 in ( absolute /\ ( tangent P2)) by A5, XBOOLE_0:def 4;

            then PT2 in {P1} & PT2 in {P2} by Th22;

            then PT2 = P1 & PT2 = P2 by TARSKI:def 1;

            hence contradiction by A2;

          end;

          

           A30: P2 <> PT1

          proof

            (P,PT1,RR) are_collinear by A26, COLLSP: 4;

            hence thesis by A5, A28, Th32;

          end;

          P1 <> PT1

          proof

            assume

             A31: P1 = PT1;

            consider p1 be Element of real_projective_plane such that

             A32: p1 = P1 & ( tangent P1) = ( Line (p1,( pole_infty P1))) by Def04;

            reconsider pt1 = PT1, rr = RR, p = P as Element of real_projective_plane ;

            

             A33: (p1,( pole_infty P1),pt1) are_collinear & (p1,( pole_infty P1),rr) are_collinear by A31, A5, Th21, A32, COLLSP: 11;

            (rr,pt1,p) are_collinear by A26, COLLSP: 8;

            then P in ( tangent P1) & P in BK_model by A28, A33, A32, COLLSP: 9, COLLSP: 11;

            then ( tangent P1) meets BK_model by XBOOLE_0:def 4;

            hence contradiction by Th30;

          end;

          hence (P1,P2,PT1) are_mutually_distinct by A30, A2;

          

           A34: P1 <> PT2

          proof

            (Q,PT2,RR) are_collinear by A27, COLLSP: 4;

            hence thesis by A5, A29, Th32;

          end;

          P2 <> PT2

          proof

            assume

             A35: P2 = PT2;

            consider p2 be Element of real_projective_plane such that

             A36: p2 = P2 & ( tangent P2) = ( Line (p2,( pole_infty P2))) by Def04;

            reconsider pt2 = PT2, rr = RR, q = Q as Element of real_projective_plane ;

            

             A37: (p2,( pole_infty P2),pt2) are_collinear & (p2,( pole_infty P2),rr) are_collinear by A35, A5, Th21, A36, COLLSP: 11;

            (rr,pt2,q) are_collinear by A27, COLLSP: 8;

            then Q in ( tangent P2) by A36, A29, A37, COLLSP: 9, COLLSP: 11;

            then ( tangent P2) meets BK_model by XBOOLE_0:def 4;

            hence contradiction by Th30;

          end;

          hence (P1,P2,PT2) are_mutually_distinct by A34, A2;

          thus RR in (( tangent P1) /\ ( tangent P2)) by A5, XBOOLE_0:def 4;

          thus (P,RR,PT1) are_collinear by A26;

          thus (Q,RR,PT2) are_collinear by A27;

        end;

        hence thesis;

      end;

    end;

    theorem :: BKMODEL2:59

    

     Th45: for P,Q be Element of BK_model st P <> Q holds ex N be invertible Matrix of 3, F_Real st (( homography N) .: absolute ) = absolute & (( homography N) . P) = Q & (( homography N) . Q) = P & (ex P1,P2 be Element of absolute st P1 <> P2 & (P,Q,P1) are_collinear & (P,Q,P2) are_collinear & (( homography N) . P1) = P2 & (( homography N) . P2) = P1)

    proof

      let P,Q be Element of BK_model ;

      assume

       A1: P <> Q;

      consider P1,P2,P3,P4 be Element of absolute , P5 be Element of ( ProjectiveSpace ( TOP-REAL 3)) such that

       A2: P1 <> P2 and

       A3: (P,Q,P1) are_collinear and

       A4: (P,Q,P2) are_collinear and

       A5: (P,P5,P3) are_collinear and

       A6: (Q,P5,P4) are_collinear and

       A7: (P1,P2,P3) are_mutually_distinct and

       A8: (P1,P2,P4) are_mutually_distinct and

       A9: P5 in (( tangent P1) /\ ( tangent P2)) by A1, Th44;

      consider N1 be invertible Matrix of 3, F_Real such that

       A10: (( homography N1) .: absolute ) = absolute and

       A11: (( homography N1) . Dir101 ) = P1 and

       A12: (( homography N1) . Dirm101 ) = P2 and

       A13: (( homography N1) . Dir011 ) = P3 and

       A14: (( homography N1) . Dir010 ) = P5 by A7, A9, Th37;

