%------------------------------------------------------------------------------
% File     : Zenon---0.7.1
% Problem  : NUM859+1 : TPTP v8.1.0. Released v4.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_zenon %s %d

% Computer : n025.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 : Mon Jul 18 16:00:03 EDT 2022

% Result   : Theorem 4.25s 4.40s
% Output   : Proof 4.25s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12  % Problem  : NUM859+1 : TPTP v8.1.0. Released v4.1.0.
% 0.11/0.13  % Command  : run_zenon %s %d
% 0.14/0.34  % Computer : n025.cluster.edu
% 0.14/0.34  % Model    : x86_64 x86_64
% 0.14/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34  % Memory   : 8042.1875MB
% 0.14/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34  % CPULimit : 300
% 0.14/0.34  % WCLimit  : 600
% 0.14/0.34  % DateTime : Tue Jul  5 13:29:27 EDT 2022
% 0.14/0.34  % CPUTime  : 
% 4.25/4.40  (* PROOF-FOUND *)
% 4.25/4.40  % SZS status Theorem
% 4.25/4.40  (* BEGIN-PROOF *)
% 4.25/4.40  % SZS output start Proof
% 4.25/4.40  Theorem max_is_ub_1 : (forall X : zenon_U, (forall Y : zenon_U, (forall Z : zenon_U, ((minsol_model_ub X Y Z)<->(minsol_model_max X Y Z))))).
% 4.25/4.40  Proof.
% 4.25/4.40  assert (zenon_L1_ : forall (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (lesseq zenon_TY_s zenon_TX_t) -> False).
% 4.25/4.40  do 2 intro. intros zenon_Hf zenon_H10 zenon_H11.
% 4.25/4.40  generalize (zenon_Hf zenon_TY_s). zenon_intro zenon_H14.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H14); [ zenon_intro zenon_H16 | zenon_intro zenon_H15 ].
% 4.25/4.40  apply zenon_H10. apply sym_equal. exact zenon_H16.
% 4.25/4.40  exact (zenon_H15 zenon_H11).
% 4.25/4.40  (* end of lemma zenon_L1_ *)
% 4.25/4.40  assert (zenon_L2_ : forall (zenon_TX_t : zenon_U) (zenon_TY_s : zenon_U), (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (~(lesseq zenon_TX_t zenon_TY_s)) -> False).
% 4.25/4.40  do 2 intro. intros zenon_H17 zenon_H10 zenon_H18.
% 4.25/4.40  generalize (zenon_H17 zenon_TX_t). zenon_intro zenon_H19.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H19); [ zenon_intro zenon_H11 | zenon_intro zenon_H1a ].
% 4.25/4.40  generalize (max_1 zenon_TX_t). zenon_intro zenon_Hf.
% 4.25/4.40  apply (zenon_L1_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  exact (zenon_H18 zenon_H1a).
% 4.25/4.40  (* end of lemma zenon_L2_ *)
% 4.25/4.40  assert (zenon_L3_ : forall (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (~(zenon_TY_s = (max zenon_TX_t zenon_TY_s))) -> (lesseq zenon_TX_t zenon_TY_s) -> False).
% 4.25/4.40  do 2 intro. intros zenon_H1b zenon_H1c zenon_H1a.
% 4.25/4.40  generalize (zenon_H1b zenon_TY_s). zenon_intro zenon_H1d.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H1d); [ zenon_intro zenon_H1e | zenon_intro zenon_H18 ].
% 4.25/4.40  apply zenon_H1c. apply sym_equal. exact zenon_H1e.
% 4.25/4.40  exact (zenon_H18 zenon_H1a).
% 4.25/4.40  (* end of lemma zenon_L3_ *)
% 4.25/4.40  assert (zenon_L4_ : forall (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (lesseq zenon_TX_t zenon_TY_s) -> (~(zenon_TY_s = (max zenon_TX_t zenon_TY_s))) -> False).
% 4.25/4.40  do 2 intro. intros zenon_H1a zenon_H1c.
% 4.25/4.40  generalize (max_2 zenon_TX_t). zenon_intro zenon_H1b.
% 4.25/4.40  apply (zenon_L3_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  (* end of lemma zenon_L4_ *)
% 4.25/4.40  assert (zenon_L5_ : forall (zenon_TX_t : zenon_U) (zenon_TY_s : zenon_U), (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (~(lesseq zenon_TY_s zenon_TX_t)) -> (~(zenon_TY_s = (max zenon_TX_t zenon_TY_s))) -> False).
