TSTP Solution File: GEO223+1 by Zenon---0.7.1
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%------------------------------------------------------------------------------
% 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
%------------------------------------------------------------------------------