      (P2,P1,P4) are_mutually_distinct by A8;

      then

      consider N2 be invertible Matrix of 3, F_Real such that

       A15: (( homography N2) .: absolute ) = absolute and

       A16: (( homography N2) . Dir101 ) = P2 and

       A17: (( homography N2) . Dirm101 ) = P1 and

       A18: (( homography N2) . Dir011 ) = P4 and

       A19: (( homography N2) . Dir010 ) = P5 by A9, Th37;

      reconsider N = (N2 * (N1 ~ )) as invertible Matrix of 3, F_Real ;

      

       A20: (( homography N) . P1) = (( homography N2) . (( homography (N1 ~ )) . P1)) by ANPROJ_9: 13

      .= P2 by A11, A16, ANPROJ_9: 15;

      

       A21: (( homography N) . P2) = (( homography N2) . (( homography (N1 ~ )) . P2)) by ANPROJ_9: 13

      .= P1 by A12, A17, ANPROJ_9: 15;

      

       A22: (( homography N) . P3) = (( homography N2) . (( homography (N1 ~ )) . P3)) by ANPROJ_9: 13

      .= P4 by A13, A18, ANPROJ_9: 15;

      

       A23: (( homography N) . P5) = (( homography N2) . (( homography (N1 ~ )) . P5)) by ANPROJ_9: 13

      .= P5 by A14, A19, ANPROJ_9: 15;

      ( homography N1) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h1 = ( homography N1) as Element of EnsHomography3 ;

      h1 is_K-isometry by A10;

      then h1 in EnsK-isometry ;

      then

      reconsider hsg1 = h1 as Element of SubGroupK-isometry by Def05;

      ( homography N2) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h2 = ( homography N2) as Element of EnsHomography3 ;

      h2 is_K-isometry by A15;

      then h2 in EnsK-isometry ;

      then

      reconsider hsg2 = h2 as Element of SubGroupK-isometry by Def05;

      ( homography (N1 ~ )) in EnsHomography3 by ANPROJ_9:def 1;

      then

      reconsider h3 = ( homography (N1 ~ )) as Element of EnsHomography3 ;

      

       A24: (hsg1 " ) = h3 by Th36;

      set H = EnsK-isometry , G = GroupHomography3 ;

      reconsider hg1 = hsg1, hg2 = hsg2, hg3 = (hsg1 " ) as Element of G by A24, ANPROJ_9:def 4;

      reconsider hsg3 = h3 as Element of SubGroupK-isometry by A24;

      reconsider h4 = (hsg2 * hsg3) as Element of SubGroupK-isometry ;

      

       A25: h4 = (hg2 * hg3) by A24, GROUP_2: 43

      .= (h2 (*) h3) by A24, ANPROJ_9:def 3, ANPROJ_9:def 4

      .= ( homography N) by ANPROJ_9: 18;

      h4 in the carrier of SubGroupK-isometry ;

      then h4 in EnsK-isometry by Def05;

      then

      consider h be Element of EnsHomography3 such that

       A26: h4 = h and

       A27: h is_K-isometry ;

      take N;

      thus (( homography N) .: absolute ) = absolute by A25, A26, A27;

      set NP = (( homography N) . P), NQ = (( homography N) . Q), NP1 = (( homography N) . P1), NP2 = (( homography N) . P2), NP3 = (( homography N) . P3), NP4 = (( homography N) . P4), NP5 = (( homography N) . P5);

      

       A28: (P,P1,P2) are_collinear by A1, A3, A4, ANPROJ_8: 57, HESSENBE: 2;