% 4.25/4.40  do 2 intro. intros zenon_H17 zenon_H15 zenon_H1c.
% 4.25/4.40  generalize (zenon_H17 zenon_TX_t). zenon_intro zenon_H19.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H19); [ zenon_intro zenon_H11 | zenon_intro zenon_H1a ].
% 4.25/4.40  exact (zenon_H15 zenon_H11).
% 4.25/4.40  apply (zenon_L4_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  (* end of lemma zenon_L5_ *)
% 4.25/4.40  assert (zenon_L6_ : forall (zenon_TZ_bi : zenon_U) (zenon_TX_t : zenon_U) (zenon_TY_s : zenon_U), (~(zenon_TY_s = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (~(lesseq zenon_TX_t zenon_TZ_bi)) -> (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_bi) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H1c zenon_H17 zenon_H1f zenon_H20 zenon_H21.
% 4.25/4.40  elim (classic ((~(zenon_TX_t = (max zenon_TX_t zenon_TY_s)))/\(~(lesseq zenon_TX_t (max zenon_TX_t zenon_TY_s))))); [ zenon_intro zenon_H23 | zenon_intro zenon_H24 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H23). zenon_intro zenon_H10. zenon_intro zenon_H25.
% 4.25/4.40  generalize (sum_monotone_1 zenon_TX_t). zenon_intro zenon_H26.
% 4.25/4.40  generalize (zenon_H26 zenon_TY_s). zenon_intro zenon_H0.
% 4.25/4.40  generalize (zenon_H0 zenon_E). zenon_intro zenon_H27.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H27); [ zenon_intro zenon_H18; zenon_intro zenon_H29 | zenon_intro zenon_H1a; zenon_intro zenon_H28 ].
% 4.25/4.40  apply (zenon_L2_ zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  apply (zenon_L4_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_bi) = (lesseq zenon_TX_t zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H20.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H21.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H2b].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H24); [ zenon_intro zenon_H2d | zenon_intro zenon_H2c ].
% 4.25/4.40  apply zenon_H2d. zenon_intro zenon_H2e.
% 4.25/4.40  elim (classic (zenon_TX_t = zenon_TX_t)); [ zenon_intro zenon_H2f | zenon_intro zenon_H30 ].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t) = ((max zenon_TX_t zenon_TY_s) = zenon_TX_t)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H2b.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H2f.
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H30].
% 4.25/4.40  cut ((zenon_TX_t = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H10].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H10 zenon_H2e).
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H2c. zenon_intro zenon_H31.
% 4.25/4.40  generalize (zenon_H1f zenon_TX_t). zenon_intro zenon_H32.
% 4.25/4.40  generalize (zenon_H32 (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H33.
% 4.25/4.40  generalize (zenon_H33 zenon_TZ_bi). zenon_intro zenon_H34.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H34); [ zenon_intro zenon_H25 | zenon_intro zenon_H35 ].
% 4.25/4.40  exact (zenon_H25 zenon_H31).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H35); [ zenon_intro zenon_H37 | zenon_intro zenon_H36 ].
% 4.25/4.40  exact (zenon_H37 zenon_H21).
% 4.25/4.40  exact (zenon_H20 zenon_H36).
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  (* end of lemma zenon_L6_ *)
% 4.25/4.40  assert (zenon_L7_ : forall (zenon_TX_t : zenon_U) (zenon_TZ_bi : zenon_U) (zenon_TY_s : zenon_U), (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (~(lesseq zenon_TY_s zenon_TZ_bi)) -> (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_bi) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H1f zenon_H38 zenon_H21.
% 4.25/4.40  elim (classic ((~(zenon_TY_s = (max zenon_TX_t zenon_TY_s)))/\(~(lesseq zenon_TY_s (max zenon_TX_t zenon_TY_s))))); [ zenon_intro zenon_H39 | zenon_intro zenon_H3a ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H39). zenon_intro zenon_H1c. zenon_intro zenon_H3b.
% 4.25/4.40  generalize (summation_monotone zenon_TY_s). zenon_intro zenon_H3c.
% 4.25/4.40  generalize (lesseq_total zenon_TY_s). zenon_intro zenon_H17.
