%------------------------------------------------------------------------------ % File : Zenon---0.7.1 % Problem : GEO223+1 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : run_zenon %s %d % Computer : n006.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 07:01:03 EDT 2022 % Result : Theorem 10.40s 10.58s % Output : Proof 10.40s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GEO223+1 : TPTP v8.1.0. Released v3.3.0. % 0.07/0.13 % Command : run_zenon %s %d % 0.12/0.34 % Computer : n006.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 600 % 0.12/0.34 % DateTime : Sat Jun 18 11:55:40 EDT 2022 % 0.12/0.34 % CPUTime : % 10.40/10.58 (* PROOF-FOUND *) % 10.40/10.58 % SZS status Theorem % 10.40/10.58 (* BEGIN-PROOF *) % 10.40/10.58 % SZS output start Proof % 10.40/10.58 Theorem con : (forall A : zenon_U, (forall L : zenon_U, (~(distinct_lines (parallel_through_point L A) (orthogonal_through_point (orthogonal_through_point L A) A))))). % 10.40/10.58 Proof. % 10.40/10.58 assert (zenon_L1_ : forall (zenon_TL_y : zenon_U) (zenon_TA_z : zenon_U), (apart_point_and_line zenon_TA_z (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H17. % 10.40/10.58 generalize (ooc2 zenon_TA_z). zenon_intro zenon_H1a. % 10.40/10.58 generalize (zenon_H1a (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H1b. % 10.40/10.58 exact (zenon_H1b zenon_H17). % 10.40/10.58 (* end of lemma zenon_L1_ *) % 10.40/10.58 assert (zenon_L2_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall Y : zenon_U, (forall Z : zenon_U, ((convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) Y)->((distinct_lines Y Z)\/(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) Z))))) -> (~(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z))) -> (~(distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z))) -> (convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H1c zenon_H1d zenon_H1e zenon_H1f. % 10.40/10.58 generalize (zenon_H1c (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H20. % 10.40/10.58 generalize (zenon_H20 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H21. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H21); [ zenon_intro zenon_H23 | zenon_intro zenon_H22 ]. % 10.40/10.58 exact (zenon_H23 zenon_H1f). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H22); [ zenon_intro zenon_H25 | zenon_intro zenon_H24 ]. % 10.40/10.58 exact (zenon_H1e zenon_H25). % 10.40/10.58 exact (zenon_H1d zenon_H24). % 10.40/10.58 (* end of lemma zenon_L2_ *) % 10.40/10.58 assert (zenon_L3_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall Z : zenon_U, ((distinct_lines (parallel_through_point zenon_TL_y zenon_TA_z) Z)->((apart_point_and_line zenon_TA_z (parallel_through_point zenon_TL_y zenon_TA_z))\/((apart_point_and_line zenon_TA_z Z)\/(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) Z))))) -> (distinct_lines (parallel_through_point zenon_TL_y zenon_TA_z) (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)) -> (~(apart_point_and_line zenon_TA_z (parallel_through_point zenon_TL_y zenon_TA_z))) -> (~(apart_point_and_line zenon_TA_z (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))) -> (forall Y : zenon_U, (forall Z : zenon_U, ((convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) Y)->((distinct_lines Y Z)\/(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) Z))))) -> (~(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z))) -> (~(distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z))) -> False). % 10.40/10.58 do 2 intro. intros zenon_H26 zenon_H27 zenon_H28 zenon_H29 zenon_H1c zenon_H1d zenon_H1e. % 10.40/10.58 generalize (zenon_H26 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H2a. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H2a); [ zenon_intro zenon_H2c | zenon_intro zenon_H2b ]. % 10.40/10.58 exact (zenon_H2c zenon_H27). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H2b); [ zenon_intro zenon_H2e | zenon_intro zenon_H2d ]. % 10.40/10.58 exact (zenon_H28 zenon_H2e). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H2d); [ zenon_intro zenon_H2f | zenon_intro zenon_H1f ]. % 10.40/10.58 exact (zenon_H29 zenon_H2f). % 10.40/10.58 apply (zenon_L2_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 (* end of lemma zenon_L3_ *) % 10.40/10.58 assert (zenon_L4_ : forall (zenon_TL_y : zenon_U) (zenon_TA_z : zenon_U), (forall Y : zenon_U, (~(apart_point_and_line zenon_TA_z (parallel_through_point Y zenon_TA_z)))) -> (~(distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z))) -> (~(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z))) -> (forall Y : zenon_U, (forall Z : zenon_U, ((convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) Y)->((distinct_lines Y Z)\/(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) Z))))) -> (~(apart_point_and_line zenon_TA_z (parallel_through_point zenon_TL_y zenon_TA_z))) -> (distinct_lines (parallel_through_point zenon_TL_y zenon_TA_z) (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)) -> (forall Z : zenon_U, ((distinct_lines (parallel_through_point zenon_TL_y zenon_TA_z) Z)->((apart_point_and_line zenon_TA_z (parallel_through_point zenon_TL_y zenon_TA_z))\/((apart_point_and_line zenon_TA_z Z)\/(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) Z))))) -> False). % 10.40/10.58 do 2 intro. intros zenon_H30 zenon_H1e zenon_H1d zenon_H1c zenon_H28 zenon_H27 zenon_H26. % 10.40/10.58 generalize (zenon_H30 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H29. % 10.40/10.58 apply (zenon_L3_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 (* end of lemma zenon_L4_ *) % 10.40/10.58 assert (zenon_L5_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall M : zenon_U, (forall N : zenon_U, ((distinct_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) M)->((apart_point_and_line zenon_TA_z (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))\/((apart_point_and_line zenon_TA_z M)\/((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N)\/(unorthogonal_lines M N))))))) -> (~(unorthogonal_lines (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)) -> (~(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)) -> (~(apart_point_and_line zenon_TA_z (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> (~(apart_point_and_line zenon_TA_z (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))) -> (distinct_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H31 zenon_H32 zenon_H33 zenon_H34 zenon_H1b zenon_H35. % 10.40/10.58 generalize (zenon_H31 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H36. % 10.40/10.58 generalize (zenon_H36 zenon_TL_y). zenon_intro zenon_H37. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H37); [ zenon_intro zenon_H39 | zenon_intro zenon_H38 ]. % 10.40/10.58 exact (zenon_H39 zenon_H35). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H38); [ zenon_intro zenon_H17 | zenon_intro zenon_H3a ]. % 10.40/10.58 exact (zenon_H1b zenon_H17). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H3a); [ zenon_intro zenon_H3c | zenon_intro zenon_H3b ]. % 10.40/10.58 exact (zenon_H34 zenon_H3c). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H3b); [ zenon_intro zenon_H3e | zenon_intro zenon_H3d ]. % 10.40/10.58 exact (zenon_H33 zenon_H3e). % 10.40/10.58 exact (zenon_H32 zenon_H3d). % 10.40/10.58 (* end of lemma zenon_L5_ *) % 10.40/10.58 assert (zenon_L6_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), ((convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)/\(unorthogonal_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)) -> (~(convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H3f zenon_H40. % 10.40/10.58 apply (zenon_and_s _ _ zenon_H3f). zenon_intro zenon_H42. zenon_intro zenon_H41. % 10.40/10.58 exact (zenon_H40 zenon_H42). % 10.40/10.58 (* end of lemma zenon_L6_ *) % 10.40/10.58 assert (zenon_L7_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), ((convergent_lines (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)/\(unorthogonal_lines (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)) -> (~(unorthogonal_lines (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H43 zenon_H32. % 10.40/10.58 apply (zenon_and_s _ _ zenon_H43). zenon_intro zenon_H44. zenon_intro zenon_H3d. % 10.40/10.58 exact (zenon_H32 zenon_H3d). % 10.40/10.58 (* end of lemma zenon_L7_ *) % 10.40/10.58 assert (zenon_L8_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall M : zenon_U, (forall N : zenon_U, (((convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) M)/\(unorthogonal_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) M))->(((convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N)/\(unorthogonal_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N))\/((convergent_lines M N)/\(unorthogonal_lines M N)))))) -> (~(unorthogonal_lines (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)) -> (~(convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)) -> (unorthogonal_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z)) -> (convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H45 zenon_H32 zenon_H40 zenon_H46 zenon_H47. % 10.40/10.58 generalize (zenon_H45 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H48. % 10.40/10.58 generalize (zenon_H48 zenon_TL_y). zenon_intro zenon_H49. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H49); [ zenon_intro zenon_H4b | zenon_intro zenon_H4a ]. % 10.40/10.58 apply (zenon_notand_s _ _ zenon_H4b); [ zenon_intro zenon_H4d | zenon_intro zenon_H4c ]. % 10.40/10.58 exact (zenon_H4d zenon_H47). % 10.40/10.58 exact (zenon_H4c zenon_H46). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H4a); [ zenon_intro zenon_H3f | zenon_intro zenon_H43 ]. % 10.40/10.58 apply (zenon_L6_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_L7_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 (* end of lemma zenon_L8_ *) % 10.40/10.58 assert (zenon_L9_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall N : zenon_U, ((distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))->((apart_point_and_line zenon_TA_z (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))\/((apart_point_and_line zenon_TA_z (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))\/((unorthogonal_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N)\/(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N)))))) -> (distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)) -> (~(apart_point_and_line zenon_TA_z (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))) -> (~(apart_point_and_line zenon_TA_z (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))) -> (forall Z : zenon_U, ((distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z)->((apart_point_and_line zenon_TA_z (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))\/((apart_point_and_line zenon_TA_z Z)\/(convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z))))) -> (forall Z : zenon_U, ((distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))->((distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z)\/(distinct_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z)))) -> (forall M : zenon_U, (forall N : zenon_U, ((distinct_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) M)->((apart_point_and_line zenon_TA_z (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))\/((apart_point_and_line zenon_TA_z M)\/((unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N)\/(unorthogonal_lines M N))))))) -> (~(unorthogonal_lines (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)) -> (~(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)) -> (~(apart_point_and_line zenon_TA_z (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> (forall M : zenon_U, (forall N : zenon_U, (((convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) M)/\(unorthogonal_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) M))->(((convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N)/\(unorthogonal_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N))\/((convergent_lines M N)/\(unorthogonal_lines M N)))))) -> (~(convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)) -> (~(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> False). % 10.40/10.58 do 2 intro. intros zenon_H4e zenon_H4f zenon_H29 zenon_H1b zenon_H50 zenon_H51 zenon_H31 zenon_H32 zenon_H33 zenon_H34 zenon_H45 zenon_H40 zenon_H52. % 10.40/10.58 generalize (zenon_H4e (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H53. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H53); [ zenon_intro zenon_H55 | zenon_intro zenon_H54 ]. % 10.40/10.58 exact (zenon_H55 zenon_H4f). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H54); [ zenon_intro zenon_H2f | zenon_intro zenon_H56 ]. % 10.40/10.58 exact (zenon_H29 zenon_H2f). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H56); [ zenon_intro zenon_H17 | zenon_intro zenon_H57 ]. % 10.40/10.58 exact (zenon_H1b zenon_H17). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H57); [ zenon_intro zenon_H46 | zenon_intro zenon_H58 ]. % 10.40/10.58 generalize (zenon_H50 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H59. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H59); [ zenon_intro zenon_H5b | zenon_intro zenon_H5a ]. % 10.40/10.58 generalize (zenon_H51 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H5c. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H5c); [ zenon_intro zenon_H55 | zenon_intro zenon_H5d ]. % 10.40/10.58 exact (zenon_H55 zenon_H4f). