%------------------------------------------------------------------------------ % File : Zenon---0.7.1 % Problem : GEO196+3 : TPTP v8.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : run_zenon %s %d % Computer : n017.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8042.1875MB % OS : Linux 3.10.0-693.el7.x86_64 % CPULimit : 300s % WCLimit : 600s % DateTime : Sat Jul 16 07:00:41 EDT 2022 % Result : Theorem 1.18s 1.34s % Output : Proof 1.18s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : GEO196+3 : TPTP v8.1.0. Released v4.0.0. % 0.00/0.12 % Command : run_zenon %s %d % 0.12/0.33 % Computer : n017.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Sat Jun 18 09:26:57 EDT 2022 % 0.12/0.34 % CPUTime : % 1.18/1.34 (* PROOF-FOUND *) % 1.18/1.34 % SZS status Theorem % 1.18/1.34 (* BEGIN-PROOF *) % 1.18/1.34 % SZS output start Proof % 1.18/1.34 Theorem con : (forall X : zenon_U, (forall Y : zenon_U, (forall U : zenon_U, (forall V : zenon_U, (((convergent_lines X Y)/\((convergent_lines U V)/\((incident_point_and_line (intersection_point X Y) U)/\(incident_point_and_line (intersection_point X Y) V))))->((incident_point_and_line (intersection_point U V) X)/\(incident_point_and_line (intersection_point U V) Y))))))). % 1.18/1.34 Proof. % 1.18/1.34 assert (zenon_L1_ : forall (zenon_TV_bn : zenon_U) (zenon_TX_bo : zenon_U), (forall Y : zenon_U, ((distinct_lines zenon_TX_bo Y)->(convergent_lines zenon_TX_bo Y))) -> (distinct_lines zenon_TX_bo zenon_TV_bn) -> (~(convergent_lines zenon_TX_bo zenon_TV_bn)) -> False). % 1.18/1.34 do 2 intro. intros zenon_H24 zenon_H25 zenon_H26. % 1.18/1.34 generalize (zenon_H24 zenon_TV_bn). zenon_intro zenon_H29. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H29); [ zenon_intro zenon_H2b | zenon_intro zenon_H2a ]. % 1.18/1.34 exact (zenon_H2b zenon_H25). % 1.18/1.34 exact (zenon_H26 zenon_H2a). % 1.18/1.34 (* end of lemma zenon_L1_ *) % 1.18/1.34 assert (zenon_L2_ : forall (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TV_bn) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> False). % 1.18/1.34 do 2 intro. intros zenon_H2c zenon_H2d. % 1.18/1.34 generalize (ci4 zenon_TU_bu). zenon_intro zenon_H2f. % 1.18/1.34 generalize (zenon_H2f zenon_TV_bn). zenon_intro zenon_H30. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H30); [ zenon_intro zenon_H32 | zenon_intro zenon_H31 ]. % 1.18/1.34 exact (zenon_H32 zenon_H2d). % 1.18/1.34 exact (zenon_H31 zenon_H2c). % 1.18/1.34 (* end of lemma zenon_L2_ *) % 1.18/1.34 assert (zenon_L3_ : forall (zenon_TX_bo : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TX_bo) -> (~(convergent_lines zenon_TX_bo zenon_TV_bn)) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> False). % 1.18/1.34 do 3 intro. intros zenon_H33 zenon_H26 zenon_H2d. % 1.18/1.34 generalize (ceq2 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H34. % 1.18/1.34 generalize (zenon_H34 zenon_TX_bo). zenon_intro zenon_H35. % 1.18/1.34 generalize (zenon_H35 zenon_TV_bn). zenon_intro zenon_H36. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H36); [ zenon_intro zenon_H38 | zenon_intro zenon_H37 ]. % 1.18/1.34 exact (zenon_H38 zenon_H33). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H37); [ zenon_intro zenon_H25 | zenon_intro zenon_H2c ]. % 1.18/1.34 generalize (p1 zenon_TX_bo). zenon_intro zenon_H24. % 1.18/1.34 apply (zenon_L1_ zenon_TV_bn zenon_TX_bo); trivial. % 1.18/1.34 apply (zenon_L2_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 (* end of lemma zenon_L3_ *) % 1.18/1.34 assert (zenon_L4_ : forall (zenon_TV_bn : zenon_U), (distinct_lines zenon_TV_bn zenon_TV_bn) -> False). % 1.18/1.34 do 1 intro. intros zenon_H39. % 1.18/1.34 generalize (apart2 zenon_TV_bn). zenon_intro zenon_H3a. % 1.18/1.34 exact (zenon_H3a zenon_H39). % 1.18/1.34 (* end of lemma zenon_L4_ *) % 1.18/1.34 assert (zenon_L5_ : forall (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (forall Y : zenon_U, ((convergent_lines zenon_TU_bu Y)->(~(apart_point_and_line (intersection_point zenon_TU_bu Y) zenon_TU_bu)))) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TU_bu) -> False). % 1.18/1.34 do 2 intro. intros zenon_H3b zenon_H2d zenon_H3c. % 1.18/1.34 generalize (zenon_H3b zenon_TV_bn). zenon_intro zenon_H3d. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H3d); [ zenon_intro zenon_H32 | zenon_intro zenon_H3e ]. % 1.18/1.34 exact (zenon_H32 zenon_H2d). % 1.18/1.34 exact (zenon_H3e zenon_H3c). % 1.18/1.34 (* end of lemma zenon_L5_ *) % 1.18/1.34 assert (zenon_L6_ : forall (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TU_bu) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> False). % 1.18/1.34 do 2 intro. intros zenon_H3c zenon_H2d. % 1.18/1.34 generalize (ci3 zenon_TU_bu). zenon_intro zenon_H3b. % 1.18/1.34 apply (zenon_L5_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 (* end of lemma zenon_L6_ *) % 1.18/1.34 assert (zenon_L7_ : forall (zenon_TV_bn : zenon_U) (zenon_TX_bo : zenon_U), (apart_point_and_line (intersection_point zenon_TX_bo zenon_TV_bn) zenon_TV_bn) -> (convergent_lines zenon_TX_bo zenon_TV_bn) -> False). % 1.18/1.34 do 2 intro. intros zenon_H3f zenon_H2a. % 1.18/1.34 generalize (ci4 zenon_TX_bo). zenon_intro zenon_H40. % 1.18/1.34 generalize (zenon_H40 zenon_TV_bn). zenon_intro zenon_H41. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H41); [ zenon_intro zenon_H26 | zenon_intro zenon_H42 ]. % 1.18/1.34 exact (zenon_H26 zenon_H2a). % 1.18/1.34 exact (zenon_H42 zenon_H3f). % 1.18/1.34 (* end of lemma zenon_L7_ *) % 1.18/1.34 assert (zenon_L8_ : forall (zenon_TY_cs : zenon_U) (zenon_TX_bo : zenon_U), (forall Y : zenon_U, ((convergent_lines zenon_TX_bo Y)->(~(apart_point_and_line (intersection_point zenon_TX_bo Y) zenon_TX_bo)))) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TX_bo) -> False). % 1.18/1.34 do 2 intro. intros zenon_H43 zenon_H44 zenon_H45. % 1.18/1.34 generalize (zenon_H43 zenon_TY_cs). zenon_intro zenon_H47. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H47); [ zenon_intro zenon_H49 | zenon_intro zenon_H48 ]. % 1.18/1.34 exact (zenon_H49 zenon_H44). % 1.18/1.34 exact (zenon_H48 zenon_H45). % 1.18/1.34 (* end of lemma zenon_L8_ *) % 1.18/1.34 assert (zenon_L9_ : forall (zenon_TY_cs : zenon_U) (zenon_TX_bo : zenon_U), (apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TX_bo) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> False). % 1.18/1.34 do 2 intro. intros zenon_H45 zenon_H44. % 1.18/1.34 generalize (ci3 zenon_TX_bo). zenon_intro zenon_H43. % 1.18/1.34 apply (zenon_L8_ zenon_TY_cs zenon_TX_bo); trivial. % 1.18/1.34 (* end of lemma zenon_L9_ *) % 1.18/1.34 assert (zenon_L10_ : forall (zenon_TY_cs : zenon_U) (zenon_TU_bu : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TX_bo : zenon_U), (forall Z : zenon_U, ((apart_point_and_line (intersection_point zenon_TX_bo zenon_TV_bn) zenon_TU_bu)->((distinct_points (intersection_point zenon_TX_bo zenon_TV_bn) Z)\/(apart_point_and_line Z zenon_TU_bu)))) -> (apart_point_and_line (intersection_point zenon_TX_bo zenon_TV_bn) zenon_TU_bu) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TV_bn) zenon_TX_bo)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TV_bn) zenon_TV_bn)) -> (distinct_lines zenon_TV_bn zenon_TX_bo) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> False). % 1.18/1.34 do 4 intro. intros zenon_H4a zenon_H4b zenon_H44 zenon_H4c zenon_H4d zenon_H42 zenon_H4e zenon_H4f. % 1.18/1.34 generalize (zenon_H4a (intersection_point zenon_TX_bo zenon_TY_cs)). zenon_intro zenon_H50. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H50); [ zenon_intro zenon_H52 | zenon_intro zenon_H51 ]. % 1.18/1.34 exact (zenon_H52 zenon_H4b). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H51); [ zenon_intro zenon_H54 | zenon_intro zenon_H53 ]. % 1.18/1.34 generalize (cu1 (intersection_point zenon_TX_bo zenon_TV_bn)). zenon_intro zenon_H55. % 1.18/1.34 generalize (zenon_H55 (intersection_point zenon_TX_bo zenon_TY_cs)). zenon_intro zenon_H56. % 1.18/1.34 generalize (zenon_H56 zenon_TV_bn). zenon_intro zenon_H57. % 1.18/1.34 generalize (zenon_H57 zenon_TX_bo). zenon_intro zenon_H58. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H58); [ zenon_intro zenon_H5a | zenon_intro zenon_H59 ]. % 1.18/1.34 apply (zenon_notand_s _ _ zenon_H5a); [ zenon_intro zenon_H5c | zenon_intro zenon_H5b ]. % 1.18/1.34 exact (zenon_H5c zenon_H54). % 1.18/1.34 exact (zenon_H5b zenon_H4e). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H59); [ zenon_intro zenon_H3f | zenon_intro zenon_H5d ]. % 1.18/1.34 exact (zenon_H42 zenon_H3f). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H5d); [ zenon_intro zenon_H5f | zenon_intro zenon_H5e ]. % 1.18/1.34 exact (zenon_H4d zenon_H5f). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H5e); [ zenon_intro zenon_H60 | zenon_intro zenon_H45 ]. % 1.18/1.34 exact (zenon_H4c zenon_H60). % 1.18/1.34 apply (zenon_L9_ zenon_TY_cs zenon_TX_bo); trivial. % 1.18/1.34 exact (zenon_H4f zenon_H53). % 1.18/1.34 (* end of lemma zenon_L10_ *) % 1.18/1.34 assert (zenon_L11_ : forall (zenon_TY_cs : zenon_U) (zenon_TU_bu : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TX_bo : zenon_U), (apart_point_and_line (intersection_point zenon_TX_bo zenon_TV_bn) zenon_TU_bu) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TV_bn) zenon_TX_bo)) -> (distinct_lines zenon_TV_bn zenon_TX_bo) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (convergent_lines zenon_TX_bo zenon_TV_bn) -> False). % 1.18/1.34 do 4 intro. intros zenon_H4b zenon_H44 zenon_H4c zenon_H4d zenon_H4e zenon_H4f zenon_H2a. % 1.18/1.34 generalize (ceq1 (intersection_point zenon_TX_bo zenon_TV_bn)). zenon_intro zenon_H61. % 1.18/1.34 generalize (zenon_H61 zenon_TU_bu). zenon_intro zenon_H4a. % 1.18/1.34 generalize (ceq2 (intersection_point zenon_TX_bo zenon_TV_bn)). zenon_intro zenon_H62. % 1.18/1.34 generalize (zenon_H62 zenon_TV_bn). zenon_intro zenon_H63. % 1.18/1.34 generalize (zenon_H63 zenon_TV_bn). zenon_intro zenon_H64. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H64); [ zenon_intro zenon_H42 | zenon_intro zenon_H65 ]. % 1.18/1.34 apply (zenon_L10_ zenon_TY_cs zenon_TU_bu zenon_TV_bn zenon_TX_bo); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_H65); [ zenon_intro zenon_H39 | zenon_intro zenon_H3f ]. % 1.18/1.34 apply (zenon_L4_ zenon_TV_bn); trivial. % 1.18/1.34 apply (zenon_L7_ zenon_TV_bn zenon_TX_bo); trivial. % 1.18/1.34 (* end of lemma zenon_L11_ *) % 1.18/1.34 assert (zenon_L12_ : forall (zenon_TX_bo : zenon_U), (convergent_lines zenon_TX_bo zenon_TX_bo) -> False). % 1.18/1.34 do 1 intro. intros zenon_H66. % 1.18/1.34 generalize (apart3 zenon_TX_bo). zenon_intro zenon_H67. % 1.18/1.34 exact (zenon_H67 zenon_H66). % 1.18/1.34 (* end of lemma zenon_L12_ *) % 1.18/1.34 assert (zenon_L13_ : forall (zenon_TU_bu : zenon_U), (convergent_lines zenon_TU_bu zenon_TU_bu) -> False). % 1.18/1.34 do 1 intro. intros zenon_H68. % 1.18/1.34 generalize (apart3 zenon_TU_bu). zenon_intro zenon_H69. % 1.18/1.34 exact (zenon_H69 zenon_H68). % 1.18/1.34 (* end of lemma zenon_L13_ *) % 1.18/1.34 assert (zenon_L14_ : forall (zenon_TY_cs : zenon_U) (zenon_TX_bo : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TX_bo) -> (forall Z : zenon_U, ((convergent_lines zenon_TU_bu zenon_TV_bn)->((distinct_lines zenon_TV_bn Z)\/(convergent_lines zenon_TU_bu Z)))) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (forall Z : zenon_U, ((convergent_lines zenon_TX_bo zenon_TV_bn)->((distinct_lines zenon_TV_bn Z)\/(convergent_lines zenon_TX_bo Z)))) -> (convergent_lines zenon_TX_bo zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TV_bn) zenon_TX_bo)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> False). % 1.18/1.34 do 4 intro. intros zenon_H33 zenon_H6a zenon_H2d zenon_H6b zenon_H2a zenon_H4f zenon_H4d zenon_H4c zenon_H44. % 1.18/1.34 generalize (ceq1 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H6c. % 1.18/1.34 generalize (zenon_H6c zenon_TX_bo). zenon_intro zenon_H6d. % 1.18/1.34 generalize (zenon_H6d (intersection_point zenon_TX_bo zenon_TV_bn)). zenon_intro zenon_H6e. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H6e); [ zenon_intro zenon_H38 | zenon_intro zenon_H6f ]. % 1.18/1.34 exact (zenon_H38 zenon_H33). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H6f); [ zenon_intro zenon_H70 | zenon_intro zenon_H5f ]. % 1.18/1.34 generalize (cu1 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H71. % 1.18/1.34 generalize (zenon_H71 (intersection_point zenon_TX_bo zenon_TV_bn)). zenon_intro zenon_H72. % 1.18/1.34 generalize (zenon_H6a zenon_TU_bu). zenon_intro zenon_H73. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H73); [ zenon_intro zenon_H32 | zenon_intro zenon_H74 ]. % 1.18/1.34 exact (zenon_H32 zenon_H2d). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H74); [ zenon_intro zenon_H75 | zenon_intro zenon_H68 ]. % 1.18/1.34 generalize (zenon_H6b zenon_TX_bo). zenon_intro zenon_H76. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H76); [ zenon_intro zenon_H26 | zenon_intro zenon_H77 ]. % 1.18/1.34 exact (zenon_H26 zenon_H2a). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H77); [ zenon_intro zenon_H4e | zenon_intro zenon_H66 ]. % 1.18/1.34 generalize (zenon_H72 zenon_TV_bn). zenon_intro zenon_H78. % 1.18/1.34 generalize (zenon_H78 zenon_TU_bu). zenon_intro zenon_H79. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H79); [ zenon_intro zenon_H7b | zenon_intro zenon_H7a ]. % 1.18/1.34 apply (zenon_notand_s _ _ zenon_H7b); [ zenon_intro zenon_H7d | zenon_intro zenon_H7c ]. % 1.18/1.34 exact (zenon_H7d zenon_H70). % 1.18/1.34 exact (zenon_H7c zenon_H75). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H7a); [ zenon_intro zenon_H2c | zenon_intro zenon_H7e ]. % 1.18/1.34 apply (zenon_L2_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_H7e); [ zenon_intro zenon_H3c | zenon_intro zenon_H7f ]. % 1.18/1.34 apply (zenon_L6_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_H7f); [ zenon_intro zenon_H3f | zenon_intro zenon_H4b ]. % 1.18/1.34 apply (zenon_L7_ zenon_TV_bn zenon_TX_bo); trivial. % 1.18/1.34 apply (zenon_L11_ zenon_TY_cs zenon_TU_bu zenon_TV_bn zenon_TX_bo); trivial. % 1.18/1.34 apply (zenon_L12_ zenon_TX_bo); trivial. % 1.18/1.34 apply (zenon_L13_ zenon_TU_bu); trivial. % 1.18/1.34 exact (zenon_H4d zenon_H5f). % 1.18/1.34 (* end of lemma zenon_L14_ *) % 1.18/1.34 assert (zenon_L15_ : forall (zenon_TV_bn : zenon_U) (zenon_TY_cs : zenon_U), (~(convergent_lines zenon_TY_cs zenon_TV_bn)) -> (distinct_lines zenon_TY_cs zenon_TV_bn) -> False). % 1.18/1.34 do 2 intro. intros zenon_H80 zenon_H81. % 1.18/1.34 generalize (p1 zenon_TY_cs). zenon_intro zenon_H82. % 1.18/1.34 generalize (zenon_H82 zenon_TV_bn). zenon_intro zenon_H83. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H83); [ zenon_intro zenon_H85 | zenon_intro zenon_H84 ]. % 1.18/1.34 exact (zenon_H85 zenon_H81). % 1.18/1.34 exact (zenon_H80 zenon_H84). % 1.18/1.34 (* end of lemma zenon_L15_ *) % 1.18/1.34 assert (zenon_L16_ : forall (zenon_TY_cs : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (forall Z : zenon_U, ((apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs)->((distinct_lines zenon_TY_cs Z)\/(apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) Z)))) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (~(convergent_lines zenon_TY_cs zenon_TV_bn)) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> False). % 1.18/1.34 do 3 intro. intros zenon_H86 zenon_H87 zenon_H80 zenon_H2d. % 1.18/1.34 generalize (zenon_H86 zenon_TV_bn). zenon_intro zenon_H88. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H88); [ zenon_intro zenon_H8a | zenon_intro zenon_H89 ]. % 1.18/1.34 exact (zenon_H8a zenon_H87). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H89); [ zenon_intro zenon_H81 | zenon_intro zenon_H2c ]. % 1.18/1.34 apply (zenon_L15_ zenon_TV_bn zenon_TY_cs); trivial. % 1.18/1.34 apply (zenon_L2_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 (* end of lemma zenon_L16_ *) % 1.18/1.34 assert (zenon_L17_ : forall (zenon_TY_cs : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), ((apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs)->((distinct_lines zenon_TY_cs zenon_TV_bn)\/(apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TV_bn))) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (~(convergent_lines zenon_TY_cs zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TV_bn)) -> False). % 1.18/1.34 do 3 intro. intros zenon_H88 zenon_H87 zenon_H80 zenon_H31. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H88); [ zenon_intro zenon_H8a | zenon_intro zenon_H89 ]. % 1.18/1.34 exact (zenon_H8a zenon_H87). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H89); [ zenon_intro zenon_H81 | zenon_intro zenon_H2c ]. % 1.18/1.34 apply (zenon_L15_ zenon_TV_bn zenon_TY_cs); trivial. % 1.18/1.34 exact (zenon_H31 zenon_H2c). % 1.18/1.34 (* end of lemma zenon_L17_ *) % 1.18/1.34 assert (zenon_L18_ : forall (zenon_TY_cs : zenon_U) (zenon_TX_bo : zenon_U), (apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TY_cs) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> False). % 1.18/1.34 do 2 intro. intros zenon_H8b zenon_H44. % 1.18/1.34 generalize (ci4 zenon_TX_bo). zenon_intro zenon_H40. % 1.18/1.34 generalize (zenon_H40 zenon_TY_cs). zenon_intro zenon_H8c. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H8c); [ zenon_intro zenon_H49 | zenon_intro zenon_H8d ]. % 1.18/1.34 exact (zenon_H49 zenon_H44). % 1.18/1.34 exact (zenon_H8d zenon_H8b). % 1.18/1.34 (* end of lemma zenon_L18_ *) % 1.18/1.34 assert (zenon_L19_ : forall (zenon_TV_bn : zenon_U) (zenon_TY_cs : zenon_U), (apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TV_bn) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> False). % 1.18/1.34 do 2 intro. intros zenon_H8e zenon_H84. % 1.18/1.34 generalize (ci4 zenon_TY_cs). zenon_intro zenon_H8f. % 1.18/1.34 generalize (zenon_H8f zenon_TV_bn). zenon_intro zenon_H90. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H90); [ zenon_intro zenon_H80 | zenon_intro zenon_H91 ]. % 1.18/1.34 exact (zenon_H80 zenon_H84). % 1.18/1.34 exact (zenon_H91 zenon_H8e). % 1.18/1.34 (* end of lemma zenon_L19_ *) % 1.18/1.34 assert (zenon_L20_ : forall (zenon_TX_bo : zenon_U) (zenon_TU_bu : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TY_cs : zenon_U), (apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TU_bu) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (distinct_lines zenon_TY_cs zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (~(distinct_lines zenon_TV_bn zenon_TV_bn)) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> False). % 1.18/1.34 do 4 intro. intros zenon_H92 zenon_H4c zenon_H44 zenon_H93 zenon_H81 zenon_H4f zenon_H3a zenon_H84. % 1.18/1.34 generalize (ceq2 (intersection_point zenon_TY_cs zenon_TV_bn)). zenon_intro zenon_H94. % 1.18/1.34 generalize (zenon_H94 zenon_TV_bn). zenon_intro zenon_H95. % 1.18/1.34 generalize (zenon_H95 zenon_TV_bn). zenon_intro zenon_H96. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H96); [ zenon_intro zenon_H91 | zenon_intro zenon_H97 ]. % 1.18/1.34 generalize (ceq1 (intersection_point zenon_TY_cs zenon_TV_bn)). zenon_intro zenon_H98. % 1.18/1.34 generalize (zenon_H98 zenon_TU_bu). zenon_intro zenon_H99. % 1.18/1.34 generalize (zenon_H99 (intersection_point zenon_TX_bo zenon_TY_cs)). zenon_intro zenon_H9a. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H9a); [ zenon_intro zenon_H9c | zenon_intro zenon_H9b ]. % 1.18/1.34 exact (zenon_H9c zenon_H92). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H9b); [ zenon_intro zenon_H9d | zenon_intro zenon_H53 ]. % 1.18/1.34 generalize (cu1 (intersection_point zenon_TY_cs zenon_TV_bn)). zenon_intro zenon_H9e. % 1.18/1.34 generalize (zenon_H9e (intersection_point zenon_TX_bo zenon_TY_cs)). zenon_intro zenon_H9f. % 1.18/1.34 generalize (zenon_H9f zenon_TY_cs). zenon_intro zenon_Ha0. % 1.18/1.34 generalize (zenon_Ha0 zenon_TV_bn). zenon_intro zenon_Ha1. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Ha1); [ zenon_intro zenon_Ha3 | zenon_intro zenon_Ha2 ]. % 1.18/1.34 apply (zenon_notand_s _ _ zenon_Ha3); [ zenon_intro zenon_Ha4 | zenon_intro zenon_H85 ]. % 1.18/1.34 exact (zenon_Ha4 zenon_H9d). % 1.18/1.34 exact (zenon_H85 zenon_H81). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Ha2); [ zenon_intro zenon_Ha6 | zenon_intro zenon_Ha5 ]. % 1.18/1.34 exact (zenon_H93 zenon_Ha6). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Ha5); [ zenon_intro zenon_H8e | zenon_intro zenon_Ha7 ]. % 1.18/1.34 exact (zenon_H91 zenon_H8e). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Ha7); [ zenon_intro zenon_H8b | zenon_intro zenon_H60 ]. % 1.18/1.34 apply (zenon_L18_ zenon_TY_cs zenon_TX_bo); trivial. % 1.18/1.34 exact (zenon_H4c zenon_H60). % 1.18/1.34 exact (zenon_H4f zenon_H53). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H97); [ zenon_intro zenon_H39 | zenon_intro zenon_H8e ]. % 1.18/1.34 exact (zenon_H3a zenon_H39). % 1.18/1.34 apply (zenon_L19_ zenon_TV_bn zenon_TY_cs); trivial. % 1.18/1.34 (* end of lemma zenon_L20_ *) % 1.18/1.34 assert (zenon_L21_ : forall (zenon_TX_bo : zenon_U) (zenon_TY_cs : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (forall U : zenon_U, (forall V : zenon_U, (((distinct_points (intersection_point zenon_TU_bu zenon_TV_bn) (intersection_point zenon_TY_cs zenon_TV_bn))/\(distinct_lines U V))->((apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) U)\/((apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) V)\/((apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) U)\/(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) V))))))) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> (~(distinct_lines zenon_TV_bn zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (distinct_lines zenon_TY_cs zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TU_bu)) -> (distinct_lines zenon_TU_bu zenon_TV_bn) -> (distinct_points (intersection_point zenon_TU_bu zenon_TV_bn) (intersection_point zenon_TY_cs zenon_TV_bn)) -> False). % 1.18/1.34 do 4 intro. intros zenon_Ha8 zenon_H84 zenon_H3a zenon_H4f zenon_H81 zenon_H93 zenon_H44 zenon_H4c zenon_H31 zenon_H3e zenon_Ha9 zenon_Haa. % 1.18/1.34 generalize (zenon_Ha8 zenon_TU_bu). zenon_intro zenon_Hab. % 1.18/1.34 generalize (zenon_Hab zenon_TV_bn). zenon_intro zenon_Hac. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hac); [ zenon_intro zenon_Hae | zenon_intro zenon_Had ]. % 1.18/1.34 apply (zenon_notand_s _ _ zenon_Hae); [ zenon_intro zenon_Hb0 | zenon_intro zenon_Haf ]. % 1.18/1.34 exact (zenon_Hb0 zenon_Haa). % 1.18/1.34 exact (zenon_Haf zenon_Ha9). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Had); [ zenon_intro zenon_H3c | zenon_intro zenon_Hb1 ]. % 1.18/1.34 exact (zenon_H3e zenon_H3c). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hb1); [ zenon_intro zenon_H2c | zenon_intro zenon_Hb2 ]. % 1.18/1.34 exact (zenon_H31 zenon_H2c). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hb2); [ zenon_intro zenon_H92 | zenon_intro zenon_H8e ]. % 1.18/1.34 apply (zenon_L20_ zenon_TX_bo zenon_TU_bu zenon_TV_bn zenon_TY_cs); trivial. % 1.18/1.34 apply (zenon_L19_ zenon_TV_bn zenon_TY_cs); trivial. % 1.18/1.34 (* end of lemma zenon_L21_ *) % 1.18/1.34 assert (zenon_L22_ : forall (zenon_TY_cs : zenon_U) (zenon_TX_bo : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (forall Y : zenon_U, ((convergent_lines zenon_TU_bu Y)->(~(apart_point_and_line (intersection_point zenon_TU_bu Y) zenon_TU_bu)))) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (distinct_lines zenon_TU_bu zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (distinct_lines zenon_TY_cs zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (~(distinct_lines zenon_TV_bn zenon_TV_bn)) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> False). % 1.18/1.34 do 4 intro. intros zenon_H3b zenon_H2d zenon_Ha9 zenon_H31 zenon_H4c zenon_H44 zenon_H93 zenon_H81 zenon_H4f zenon_H3a zenon_H84 zenon_H87. % 1.18/1.34 generalize (ceq1 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H6c. % 1.18/1.34 generalize (zenon_H6c zenon_TY_cs). zenon_intro zenon_Hb3. % 1.18/1.34 generalize (zenon_H6c zenon_TU_bu). zenon_intro zenon_Hb4. % 1.18/1.34 generalize (zenon_Hb3 (intersection_point zenon_TY_cs zenon_TV_bn)). zenon_intro zenon_Hb5. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hb5); [ zenon_intro zenon_H8a | zenon_intro zenon_Hb6 ]. % 1.18/1.34 exact (zenon_H8a zenon_H87). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hb6); [ zenon_intro zenon_Haa | zenon_intro zenon_Ha6 ]. % 1.18/1.34 generalize (zenon_Hb4 (intersection_point zenon_TX_bo zenon_TY_cs)). zenon_intro zenon_Hb7. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hb7); [ zenon_intro zenon_H3e | zenon_intro zenon_Hb8 ]. % 1.18/1.34 generalize (cu1 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H71. % 1.18/1.34 generalize (zenon_H71 (intersection_point zenon_TY_cs zenon_TV_bn)). zenon_intro zenon_Ha8. % 1.18/1.34 apply (zenon_L21_ zenon_TX_bo zenon_TY_cs zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hb8); [ zenon_intro zenon_Hb9 | zenon_intro zenon_H53 ]. % 1.18/1.34 generalize (cu1 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H71. % 1.18/1.34 generalize (zenon_H71 (intersection_point zenon_TX_bo zenon_TY_cs)). zenon_intro zenon_Hba. % 1.18/1.34 generalize (zenon_Hba zenon_TU_bu). zenon_intro zenon_Hbb. % 1.18/1.34 generalize (zenon_Hbb zenon_TV_bn). zenon_intro zenon_Hbc. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hbc); [ zenon_intro zenon_Hbe | zenon_intro zenon_Hbd ]. % 1.18/1.34 apply (zenon_notand_s _ _ zenon_Hbe); [ zenon_intro zenon_Hbf | zenon_intro zenon_Haf ]. % 1.18/1.34 exact (zenon_Hbf zenon_Hb9). % 1.18/1.34 exact (zenon_Haf zenon_Ha9). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hbd); [ zenon_intro zenon_H3c | zenon_intro zenon_Hc0 ]. % 1.18/1.34 apply (zenon_L5_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hc0); [ zenon_intro zenon_H2c | zenon_intro zenon_Hc1 ]. % 1.18/1.34 exact (zenon_H31 zenon_H2c). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hc1); [ zenon_intro zenon_H53 | zenon_intro zenon_H60 ]. % 1.18/1.34 exact (zenon_H4f zenon_H53). % 1.18/1.34 exact (zenon_H4c zenon_H60). % 1.18/1.34 exact (zenon_H4f zenon_H53). % 1.18/1.34 exact (zenon_H93 zenon_Ha6). % 1.18/1.34 (* end of lemma zenon_L22_ *) % 1.18/1.34 assert (zenon_L23_ : forall (zenon_TX_bo : zenon_U) (zenon_TY_cs : zenon_U) (zenon_TU_bu : zenon_U) (zenon_TV_bn : zenon_U), (forall Y : zenon_U, (forall Z : zenon_U, ((convergent_lines zenon_TV_bn Y)->((distinct_lines Y Z)\/(convergent_lines zenon_TV_bn Z))))) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> (~(distinct_lines zenon_TV_bn zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (distinct_lines zenon_TY_cs zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (forall Y : zenon_U, ((convergent_lines zenon_TU_bu Y)->(~(apart_point_and_line (intersection_point zenon_TU_bu Y) zenon_TU_bu)))) -> (convergent_lines zenon_TV_bn zenon_TU_bu) -> False). % 1.18/1.34 do 4 intro. intros zenon_Hc2 zenon_Hc3 zenon_H87 zenon_H84 zenon_H3a zenon_H4f zenon_H81 zenon_H93 zenon_H44 zenon_H4c zenon_H2d zenon_H3b zenon_Hc4. % 1.18/1.34 generalize (zenon_Hc2 zenon_TU_bu). zenon_intro zenon_Hc5. % 1.18/1.34 generalize (zenon_Hc5 zenon_TV_bn). zenon_intro zenon_Hc6. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hc6); [ zenon_intro zenon_Hc8 | zenon_intro zenon_Hc7 ]. % 1.18/1.34 exact (zenon_Hc8 zenon_Hc4). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hc7); [ zenon_intro zenon_Ha9 | zenon_intro zenon_Hc9 ]. % 1.18/1.34 generalize (ceq2 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H34. % 1.18/1.34 generalize (zenon_H34 zenon_TV_bn). zenon_intro zenon_Hca. % 1.18/1.34 generalize (zenon_Hca zenon_TV_bn). zenon_intro zenon_Hcb. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hcb); [ zenon_intro zenon_H31 | zenon_intro zenon_Hcc ]. % 1.18/1.34 apply (zenon_L22_ zenon_TY_cs zenon_TX_bo zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hcc); [ zenon_intro zenon_H39 | zenon_intro zenon_H2c ]. % 1.18/1.34 exact (zenon_H3a zenon_H39). % 1.18/1.34 apply (zenon_L2_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 exact (zenon_Hc3 zenon_Hc9). % 1.18/1.34 (* end of lemma zenon_L23_ *) % 1.18/1.34 assert (zenon_L24_ : forall (zenon_TX_bo : zenon_U) (zenon_TY_cs : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (forall Z : zenon_U, ((convergent_lines zenon_TU_bu zenon_TV_bn)->((distinct_lines zenon_TV_bn Z)\/(convergent_lines zenon_TU_bu Z)))) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (forall Y : zenon_U, ((distinct_lines zenon_TV_bn Y)->(convergent_lines zenon_TV_bn Y))) -> (forall Y : zenon_U, (forall Z : zenon_U, ((convergent_lines zenon_TV_bn Y)->((distinct_lines Y Z)\/(convergent_lines zenon_TV_bn Z))))) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> (~(distinct_lines zenon_TV_bn zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (distinct_lines zenon_TY_cs zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (forall Y : zenon_U, ((convergent_lines zenon_TU_bu Y)->(~(apart_point_and_line (intersection_point zenon_TU_bu Y) zenon_TU_bu)))) -> (~(convergent_lines zenon_TU_bu zenon_TU_bu)) -> False). % 1.18/1.34 do 4 intro. intros zenon_H6a zenon_H2d zenon_Hcd zenon_Hc2 zenon_Hc3 zenon_H87 zenon_H84 zenon_H3a zenon_H4f zenon_H81 zenon_H93 zenon_H44 zenon_H4c zenon_H3b zenon_H69. % 1.18/1.34 generalize (zenon_H6a zenon_TU_bu). zenon_intro zenon_H73. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H73); [ zenon_intro zenon_H32 | zenon_intro zenon_H74 ]. % 1.18/1.34 exact (zenon_H32 zenon_H2d). % 1.18/1.34 apply (zenon_or_s _ _ zenon_H74); [ zenon_intro zenon_H75 | zenon_intro zenon_H68 ]. % 1.18/1.34 generalize (zenon_Hcd zenon_TU_bu). zenon_intro zenon_Hce. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hce); [ zenon_intro zenon_H7c | zenon_intro zenon_Hc4 ]. % 1.18/1.34 exact (zenon_H7c zenon_H75). % 1.18/1.34 apply (zenon_L23_ zenon_TX_bo zenon_TY_cs zenon_TU_bu zenon_TV_bn); trivial. % 1.18/1.34 exact (zenon_H69 zenon_H68). % 1.18/1.34 (* end of lemma zenon_L24_ *) % 1.18/1.34 assert (zenon_L25_ : forall (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), ((convergent_lines zenon_TU_bu zenon_TV_bn)/\(unorthogonal_lines zenon_TU_bu zenon_TV_bn)) -> (~(unorthogonal_lines zenon_TU_bu zenon_TV_bn)) -> False). % 1.18/1.34 do 2 intro. intros zenon_Hcf zenon_Hd0. % 1.18/1.34 apply (zenon_and_s _ _ zenon_Hcf). zenon_intro zenon_H2d. zenon_intro zenon_Hd1. % 1.18/1.34 exact (zenon_Hd0 zenon_Hd1). % 1.18/1.34 (* end of lemma zenon_L25_ *) % 1.18/1.34 assert (zenon_L26_ : forall (zenon_TX_bo : zenon_U) (zenon_TY_cs : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U), (forall Z : zenon_U, ((convergent_lines zenon_TU_bu zenon_TV_bn)->((distinct_lines zenon_TV_bn Z)\/(convergent_lines zenon_TU_bu Z)))) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> (~(distinct_lines zenon_TV_bn zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (distinct_lines zenon_TY_cs zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (~(unorthogonal_lines zenon_TU_bu zenon_TV_bn)) -> (forall M : zenon_U, (forall N : zenon_U, (((convergent_lines zenon_TU_bu M)/\(unorthogonal_lines zenon_TU_bu M))->(((convergent_lines zenon_TU_bu N)/\(unorthogonal_lines zenon_TU_bu N))\/((convergent_lines M N)/\(unorthogonal_lines M N)))))) -> False). % 1.18/1.34 do 4 intro. intros zenon_H6a zenon_H2d zenon_Hc3 zenon_H87 zenon_H84 zenon_H3a zenon_H4f zenon_H81 zenon_H93 zenon_H44 zenon_H4c zenon_Hd0 zenon_Hd2. % 1.18/1.34 generalize (couo1 zenon_TU_bu). zenon_intro zenon_Hd3. % 1.18/1.34 generalize (p1 zenon_TV_bn). zenon_intro zenon_Hcd. % 1.18/1.34 generalize (ceq3 zenon_TV_bn). zenon_intro zenon_Hc2. % 1.18/1.34 generalize (ci3 zenon_TU_bu). zenon_intro zenon_H3b. % 1.18/1.34 generalize (zenon_Hd3 zenon_TU_bu). zenon_intro zenon_Hd4. % 1.18/1.34 generalize (zenon_Hd4 zenon_TV_bn). zenon_intro zenon_Hd5. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hd5); [ zenon_intro zenon_Hd6 | zenon_intro zenon_H32 ]. % 1.18/1.34 apply (zenon_notand_s _ _ zenon_Hd6); [ zenon_intro zenon_Hd8 | zenon_intro zenon_Hd7 ]. % 1.18/1.34 apply zenon_Hd8. zenon_intro zenon_Hd9. % 1.18/1.34 generalize (zenon_Hd2 zenon_TU_bu). zenon_intro zenon_Hda. % 1.18/1.34 generalize (zenon_Hda zenon_TV_bn). zenon_intro zenon_Hdb. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hdb); [ zenon_intro zenon_Hdd | zenon_intro zenon_Hdc ]. % 1.18/1.34 apply (zenon_notand_s _ _ zenon_Hdd); [ zenon_intro zenon_H69 | zenon_intro zenon_Hde ]. % 1.18/1.34 apply (zenon_L24_ zenon_TX_bo zenon_TY_cs zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 exact (zenon_Hde zenon_Hd9). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hdc); [ zenon_intro zenon_Hcf | zenon_intro zenon_Hcf ]. % 1.18/1.34 apply (zenon_L25_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 apply (zenon_L25_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 exact (zenon_Hd7 zenon_Hd0). % 1.18/1.34 exact (zenon_H32 zenon_H2d). % 1.18/1.34 (* end of lemma zenon_L26_ *) % 1.18/1.34 assert (zenon_L27_ : forall (zenon_TV_bn : zenon_U), (~((~(convergent_lines zenon_TV_bn zenon_TV_bn))\/(~(unorthogonal_lines zenon_TV_bn zenon_TV_bn)))) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> False). % 1.18/1.34 do 1 intro. intros zenon_Hdf zenon_Hc3. % 1.18/1.34 apply (zenon_notor_s _ _ zenon_Hdf). zenon_intro zenon_He1. zenon_intro zenon_He0. % 1.18/1.34 exact (zenon_He1 zenon_Hc3). % 1.18/1.34 (* end of lemma zenon_L27_ *) % 1.18/1.34 assert (zenon_L28_ : forall (zenon_TX_bo : zenon_U) (zenon_TY_cs : zenon_U) (zenon_TU_bu : zenon_U) (zenon_TV_bn : zenon_U), (forall Y : zenon_U, (forall Z : zenon_U, ((convergent_lines zenon_TV_bn Y)->((distinct_lines Y Z)\/(convergent_lines zenon_TV_bn Z))))) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> (~(distinct_lines zenon_TV_bn zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (distinct_lines zenon_TY_cs zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (forall Y : zenon_U, ((convergent_lines zenon_TU_bu Y)->(~(apart_point_and_line (intersection_point zenon_TU_bu Y) zenon_TU_bu)))) -> (unorthogonal_lines zenon_TU_bu zenon_TV_bn) -> False). % 1.18/1.34 do 4 intro. intros zenon_Hc2 zenon_Hc3 zenon_H87 zenon_H84 zenon_H3a zenon_H4f zenon_H81 zenon_H93 zenon_H44 zenon_H4c zenon_H2d zenon_H3b zenon_Hd1. % 1.18/1.34 generalize (cotno1 zenon_TV_bn). zenon_intro zenon_He2. % 1.18/1.34 generalize (zenon_He2 zenon_TU_bu). zenon_intro zenon_He3. % 1.18/1.34 generalize (zenon_He3 zenon_TV_bn). zenon_intro zenon_He4. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_He4); [ zenon_intro zenon_He6 | zenon_intro zenon_He5 ]. % 1.18/1.34 apply (zenon_notand_s _ _ zenon_He6); [ zenon_intro zenon_He7 | zenon_intro zenon_Hdf ]. % 1.18/1.34 apply (zenon_notor_s _ _ zenon_He7). zenon_intro zenon_He9. zenon_intro zenon_He8. % 1.18/1.34 apply zenon_He9. zenon_intro zenon_Hc4. % 1.18/1.34 apply (zenon_L23_ zenon_TX_bo zenon_TY_cs zenon_TU_bu zenon_TV_bn); trivial. % 1.18/1.34 apply (zenon_L27_ zenon_TV_bn); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_He5); [ zenon_intro zenon_H32 | zenon_intro zenon_Hd0 ]. % 1.18/1.34 exact (zenon_H32 zenon_H2d). % 1.18/1.34 exact (zenon_Hd0 zenon_Hd1). % 1.18/1.34 (* end of lemma zenon_L28_ *) % 1.18/1.34 assert (zenon_L29_ : forall (zenon_TV_bn : zenon_U), ((convergent_lines zenon_TV_bn zenon_TV_bn)/\(unorthogonal_lines zenon_TV_bn zenon_TV_bn)) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> False). % 1.18/1.34 do 1 intro. intros zenon_Hea zenon_Hc3. % 1.18/1.34 apply (zenon_and_s _ _ zenon_Hea). zenon_intro zenon_Hc9. zenon_intro zenon_Heb. % 1.18/1.34 exact (zenon_Hc3 zenon_Hc9). % 1.18/1.34 (* end of lemma zenon_L29_ *) % 1.18/1.34 assert (zenon_L30_ : forall (zenon_TU_bu : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TY_cs : zenon_U) (zenon_TX_bo : zenon_U), (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (distinct_lines zenon_TY_cs zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> (forall Z : zenon_U, ((convergent_lines zenon_TU_bu zenon_TV_bn)->((distinct_lines zenon_TV_bn Z)\/(convergent_lines zenon_TU_bu Z)))) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> False). % 1.18/1.34 do 4 intro. intros zenon_H4c zenon_H44 zenon_H93 zenon_H81 zenon_H4f zenon_H84 zenon_H87 zenon_Hc3 zenon_H6a zenon_H2d. % 1.18/1.34 generalize (apart2 zenon_TV_bn). zenon_intro zenon_H3a. % 1.18/1.34 generalize (oac1 zenon_TU_bu). zenon_intro zenon_Hd2. % 1.18/1.34 generalize (zenon_Hd2 zenon_TV_bn). zenon_intro zenon_Hec. % 1.18/1.34 generalize (ceq3 zenon_TV_bn). zenon_intro zenon_Hc2. % 1.18/1.34 generalize (ci3 zenon_TU_bu). zenon_intro zenon_H3b. % 1.18/1.34 generalize (zenon_Hec zenon_TV_bn). zenon_intro zenon_Hed. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hed); [ zenon_intro zenon_Hef | zenon_intro zenon_Hee ]. % 1.18/1.34 apply (zenon_notand_s _ _ zenon_Hef); [ zenon_intro zenon_H32 | zenon_intro zenon_Hd0 ]. % 1.18/1.34 exact (zenon_H32 zenon_H2d). % 1.18/1.34 apply (zenon_L26_ zenon_TX_bo zenon_TY_cs zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hee); [ zenon_intro zenon_Hcf | zenon_intro zenon_Hea ]. % 1.18/1.34 apply (zenon_and_s _ _ zenon_Hcf). zenon_intro zenon_H2d. zenon_intro zenon_Hd1. % 1.18/1.34 apply (zenon_L28_ zenon_TX_bo zenon_TY_cs zenon_TU_bu zenon_TV_bn); trivial. % 1.18/1.34 apply (zenon_L29_ zenon_TV_bn); trivial. % 1.18/1.34 (* end of lemma zenon_L30_ *) % 1.18/1.34 assert (zenon_L31_ : forall (zenon_TU_bu : zenon_U) (zenon_TX_bo : zenon_U) (zenon_TY_cs : zenon_U) (zenon_TV_bn : zenon_U), (convergent_lines zenon_TV_bn zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> (forall Z : zenon_U, ((convergent_lines zenon_TU_bu zenon_TV_bn)->((distinct_lines zenon_TV_bn Z)\/(convergent_lines zenon_TU_bu Z)))) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> False). % 1.18/1.34 do 4 intro. intros zenon_Hf0 zenon_H4c zenon_H44 zenon_H93 zenon_H4f zenon_H84 zenon_H87 zenon_Hc3 zenon_H6a zenon_H2d. % 1.18/1.34 generalize (ceq3 zenon_TV_bn). zenon_intro zenon_Hc2. % 1.18/1.34 generalize (zenon_Hc2 zenon_TY_cs). zenon_intro zenon_Hf1. % 1.18/1.34 generalize (zenon_Hf1 zenon_TV_bn). zenon_intro zenon_Hf2. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hf2); [ zenon_intro zenon_Hf4 | zenon_intro zenon_Hf3 ]. % 1.18/1.34 exact (zenon_Hf4 zenon_Hf0). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hf3); [ zenon_intro zenon_H81 | zenon_intro zenon_Hc9 ]. % 1.18/1.34 apply (zenon_L30_ zenon_TU_bu zenon_TV_bn zenon_TY_cs zenon_TX_bo); trivial. % 1.18/1.34 exact (zenon_Hc3 zenon_Hc9). % 1.18/1.34 (* end of lemma zenon_L31_ *) % 1.18/1.34 assert (zenon_L32_ : forall (zenon_TU_bu : zenon_U) (zenon_TX_bo : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TY_cs : zenon_U), ((convergent_lines zenon_TY_cs zenon_TV_bn)->((convergent_lines zenon_TY_cs zenon_TY_cs)\/(convergent_lines zenon_TV_bn zenon_TY_cs))) -> (~(convergent_lines zenon_TY_cs zenon_TY_cs)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (convergent_lines zenon_TY_cs zenon_TV_bn) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> (forall Z : zenon_U, ((convergent_lines zenon_TU_bu zenon_TV_bn)->((distinct_lines zenon_TV_bn Z)\/(convergent_lines zenon_TU_bu Z)))) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> False). % 1.18/1.34 do 4 intro. intros zenon_Hf5 zenon_Hf6 zenon_H4c zenon_H44 zenon_H93 zenon_H4f zenon_H84 zenon_H87 zenon_Hc3 zenon_H6a zenon_H2d. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hf5); [ zenon_intro zenon_H80 | zenon_intro zenon_Hf7 ]. % 1.18/1.34 exact (zenon_H80 zenon_H84). % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hf7); [ zenon_intro zenon_Hf8 | zenon_intro zenon_Hf0 ]. % 1.18/1.34 exact (zenon_Hf6 zenon_Hf8). % 1.18/1.34 apply (zenon_L31_ zenon_TU_bu zenon_TX_bo zenon_TY_cs zenon_TV_bn); trivial. % 1.18/1.34 (* end of lemma zenon_L32_ *) % 1.18/1.34 assert (zenon_L33_ : forall (zenon_TX_bo : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U) (zenon_TY_cs : zenon_U), (forall Y : zenon_U, (forall Z : zenon_U, ((convergent_lines zenon_TY_cs Y)->((convergent_lines zenon_TY_cs Z)\/(convergent_lines Y Z))))) -> ((apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs)->((distinct_lines zenon_TY_cs zenon_TV_bn)\/(apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TV_bn))) -> (~(apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TV_bn)) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (forall Z : zenon_U, ((convergent_lines zenon_TU_bu zenon_TV_bn)->((distinct_lines zenon_TV_bn Z)\/(convergent_lines zenon_TU_bu Z)))) -> (~(convergent_lines zenon_TV_bn zenon_TV_bn)) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (~(apart_point_and_line (intersection_point zenon_TY_cs zenon_TV_bn) zenon_TY_cs)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (~(convergent_lines zenon_TY_cs zenon_TY_cs)) -> False). % 1.18/1.34 do 4 intro. intros zenon_Hf9 zenon_H88 zenon_H31 zenon_H87 zenon_H2d zenon_H6a zenon_Hc3 zenon_H4f zenon_H93 zenon_H44 zenon_H4c zenon_Hf6. % 1.18/1.34 generalize (zenon_Hf9 zenon_TV_bn). zenon_intro zenon_Hfa. % 1.18/1.34 generalize (zenon_Hfa zenon_TY_cs). zenon_intro zenon_Hf5. % 1.18/1.34 generalize (zenon_Hfa zenon_TV_bn). zenon_intro zenon_Hfb. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hfb); [ zenon_intro zenon_H80 | zenon_intro zenon_Hfc ]. % 1.18/1.34 apply (zenon_L17_ zenon_TY_cs zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hfc); [ zenon_intro zenon_H84 | zenon_intro zenon_Hc9 ]. % 1.18/1.34 apply (zenon_L32_ zenon_TU_bu zenon_TX_bo zenon_TV_bn zenon_TY_cs); trivial. % 1.18/1.34 exact (zenon_Hc3 zenon_Hc9). % 1.18/1.34 (* end of lemma zenon_L33_ *) % 1.18/1.34 assert (zenon_L34_ : forall (zenon_TX_bo : zenon_U) (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U) (zenon_TY_cs : zenon_U), (forall Y : zenon_U, ((convergent_lines zenon_TY_cs Y)->(~(apart_point_and_line (intersection_point zenon_TY_cs Y) zenon_TY_cs)))) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> False). % 1.18/1.34 do 4 intro. intros zenon_Hfd zenon_H87 zenon_H2d zenon_H4c zenon_H44 zenon_H4f. % 1.18/1.34 generalize (apart3 zenon_TV_bn). zenon_intro zenon_Hc3. % 1.18/1.34 generalize (apart3 zenon_TY_cs). zenon_intro zenon_Hf6. % 1.18/1.34 generalize (zenon_Hfd zenon_TV_bn). zenon_intro zenon_Hfe. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hfe); [ zenon_intro zenon_H80 | zenon_intro zenon_H93 ]. % 1.18/1.34 generalize (ceq2 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H34. % 1.18/1.34 generalize (zenon_H34 zenon_TY_cs). zenon_intro zenon_H86. % 1.18/1.34 apply (zenon_L16_ zenon_TY_cs zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 generalize (ceq3 zenon_TU_bu). zenon_intro zenon_Hff. % 1.18/1.34 generalize (zenon_Hff zenon_TV_bn). zenon_intro zenon_H6a. % 1.18/1.34 generalize (ceq2 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H34. % 1.18/1.34 generalize (zenon_H34 zenon_TY_cs). zenon_intro zenon_H86. % 1.18/1.34 generalize (zenon_H34 zenon_TV_bn). zenon_intro zenon_Hca. % 1.18/1.34 generalize (ax6 zenon_TY_cs). zenon_intro zenon_Hf9. % 1.18/1.34 generalize (zenon_H86 zenon_TV_bn). zenon_intro zenon_H88. % 1.18/1.34 generalize (zenon_Hca zenon_TV_bn). zenon_intro zenon_Hcb. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_Hcb); [ zenon_intro zenon_H31 | zenon_intro zenon_Hcc ]. % 1.18/1.34 apply (zenon_L33_ zenon_TX_bo zenon_TV_bn zenon_TU_bu zenon_TY_cs); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_Hcc); [ zenon_intro zenon_H39 | zenon_intro zenon_H2c ]. % 1.18/1.34 apply (zenon_L4_ zenon_TV_bn); trivial. % 1.18/1.34 apply (zenon_L2_ zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 (* end of lemma zenon_L34_ *) % 1.18/1.34 assert (zenon_L35_ : forall (zenon_TV_bn : zenon_U) (zenon_TU_bu : zenon_U) (zenon_TY_cs : zenon_U) (zenon_TX_bo : zenon_U), (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)) -> (convergent_lines zenon_TX_bo zenon_TY_cs) -> (~(apart_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TV_bn)) -> (convergent_lines zenon_TU_bu zenon_TV_bn) -> (apart_point_and_line (intersection_point zenon_TU_bu zenon_TV_bn) zenon_TY_cs) -> False). % 1.18/1.34 do 4 intro. intros zenon_H4f zenon_H44 zenon_H4c zenon_H2d zenon_H87. % 1.18/1.34 generalize (ci3 zenon_TY_cs). zenon_intro zenon_Hfd. % 1.18/1.34 apply (zenon_L34_ zenon_TX_bo zenon_TV_bn zenon_TU_bu zenon_TY_cs); trivial. % 1.18/1.34 (* end of lemma zenon_L35_ *) % 1.18/1.34 apply NNPP. intro zenon_G. % 1.18/1.34 apply (zenon_notallex_s (fun X : zenon_U => (forall Y : zenon_U, (forall U : zenon_U, (forall V : zenon_U, (((convergent_lines X Y)/\((convergent_lines U V)/\((incident_point_and_line (intersection_point X Y) U)/\(incident_point_and_line (intersection_point X Y) V))))->((incident_point_and_line (intersection_point U V) X)/\(incident_point_and_line (intersection_point U V) Y))))))) zenon_G); [ zenon_intro zenon_H100; idtac ]. % 1.18/1.34 elim zenon_H100. zenon_intro zenon_TX_bo. zenon_intro zenon_H101. % 1.18/1.34 apply (zenon_notallex_s (fun Y : zenon_U => (forall U : zenon_U, (forall V : zenon_U, (((convergent_lines zenon_TX_bo Y)/\((convergent_lines U V)/\((incident_point_and_line (intersection_point zenon_TX_bo Y) U)/\(incident_point_and_line (intersection_point zenon_TX_bo Y) V))))->((incident_point_and_line (intersection_point U V) zenon_TX_bo)/\(incident_point_and_line (intersection_point U V) Y)))))) zenon_H101); [ zenon_intro zenon_H102; idtac ]. % 1.18/1.34 elim zenon_H102. zenon_intro zenon_TY_cs. zenon_intro zenon_H103. % 1.18/1.34 apply (zenon_notallex_s (fun U : zenon_U => (forall V : zenon_U, (((convergent_lines zenon_TX_bo zenon_TY_cs)/\((convergent_lines U V)/\((incident_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) U)/\(incident_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) V))))->((incident_point_and_line (intersection_point U V) zenon_TX_bo)/\(incident_point_and_line (intersection_point U V) zenon_TY_cs))))) zenon_H103); [ zenon_intro zenon_H104; idtac ]. % 1.18/1.34 elim zenon_H104. zenon_intro zenon_TU_bu. zenon_intro zenon_H105. % 1.18/1.34 apply (zenon_notallex_s (fun V : zenon_U => (((convergent_lines zenon_TX_bo zenon_TY_cs)/\((convergent_lines zenon_TU_bu V)/\((incident_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) zenon_TU_bu)/\(incident_point_and_line (intersection_point zenon_TX_bo zenon_TY_cs) V))))->((incident_point_and_line (intersection_point zenon_TU_bu V) zenon_TX_bo)/\(incident_point_and_line (intersection_point zenon_TU_bu V) zenon_TY_cs)))) zenon_H105); [ zenon_intro zenon_H106; idtac ]. % 1.18/1.34 elim zenon_H106. zenon_intro zenon_TV_bn. zenon_intro zenon_H107. % 1.18/1.34 apply (zenon_notimply_s _ _ zenon_H107). zenon_intro zenon_H109. zenon_intro zenon_H108. % 1.18/1.34 apply (zenon_and_s _ _ zenon_H109). zenon_intro zenon_H44. zenon_intro zenon_H10a. % 1.18/1.34 apply (zenon_and_s _ _ zenon_H10a). zenon_intro zenon_H2d. zenon_intro zenon_H10b. % 1.18/1.34 apply (zenon_and_s _ _ zenon_H10b). zenon_intro zenon_H10d. zenon_intro zenon_H10c. % 1.18/1.34 generalize (a4 (intersection_point zenon_TX_bo zenon_TY_cs)). zenon_intro zenon_H10e. % 1.18/1.34 generalize (zenon_H10e zenon_TU_bu). zenon_intro zenon_H10f. % 1.18/1.34 apply (zenon_equiv_s _ _ zenon_H10f); [ zenon_intro zenon_H111; zenon_intro zenon_H110 | zenon_intro zenon_H10d; zenon_intro zenon_H4f ]. % 1.18/1.34 exact (zenon_H111 zenon_H10d). % 1.18/1.34 generalize (a4 (intersection_point zenon_TX_bo zenon_TY_cs)). zenon_intro zenon_H10e. % 1.18/1.34 generalize (zenon_H10e zenon_TV_bn). zenon_intro zenon_H112. % 1.18/1.34 apply (zenon_equiv_s _ _ zenon_H112); [ zenon_intro zenon_H114; zenon_intro zenon_H113 | zenon_intro zenon_H10c; zenon_intro zenon_H4c ]. % 1.18/1.34 exact (zenon_H114 zenon_H10c). % 1.18/1.34 apply (zenon_notand_s _ _ zenon_H108); [ zenon_intro zenon_H116 | zenon_intro zenon_H115 ]. % 1.18/1.34 generalize (a4 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H117. % 1.18/1.34 generalize (zenon_H117 zenon_TX_bo). zenon_intro zenon_H118. % 1.18/1.34 apply (zenon_equiv_s _ _ zenon_H118); [ zenon_intro zenon_H116; zenon_intro zenon_H11a | zenon_intro zenon_H119; zenon_intro zenon_H38 ]. % 1.18/1.34 apply zenon_H11a. zenon_intro zenon_H33. % 1.18/1.34 generalize (ci3 zenon_TX_bo). zenon_intro zenon_H43. % 1.18/1.34 generalize (zenon_H43 zenon_TV_bn). zenon_intro zenon_H11b. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H11b); [ zenon_intro zenon_H26 | zenon_intro zenon_H4d ]. % 1.18/1.34 apply (zenon_L3_ zenon_TX_bo zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 generalize (ceq3 zenon_TU_bu). zenon_intro zenon_Hff. % 1.18/1.34 generalize (ceq3 zenon_TX_bo). zenon_intro zenon_H11c. % 1.18/1.34 generalize (zenon_Hff zenon_TV_bn). zenon_intro zenon_H6a. % 1.18/1.34 generalize (zenon_H11c zenon_TV_bn). zenon_intro zenon_H6b. % 1.18/1.34 generalize (zenon_H6b zenon_TV_bn). zenon_intro zenon_H11d. % 1.18/1.34 apply (zenon_imply_s _ _ zenon_H11d); [ zenon_intro zenon_H26 | zenon_intro zenon_H11e ]. % 1.18/1.34 apply (zenon_L3_ zenon_TX_bo zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 apply (zenon_or_s _ _ zenon_H11e); [ zenon_intro zenon_H39 | zenon_intro zenon_H2a ]. % 1.18/1.34 apply (zenon_L4_ zenon_TV_bn); trivial. % 1.18/1.34 apply (zenon_L14_ zenon_TY_cs zenon_TX_bo zenon_TV_bn zenon_TU_bu); trivial. % 1.18/1.34 exact (zenon_H116 zenon_H119). % 1.18/1.34 generalize (a4 (intersection_point zenon_TU_bu zenon_TV_bn)). zenon_intro zenon_H117. % 1.18/1.34 generalize (zenon_H117 zenon_TY_cs). zenon_intro zenon_H11f. % 1.18/1.34 apply (zenon_equiv_s _ _ zenon_H11f); [ zenon_intro zenon_H115; zenon_intro zenon_H121 | zenon_intro zenon_H120; zenon_intro zenon_H8a ]. % 1.18/1.34 apply zenon_H121. zenon_intro zenon_H87. % 1.18/1.34 apply (zenon_L35_ zenon_TV_bn zenon_TU_bu zenon_TY_cs zenon_TX_bo); trivial. % 1.18/1.34 exact (zenon_H115 zenon_H120). % 1.18/1.34 Qed. % 1.18/1.34 % SZS output end Proof % 1.18/1.34 (* END-PROOF *) % 1.18/1.34 nodes searched: 55409 % 1.18/1.34 max branch formulas: 3529 % 1.18/1.34 proof nodes created: 3806 % 1.18/1.34 formulas created: 127987 % 1.18/1.34 %------------------------------------------------------------------------------