% 4.25/4.40  generalize (zenon_H3c zenon_TX_t). zenon_intro zenon_H3d.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H3d); [ zenon_intro zenon_H15; zenon_intro zenon_H3f | zenon_intro zenon_H11; zenon_intro zenon_H3e ].
% 4.25/4.40  apply (zenon_L5_ zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  elim (classic (lesseq zenon_TX_t zenon_TZ_bi)); [ zenon_intro zenon_H36 | zenon_intro zenon_H20 ].
% 4.25/4.40  generalize (zenon_H1f zenon_TY_s). zenon_intro zenon_H40.
% 4.25/4.40  generalize (zenon_H40 zenon_TX_t). zenon_intro zenon_H41.
% 4.25/4.40  generalize (zenon_H41 zenon_TZ_bi). zenon_intro zenon_H42.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H42); [ zenon_intro zenon_H15 | zenon_intro zenon_H43 ].
% 4.25/4.40  exact (zenon_H15 zenon_H11).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H43); [ zenon_intro zenon_H20 | zenon_intro zenon_H44 ].
% 4.25/4.40  exact (zenon_H20 zenon_H36).
% 4.25/4.40  exact (zenon_H38 zenon_H44).
% 4.25/4.40  apply (zenon_L6_ zenon_TZ_bi zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_bi) = (lesseq zenon_TY_s zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H38.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H21.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H45].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H3a); [ zenon_intro zenon_H47 | zenon_intro zenon_H46 ].
% 4.25/4.40  apply zenon_H47. zenon_intro zenon_H48.
% 4.25/4.40  elim (classic (zenon_TY_s = zenon_TY_s)); [ zenon_intro zenon_H49 | zenon_intro zenon_H4a ].
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s) = ((max zenon_TX_t zenon_TY_s) = zenon_TY_s)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H45.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H49.
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H4a].
% 4.25/4.40  cut ((zenon_TY_s = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H1c].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H1c zenon_H48).
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  apply zenon_H46. zenon_intro zenon_H4b.
% 4.25/4.40  generalize (zenon_H1f zenon_TY_s). zenon_intro zenon_H40.
% 4.25/4.40  generalize (zenon_H40 (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H4c.
% 4.25/4.40  generalize (zenon_H4c zenon_TZ_bi). zenon_intro zenon_H4d.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H4d); [ zenon_intro zenon_H3b | zenon_intro zenon_H4e ].
% 4.25/4.40  exact (zenon_H3b zenon_H4b).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H4e); [ zenon_intro zenon_H37 | zenon_intro zenon_H44 ].
% 4.25/4.40  exact (zenon_H37 zenon_H21).
% 4.25/4.40  exact (zenon_H38 zenon_H44).
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  (* end of lemma zenon_L7_ *)
% 4.25/4.40  assert (zenon_L8_ : forall (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (~((max zenon_TX_t zenon_TY_s) = zenon_TY_s)) -> (lesseq zenon_TX_t zenon_TY_s) -> False).
% 4.25/4.40  do 2 intro. intros zenon_H1b zenon_H45 zenon_H1a.
% 4.25/4.40  generalize (zenon_H1b zenon_TY_s). zenon_intro zenon_H1d.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H1d); [ zenon_intro zenon_H1e | zenon_intro zenon_H18 ].
% 4.25/4.40  exact (zenon_H45 zenon_H1e).
% 4.25/4.40  exact (zenon_H18 zenon_H1a).
% 4.25/4.40  (* end of lemma zenon_L8_ *)
% 4.25/4.40  assert (zenon_L9_ : forall (zenon_TZ_bi : zenon_U) (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (forall Z : zenon_U, ((model_ub zenon_TX_t zenon_TY_s Z)->(lesseq zenon_TZ_bi Z))) -> (lesseq zenon_TY_s zenon_TX_t) -> (~(lesseq zenon_TZ_bi zenon_TX_t)) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H4f zenon_H11 zenon_H50.
% 4.25/4.40  generalize (zenon_H4f zenon_TX_t). zenon_intro zenon_H51.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H51); [ zenon_intro zenon_H53 | zenon_intro zenon_H52 ].
% 4.25/4.40  generalize (model_ub_2 zenon_TX_t). zenon_intro zenon_H54.
% 4.25/4.40  generalize (zenon_H54 zenon_TY_s). zenon_intro zenon_H55.
% 4.25/4.40  generalize (zenon_H55 zenon_TX_t). zenon_intro zenon_H56.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H56); [ zenon_intro zenon_H53; zenon_intro zenon_H59 | zenon_intro zenon_H58; zenon_intro zenon_H57 ].