      (Q,P,P1) are_collinear & (Q,P,P2) are_collinear by A3, A4, COLLSP: 4;

      then

       A29: (Q,P1,P2) are_collinear by A1, ANPROJ_8: 57, HESSENBE: 2;

      thus (( homography N) . P) = Q & (( homography N) . Q) = P

      proof

        

         A30: NP <> NQ

        proof

          assume

           A31: NP = NQ;

          Q = (( homography (N ~ )) . NQ) by ANPROJ_9: 15

          .= P by A31, ANPROJ_9: 15;

          hence contradiction by A1;

        end;

        

         A32: (NP,NQ,NP1) are_collinear & (NP,NQ,NP2) are_collinear & (NP,NP5,NP3) are_collinear & (NQ,NP5,NP4) are_collinear by A3, A4, A5, A6, ANPROJ_8: 102;

        then

         A33: (NP,NP1,NP2) are_collinear by ANPROJ_8: 57, A30, HESSENBE: 2;

        

         A34: (P1,P2,Q) are_collinear by A29, ANPROJ_8: 57, HESSENBE: 1;

        (P5,P4,Q) are_collinear by A6, ANPROJ_8: 57, HESSENBE: 1;

        then

         A35: Q in ( Line (P1,P2)) & Q in ( Line (P5,P4)) by A34, COLLSP: 11;

        then

         A36: Q in (( Line (P1,P2)) /\ ( Line (P5,P4))) by XBOOLE_0:def 4;

        (P1,P2,NP) are_collinear by A33, A20, A21, ANPROJ_8: 57, HESSENBE: 1;

        then

         A37: NP in ( Line (P1,P2)) by COLLSP: 11;

        (P5,P4,NP) are_collinear by A32, A22, A23, ANPROJ_8: 57, HESSENBE: 1;

        then NP in ( Line (P5,P4)) by COLLSP: 11;

        then NP in (( Line (P5,P4)) /\ ( Line (P1,P2))) by A37, XBOOLE_0:def 4;

        then

         A39: {Q, NP} c= (( Line (P1,P2)) /\ ( Line (P5,P4))) by A36, ZFMISC_1: 32;

        P4 <> P5

        proof

          assume P4 = P5;

          then P4 in ( tangent P1) & P4 in ( tangent P2) by A9, XBOOLE_0:def 4;

          then P4 in (( tangent P1) /\ absolute ) & P4 in (( tangent P2) /\ absolute ) by XBOOLE_0:def 4;

          then P4 in {P1} & P4 in {P2} by Th22;

          then P4 = P1 & P4 = P2 by TARSKI:def 1;

          hence contradiction by A2;

        end;

        then ( Line (P1,P2)) is LINE of real_projective_plane & ( Line (P5,P4)) is LINE of real_projective_plane by A2, COLLSP:def 7;

        then

         A41: ( Line (P1,P2)) = ( Line (P5,P4)) or ( Line (P1,P2)) misses ( Line (P5,P4)) or ex p be Element of real_projective_plane st (( Line (P1,P2)) /\ ( Line (P5,P4))) = {p} by COLLSP: 21;

        ( Line (P1,P2)) <> ( Line (P5,P4))

        proof

          assume ( Line (P1,P2)) = ( Line (P5,P4));

          then P4 in ( Line (P1,P2)) by COLLSP: 10;

          hence contradiction by A8, COLLSP: 11, BKMODEL1: 92;

        end;

        then

        consider p be Element of real_projective_plane such that

         A42: (( Line (P1,P2)) /\ ( Line (P5,P4))) = {p} by A35, A41, XBOOLE_0:def 4;