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H5d); [ zenon_intro zenon_H5e | zenon_intro zenon_H35 ]. % 10.40/10.58 exact (zenon_H5b zenon_H5e). % 10.40/10.58 apply (zenon_L5_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_or_s _ _ zenon_H5a); [ zenon_intro zenon_H2f | zenon_intro zenon_H5f ]. % 10.40/10.58 exact (zenon_H29 zenon_H2f). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H5f); [ zenon_intro zenon_H3c | zenon_intro zenon_H47 ]. % 10.40/10.58 exact (zenon_H34 zenon_H3c). % 10.40/10.58 apply (zenon_L8_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 exact (zenon_H52 zenon_H58). % 10.40/10.58 (* end of lemma zenon_L9_ *) % 10.40/10.58 assert (zenon_L10_ : forall (zenon_TL_y : zenon_U) (zenon_TA_z : zenon_U), (forall L : zenon_U, (~(unorthogonal_lines (orthogonal_through_point L zenon_TA_z) L))) -> (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H60 zenon_H58. % 10.40/10.58 generalize (zenon_H60 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H52. % 10.40/10.58 exact (zenon_H52 zenon_H58). % 10.40/10.58 (* end of lemma zenon_L10_ *) % 10.40/10.58 assert (zenon_L11_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), ((convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z))/\(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> (~(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> False). % 10.40/10.58 do 2 intro. intros zenon_H61 zenon_H52. % 10.40/10.58 apply (zenon_and_s _ _ zenon_H61). zenon_intro zenon_H62. zenon_intro zenon_H58. % 10.40/10.58 exact (zenon_H52 zenon_H58). % 10.40/10.58 (* end of lemma zenon_L11_ *) % 10.40/10.58 assert (zenon_L12_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall N : zenon_U, (((convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)/\(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y))->(((convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N)/\(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N))\/((convergent_lines zenon_TL_y N)/\(unorthogonal_lines zenon_TL_y N))))) -> (convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y) -> (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y) -> (~(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> (~(convergent_lines zenon_TL_y (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> False). % 10.40/10.58 do 2 intro. intros zenon_H63 zenon_H64 zenon_H3e zenon_H52 zenon_H65. % 10.40/10.58 generalize (zenon_H63 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H66. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H66); [ zenon_intro zenon_H68 | zenon_intro zenon_H67 ]. % 10.40/10.58 apply (zenon_notand_s _ _ zenon_H68); [ zenon_intro zenon_H69 | zenon_intro zenon_H33 ]. % 10.40/10.58 exact (zenon_H69 zenon_H64). % 10.40/10.58 exact (zenon_H33 zenon_H3e). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H67); [ zenon_intro zenon_H61 | zenon_intro zenon_H6a ]. % 10.40/10.58 apply (zenon_L11_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_and_s _ _ zenon_H6a). zenon_intro zenon_H6c. zenon_intro zenon_H6b. % 10.40/10.58 exact (zenon_H65 zenon_H6c). % 10.40/10.58 (* end of lemma zenon_L12_ *) % 10.40/10.58 assert (zenon_L13_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall N : zenon_U, (((convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)/\(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y))->(((convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N)/\(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N))\/((convergent_lines zenon_TL_y N)/\(unorthogonal_lines zenon_TL_y N))))) -> (convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y) -> (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y) -> (~(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> (~(unorthogonal_lines zenon_TL_y (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> False). % 10.40/10.58 do 2 intro. intros zenon_H63 zenon_H64 zenon_H3e zenon_H52 zenon_H6d. % 10.40/10.58 generalize (zenon_H63 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H66. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H66); [ zenon_intro zenon_H68 | zenon_intro zenon_H67 ]. % 10.40/10.58 apply (zenon_notand_s _ _ zenon_H68); [ zenon_intro zenon_H69 | zenon_intro zenon_H33 ]. % 10.40/10.58 exact (zenon_H69 zenon_H64). % 10.40/10.58 exact (zenon_H33 zenon_H3e). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H67); [ zenon_intro zenon_H61 | zenon_intro zenon_H6a ]. % 10.40/10.58 apply (zenon_L11_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_and_s _ _ zenon_H6a). zenon_intro zenon_H6c. zenon_intro zenon_H6b. % 10.40/10.58 exact (zenon_H6d zenon_H6b). % 10.40/10.58 (* end of lemma zenon_L13_ *) % 10.40/10.58 assert (zenon_L14_ : forall (zenon_TL_y : zenon_U), ((convergent_lines zenon_TL_y zenon_TL_y)/\(unorthogonal_lines zenon_TL_y zenon_TL_y)) -> (~(convergent_lines zenon_TL_y zenon_TL_y)) -> False). % 10.40/10.58 do 1 intro. intros zenon_H6e zenon_H6f. % 10.40/10.58 apply (zenon_and_s _ _ zenon_H6e). zenon_intro zenon_H71. zenon_intro zenon_H70. % 10.40/10.58 exact (zenon_H6f zenon_H71). % 10.40/10.58 (* end of lemma zenon_L14_ *) % 10.40/10.58 assert (zenon_L15_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall M : zenon_U, (forall N : zenon_U, (((convergent_lines zenon_TL_y M)/\(unorthogonal_lines zenon_TL_y M))->(((convergent_lines zenon_TL_y N)/\(unorthogonal_lines zenon_TL_y N))\/((convergent_lines M N)/\(unorthogonal_lines M N)))))) -> (~(unorthogonal_lines (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)) -> (~(convergent_lines zenon_TL_y zenon_TL_y)) -> (~(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point zenon_TL_y zenon_TA_z))) -> (unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y) -> (convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y) -> (forall N : zenon_U, (((convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)/\(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y))->(((convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N)/\(unorthogonal_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) N))\/((convergent_lines zenon_TL_y N)/\(unorthogonal_lines zenon_TL_y N))))) -> False). % 10.40/10.58 do 2 intro. intros zenon_H72 zenon_H32 zenon_H6f zenon_H52 zenon_H3e zenon_H64 zenon_H63. % 10.40/10.58 generalize (zenon_H72 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H73. % 10.40/10.58 generalize (zenon_H73 zenon_TL_y). zenon_intro zenon_H74. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H74); [ zenon_intro zenon_H76 | zenon_intro zenon_H75 ]. % 10.40/10.58 apply (zenon_notand_s _ _ zenon_H76); [ zenon_intro zenon_H65 | zenon_intro zenon_H6d ]. % 10.40/10.58 apply (zenon_L12_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_L13_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_or_s _ _ zenon_H75); [ zenon_intro zenon_H6e | zenon_intro zenon_H43 ]. % 10.40/10.58 apply (zenon_L14_ zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_L7_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 (* end of lemma zenon_L15_ *) % 10.40/10.58 assert (zenon_L16_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (convergent_lines zenon_TL_y (parallel_through_point zenon_TL_y zenon_TA_z)) -> (~(convergent_lines zenon_TL_y zenon_TL_y)) -> (~(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H77 zenon_H6f zenon_H78. % 10.40/10.58 generalize (ax6 zenon_TL_y). zenon_intro zenon_H79. % 10.40/10.58 generalize (zenon_H79 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H7a. % 10.40/10.58 generalize (zenon_H7a zenon_TL_y). zenon_intro zenon_H7b. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H7b); [ zenon_intro zenon_H7d | zenon_intro zenon_H7c ]. % 10.40/10.58 exact (zenon_H7d zenon_H77). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H7c); [ zenon_intro zenon_H71 | zenon_intro zenon_H7e ]. % 10.40/10.58 exact (zenon_H6f zenon_H71). % 10.40/10.58 exact (zenon_H78 zenon_H7e). % 10.40/10.58 (* end of lemma zenon_L16_ *) % 10.40/10.58 assert (zenon_L17_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall Z : zenon_U, ((convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y)->((convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z)\/(convergent_lines zenon_TL_y Z)))) -> (convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y) -> (~(convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z))) -> (~(convergent_lines zenon_TL_y zenon_TL_y)) -> (~(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H7f zenon_H42 zenon_H80 zenon_H6f zenon_H78. % 10.40/10.58 generalize (zenon_H7f (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H81. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H81); [ zenon_intro zenon_H40 | zenon_intro zenon_H82 ]. % 10.40/10.58 exact (zenon_H40 zenon_H42). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H82); [ zenon_intro zenon_H83 | zenon_intro zenon_H77 ]. % 10.40/10.58 exact (zenon_H80 zenon_H83). % 10.40/10.58 apply (zenon_L16_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 (* end of lemma zenon_L17_ *) % 10.40/10.58 assert (zenon_L18_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall Y : zenon_U, (forall Z : zenon_U, ((convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Y)->((convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z)\/(convergent_lines Y Z))))) -> (~(convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TL_y)) -> (~(convergent_lines zenon_TL_y zenon_TL_y)) -> (~(convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z))) -> (convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) zenon_TL_y) -> False). % 10.40/10.58 do 2 intro. intros zenon_H84 zenon_H78 zenon_H6f zenon_H80 zenon_H42. % 10.40/10.58 generalize (zenon_H84 zenon_TL_y). zenon_intro zenon_H7f. % 10.40/10.58 apply (zenon_L17_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 (* end of lemma zenon_L18_ *) % 10.40/10.58 assert (zenon_L19_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (forall Z : zenon_U, ((convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))->((convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) Z)\/(convergent_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z)))) -> (convergent_lines (parallel_through_point zenon_TL_y zenon_TA_z) (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)) -> (distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)) -> (forall Z : zenon_U, ((distinct_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z)->((apart_point_and_line zenon_TA_z (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))\/((apart_point_and_line zenon_TA_z Z)\/(convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z))))) -> (~(convergent_lines (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z))) -> (forall Y : zenon_U, (~(convergent_lines (parallel_through_point Y zenon_TA_z) Y))) -> (forall Y : zenon_U, (~(apart_point_and_line zenon_TA_z (parallel_through_point Y zenon_TA_z)))) -> False). % 10.40/10.58 do 2 intro. intros zenon_H85 zenon_H86 zenon_H4f zenon_H50 zenon_H80 zenon_H87 zenon_H30. % 10.40/10.58 generalize (ooc2 zenon_TA_z). zenon_intro zenon_H1a. % 10.40/10.58 generalize (zenon_H30 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H29. % 10.40/10.58 generalize (ax6 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H84. % 10.40/10.58 generalize (zenon_H1a (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H1b. % 10.40/10.58 generalize (apart5 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H88. % 10.40/10.58 generalize (ceq3 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H89. % 10.40/10.58 generalize (zenon_H88 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H51. % 10.40/10.58 generalize (zenon_H87 zenon_TL_y). zenon_intro zenon_H78. % 10.40/10.58 generalize (zenon_H85 zenon_TL_y). zenon_intro zenon_H8a. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H8a); [ zenon_intro zenon_H8c | zenon_intro zenon_H8b ]. % 10.40/10.58 exact (zenon_H8c zenon_H86). % 10.40/10.58 apply (zenon_or_s _ _ zenon_H8b); [ zenon_intro zenon_H7e | zenon_intro zenon_H64 ]. % 10.40/10.58 exact (zenon_H78 zenon_H7e). % 10.40/10.58 generalize (apart3 zenon_TL_y). zenon_intro zenon_H6f. % 10.40/10.58 generalize (apart2 zenon_TL_y). zenon_intro zenon_H8d. % 10.40/10.58 generalize (oac1 zenon_TL_y). zenon_intro zenon_H72. % 10.40/10.58 generalize (oac1 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H8e. % 10.40/10.58 generalize (oac1 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H45. % 10.40/10.58 generalize (zenon_H89 zenon_TL_y). zenon_intro zenon_H8f. % 10.40/10.58 generalize (zenon_H8f zenon_TL_y). zenon_intro zenon_H90. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H90); [ zenon_intro zenon_H40 | zenon_intro zenon_H91 ]. % 10.40/10.58 generalize (ooc1 zenon_TA_z). zenon_intro zenon_H60. % 10.40/10.58 generalize (ouo1 zenon_TA_z). zenon_intro zenon_H92. % 10.40/10.58 generalize (zenon_H92 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H93. % 10.40/10.58 generalize (zenon_H93 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H4e. % 10.40/10.58 generalize (zenon_H92 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H31. % 10.40/10.58 generalize (occu1 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H94. % 10.40/10.58 generalize (zenon_H60 zenon_TL_y). zenon_intro zenon_H32. % 10.40/10.58 generalize (zenon_H8e zenon_TL_y). zenon_intro zenon_H63. % 10.40/10.58 generalize (zenon_H63 zenon_TL_y). zenon_intro zenon_H95. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H95); [ zenon_intro zenon_H68 | zenon_intro zenon_H96 ]. % 10.40/10.58 apply (zenon_notand_s _ _ zenon_H68); [ zenon_intro zenon_H69 | zenon_intro zenon_H33 ]. % 10.40/10.58 exact (zenon_H69 zenon_H64). % 10.40/10.58 generalize (zenon_H1a zenon_TL_y). zenon_intro zenon_H34. % 10.40/10.58 generalize (zenon_H94 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H97. % 10.40/10.58 apply (zenon_or_s _ _ zenon_H97); [ zenon_intro zenon_H62 | zenon_intro zenon_H58 ]. % 10.40/10.58 generalize (zenon_H8e (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H98. % 10.40/10.58 generalize (zenon_H98 zenon_TL_y). zenon_intro zenon_H99. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_H99); [ zenon_intro zenon_H9b | zenon_intro zenon_H9a ]. % 10.40/10.58 apply (zenon_notand_s _ _ zenon_H9b); [ zenon_intro zenon_H9c | zenon_intro zenon_H52 ]. % 10.40/10.58 exact (zenon_H9c zenon_H62). % 10.40/10.58 apply (zenon_L9_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_or_s _ _ zenon_H9a); [ zenon_intro zenon_H9d | zenon_intro zenon_H43 ]. % 10.40/10.58 apply (zenon_and_s _ _ zenon_H9d). zenon_intro zenon_H64. zenon_intro zenon_H3e. % 10.40/10.58 exact (zenon_H33 zenon_H3e). % 10.40/10.58 apply (zenon_L7_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_L10_ zenon_TL_y zenon_TA_z); trivial. % 10.40/10.58 apply (zenon_or_s _ _ zenon_H96); [ zenon_intro zenon_H9d | zenon_intro zenon_H6e ]. % 10.40/10.58 apply (zenon_and_s _ _ zenon_H9d). zenon_intro zenon_H64. zenon_intro zenon_H3e. % 10.40/10.58 generalize (zenon_H60 (orthogonal_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H52. % 10.40/10.58 apply (zenon_L15_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_L14_ zenon_TL_y); trivial. % 10.40/10.58 apply (zenon_or_s _ _ zenon_H91); [ zenon_intro zenon_H9e | zenon_intro zenon_H42 ]. % 10.40/10.58 exact (zenon_H8d zenon_H9e). % 10.40/10.58 apply (zenon_L18_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 (* end of lemma zenon_L19_ *) % 10.40/10.58 assert (zenon_L20_ : forall (zenon_TL_y : zenon_U) (zenon_TA_z : zenon_U), (forall Y : zenon_U, (~(apart_point_and_line zenon_TA_z (parallel_through_point Y zenon_TA_z)))) -> (apart_point_and_line zenon_TA_z (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H30 zenon_H2f. % 10.40/10.58 generalize (zenon_H30 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H29. % 10.40/10.58 exact (zenon_H29 zenon_H2f). % 10.40/10.58 (* end of lemma zenon_L20_ *) % 10.40/10.58 assert (zenon_L21_ : forall (zenon_TA_z : zenon_U) (zenon_TL_y : zenon_U), (distinct_lines (parallel_through_point zenon_TL_y zenon_TA_z) (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)) -> (forall Y : zenon_U, (forall Z : zenon_U, ((distinct_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Y)->((distinct_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) Z)\/(distinct_lines Y Z))))) -> (~(distinct_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z))) -> (distinct_lines (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z) (parallel_through_point zenon_TL_y zenon_TA_z)) -> False). % 10.40/10.58 do 2 intro. intros zenon_H9f zenon_Ha0 zenon_Ha1 zenon_Ha2. % 10.40/10.58 generalize (zenon_Ha0 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_Ha3. % 10.40/10.58 generalize (apart3 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H1d. % 10.40/10.58 generalize (cp2 zenon_TA_z). zenon_intro zenon_H30. % 10.40/10.58 generalize (zenon_H30 zenon_TL_y). zenon_intro zenon_H28. % 10.40/10.58 generalize (cup1 zenon_TA_z). zenon_intro zenon_Ha4. % 10.40/10.58 generalize (zenon_Ha4 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H26. % 10.40/10.58 generalize (zenon_H26 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_Ha5. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_Ha5); [ zenon_intro zenon_Ha7 | zenon_intro zenon_Ha6 ]. % 10.40/10.58 exact (zenon_Ha7 zenon_H9f). % 10.40/10.58 apply (zenon_or_s _ _ zenon_Ha6); [ zenon_intro zenon_H2e | zenon_intro zenon_Ha8 ]. % 10.40/10.58 exact (zenon_H28 zenon_H2e). % 10.40/10.58 apply (zenon_or_s _ _ zenon_Ha8); [ zenon_intro zenon_H17 | zenon_intro zenon_H86 ]. % 10.40/10.58 apply (zenon_L1_ zenon_TL_y zenon_TA_z); trivial. % 10.40/10.58 generalize (ax6 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_Ha9. % 10.40/10.58 generalize (ceq3 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H1c. % 10.40/10.58 generalize (cp1 zenon_TA_z). zenon_intro zenon_H87. % 10.40/10.58 generalize (zenon_H87 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_H80. % 10.40/10.58 generalize (zenon_Ha4 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H50. % 10.40/10.58 generalize (zenon_H50 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_Haa. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_Haa); [ zenon_intro zenon_H1e | zenon_intro zenon_Hab ]. % 10.40/10.58 generalize (apart5 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H88. % 10.40/10.58 generalize (zenon_Ha9 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H85. % 10.40/10.58 generalize (zenon_H88 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_H51. % 10.40/10.58 generalize (zenon_Ha0 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_Hac. % 10.40/10.58 generalize (zenon_H51 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_Had. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_Had); [ zenon_intro zenon_H55 | zenon_intro zenon_Hae ]. % 10.40/10.58 generalize (zenon_Ha3 (parallel_through_point (parallel_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_Haf. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_Haf); [ zenon_intro zenon_Hb1 | zenon_intro zenon_Hb0 ]. % 10.40/10.58 exact (zenon_Hb1 zenon_Ha2). % 10.40/10.58 apply (zenon_or_s _ _ zenon_Hb0); [ zenon_intro zenon_Hb2 | zenon_intro zenon_H27 ]. % 10.40/10.58 generalize (zenon_Hac (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_Hb3. % 10.40/10.58 apply (zenon_imply_s _ _ zenon_Hb3); [ zenon_intro zenon_Hb5 | zenon_intro zenon_Hb4 ]. % 10.40/10.58 exact (zenon_Hb5 zenon_Hb2). % 10.40/10.58 apply (zenon_or_s _ _ zenon_Hb4); [ zenon_intro zenon_Hb6 | zenon_intro zenon_H4f ]. % 10.40/10.58 exact (zenon_Ha1 zenon_Hb6). % 10.40/10.58 exact (zenon_H55 zenon_H4f). % 10.40/10.58 apply (zenon_L4_ zenon_TL_y zenon_TA_z); trivial. % 10.40/10.58 apply (zenon_or_s _ _ zenon_Hae); [ zenon_intro zenon_H4f | zenon_intro zenon_Hb6 ]. % 10.40/10.58 apply (zenon_L19_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.58 exact (zenon_Ha1 zenon_Hb6). % 10.40/10.58 apply (zenon_or_s _ _ zenon_Hab); [ zenon_intro zenon_H2f | zenon_intro zenon_Hb7 ]. % 10.40/10.58 apply (zenon_L20_ zenon_TL_y zenon_TA_z); trivial. % 10.40/10.58 apply (zenon_or_s _ _ zenon_Hb7); [ zenon_intro zenon_H2e | zenon_intro zenon_H83 ]. % 10.40/10.58 exact (zenon_H28 zenon_H2e). % 10.40/10.58 exact (zenon_H80 zenon_H83). % 10.40/10.58 (* end of lemma zenon_L21_ *) % 10.40/10.58 apply NNPP. intro zenon_G. % 10.40/10.58 apply (zenon_notallex_s (fun A : zenon_U => (forall L : zenon_U, (~(distinct_lines (parallel_through_point L A) (orthogonal_through_point (orthogonal_through_point L A) A))))) zenon_G); [ zenon_intro zenon_Hb8; idtac ]. % 10.40/10.58 elim zenon_Hb8. zenon_intro zenon_TA_z. zenon_intro zenon_Hb9. % 10.40/10.59 apply (zenon_notallex_s (fun L : zenon_U => (~(distinct_lines (parallel_through_point L zenon_TA_z) (orthogonal_through_point (orthogonal_through_point L zenon_TA_z) zenon_TA_z)))) zenon_Hb9); [ zenon_intro zenon_Hba; idtac ]. % 10.40/10.59 elim zenon_Hba. zenon_intro zenon_TL_y. zenon_intro zenon_Hbb. % 10.40/10.59 apply zenon_Hbb. zenon_intro zenon_H9f. % 10.40/10.59 generalize (apart2 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_Ha1. % 10.40/10.59 generalize (apart5 (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_Ha0. % 10.40/10.59 generalize (apart2 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_Hbc. % 10.40/10.59 generalize (apart5 (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_Hbd. % 10.40/10.59 generalize (zenon_Hbd (orthogonal_through_point (orthogonal_through_point zenon_TL_y zenon_TA_z) zenon_TA_z)). zenon_intro zenon_Hbe. % 10.40/10.59 generalize (zenon_Hbe (parallel_through_point zenon_TL_y zenon_TA_z)). zenon_intro zenon_Hbf. % 10.40/10.59 apply (zenon_imply_s _ _ zenon_Hbf); [ zenon_intro zenon_Ha7 | zenon_intro zenon_Hc0 ]. % 10.40/10.59 exact (zenon_Ha7 zenon_H9f). % 10.40/10.59 apply (zenon_or_s _ _ zenon_Hc0); [ zenon_intro zenon_Hc1 | zenon_intro zenon_Ha2 ]. % 10.40/10.59 exact (zenon_Hbc zenon_Hc1). % 10.40/10.59 apply (zenon_L21_ zenon_TA_z zenon_TL_y); trivial. % 10.40/10.59 Qed. % 10.40/10.59 % SZS output end Proof % 10.40/10.59 (* END-PROOF *) % 10.40/10.59 nodes searched: 1429530 % 10.40/10.59 max branch formulas: 16525 % 10.40/10.59 proof nodes created: 13150 % 10.40/10.59 formulas created: 1414998 % 10.40/10.59 %------------------------------------------------------------------------------