        Q = p & NP = p by A42, A39, ZFMISC_1: 20;

        hence thesis by A28, A20, A21, A2, COLLSP: 8, BKMODEL1: 69;

      end;

      thus ex P1,P2 be Element of absolute st P1 <> P2 & (P,Q,P1) are_collinear & (P,Q,P2) are_collinear & (( homography N) . P1) = P2 & (( homography N) . P2) = P1 by A2, A3, A4, A20, A21;

    end;

    begin

    theorem :: BKMODEL2:60

    for P,Q be Element of BK_model holds ex h be Element of SubGroupK-isometry , N be invertible Matrix of 3, F_Real st h = ( homography N) & (( homography N) . P) = Q & (( homography N) . Q) = P

    proof

      let P,Q be Element of BK_model ;

      per cases ;

        suppose

         A1: P = Q;

        reconsider N = ( 1. ( F_Real ,3)) as invertible Matrix of 3, F_Real ;

        ( homography N) in the set of all ( homography N) where N be invertible Matrix of 3, F_Real ;

        then

        reconsider h = ( homography N) as Element of EnsHomography3 by ANPROJ_9:def 1;

        h is_K-isometry by Th33;

        then h in EnsK-isometry ;

        then

        reconsider h as Element of SubGroupK-isometry by Def05;

        take h;

        (( homography N) . P) = Q & (( homography N) . Q) = P by A1, ANPROJ_9: 14;

        hence thesis;

      end;

        suppose P <> Q;

        then

        consider N be invertible Matrix of 3, F_Real such that

         A2: (( homography N) .: absolute ) = absolute and

         A3: (( homography N) . P) = Q and

         A4: (( homography N) . Q) = P and (ex P1,P2 be Element of absolute st P1 <> P2 & (P,Q,P1) are_collinear & (P,Q,P2) are_collinear & (( homography N) . P1) = P2 & (( homography N) . P2) = P1) by Th45;

        ( homography N) in the set of all ( homography N) where N be invertible Matrix of 3, F_Real ;

        then

        reconsider h = ( homography N) as Element of EnsHomography3 by ANPROJ_9:def 1;

        h is_K-isometry by A2;

        then h in EnsK-isometry ;

        then

        reconsider h as Element of SubGroupK-isometry by Def05;

        take h;

        thus thesis by A3, A4;

      end;

    end;

    theorem :: BKMODEL2:61

    for P,Q,R,S,T,U be Element of BK_model st ex h1,h2 be Element of SubGroupK-isometry , N1,N2 be invertible Matrix of 3, F_Real st h1 = ( homography N1) & h2 = ( homography N2) & (( homography N1) . P) = R & (( homography N1) . Q) = S & (( homography N2) . R) = T & (( homography N2) . S) = U holds ex h3 be Element of SubGroupK-isometry , N3 be invertible Matrix of 3, F_Real st h3 = ( homography N3) & (( homography N3) . P) = T & (( homography N3) . Q) = U

    proof

      let P,Q,R,S,T,U be Element of BK_model ;

      assume ex h1,h2 be Element of SubGroupK-isometry , N1,N2 be invertible Matrix of 3, F_Real st h1 = ( homography N1) & h2 = ( homography N2) & (( homography N1) . P) = R & (( homography N1) . Q) = S & (( homography N2) . R) = T & (( homography N2) . S) = U;

      then

      consider h1,h2 be Element of SubGroupK-isometry , N1,N2 be invertible Matrix of 3, F_Real such that

       A1: h1 = ( homography N1) & h2 = ( homography N2) & (( homography N1) . P) = R & (( homography N1) . Q) = S & (( homography N2) . R) = T & (( homography N2) . S) = U;

      reconsider N3 = (N2 * N1) as invertible Matrix of 3, F_Real ;

      (h2 * h1) = ( homography (N2 * N1)) by A1, Th35;

      then

      reconsider h3 = ( homography N3) as Element of SubGroupK-isometry ;

      take h3;

      (( homography N3) . P) = T & (( homography N3) . Q) = U by A1, ANPROJ_9: 13;

      hence thesis;

    end;

    theorem :: BKMODEL2:62

    for P,Q,R be Element of BK_model , h be Element of SubGroupK-isometry , N be invertible Matrix of 3, F_Real st h = ( homography N) & (( homography N) . P) = R & (( homography N) . Q) = R holds P = Q by ANPROJ_9: 16;