% 4.25/4.40  apply zenon_H59. exists zenon_TX_t. apply NNPP. zenon_intro zenon_H5a.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H5a); [ zenon_intro zenon_H5c | zenon_intro zenon_H5b ].
% 4.25/4.40  generalize (ub zenon_TX_t). zenon_intro zenon_H5d.
% 4.25/4.40  generalize (zenon_H5d zenon_TY_s). zenon_intro zenon_H5e.
% 4.25/4.40  generalize (zenon_H5e zenon_TX_t). zenon_intro zenon_H5f.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H5f); [ zenon_intro zenon_H5c; zenon_intro zenon_H62 | zenon_intro zenon_H61; zenon_intro zenon_H60 ].
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H62); [ zenon_intro zenon_H5b | zenon_intro zenon_H15 ].
% 4.25/4.40  generalize (lesseq_ref zenon_TX_t). zenon_intro zenon_H63.
% 4.25/4.40  exact (zenon_H5b zenon_H63).
% 4.25/4.40  exact (zenon_H15 zenon_H11).
% 4.25/4.40  exact (zenon_H5c zenon_H61).
% 4.25/4.40  generalize (lesseq_ref zenon_TX_t). zenon_intro zenon_H63.
% 4.25/4.40  exact (zenon_H5b zenon_H63).
% 4.25/4.40  exact (zenon_H53 zenon_H58).
% 4.25/4.40  exact (zenon_H50 zenon_H52).
% 4.25/4.40  (* end of lemma zenon_L9_ *)
% 4.25/4.40  assert (zenon_L10_ : forall (zenon_TZ_bi : zenon_U) (zenon_TZ_dy : zenon_U) (zenon_TX_t : zenon_U), (lesseq zenon_TX_t zenon_TZ_dy) -> (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (~(lesseq zenon_TX_t zenon_TZ_bi)) -> (lesseq zenon_TZ_dy zenon_TZ_bi) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H64 zenon_H1f zenon_H20 zenon_H65.
% 4.25/4.40  elim (classic ((~(zenon_TX_t = zenon_TZ_dy))/\(~(lesseq zenon_TX_t zenon_TZ_dy)))); [ zenon_intro zenon_H67 | zenon_intro zenon_H68 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H67). zenon_intro zenon_H6a. zenon_intro zenon_H69.
% 4.25/4.40  exact (zenon_H69 zenon_H64).
% 4.25/4.40  cut ((lesseq zenon_TZ_dy zenon_TZ_bi) = (lesseq zenon_TX_t zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H20.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H65.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut ((zenon_TZ_dy = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H6b].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H68); [ zenon_intro zenon_H6d | zenon_intro zenon_H6c ].
% 4.25/4.40  apply zenon_H6d. zenon_intro zenon_H6e.
% 4.25/4.40  elim (classic (zenon_TX_t = zenon_TX_t)); [ zenon_intro zenon_H2f | zenon_intro zenon_H30 ].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t) = (zenon_TZ_dy = zenon_TX_t)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H6b.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H2f.
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H30].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TZ_dy)); [idtac | apply NNPP; zenon_intro zenon_H6a].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H6a zenon_H6e).
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H6c. zenon_intro zenon_H64.
% 4.25/4.40  generalize (zenon_H1f zenon_TX_t). zenon_intro zenon_H32.
% 4.25/4.40  generalize (zenon_H32 zenon_TZ_dy). zenon_intro zenon_H6f.
% 4.25/4.40  generalize (zenon_H6f zenon_TZ_bi). zenon_intro zenon_H70.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H70); [ zenon_intro zenon_H69 | zenon_intro zenon_H71 ].
% 4.25/4.40  exact (zenon_H69 zenon_H64).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H71); [ zenon_intro zenon_H72 | zenon_intro zenon_H36 ].
% 4.25/4.40  exact (zenon_H72 zenon_H65).
% 4.25/4.40  exact (zenon_H20 zenon_H36).
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  (* end of lemma zenon_L10_ *)
% 4.25/4.40  assert (zenon_L11_ : forall (zenon_TZ_bi : zenon_U) (zenon_TZ_dy : zenon_U) (zenon_TY_s : zenon_U), (lesseq zenon_TY_s zenon_TZ_dy) -> (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (~(lesseq zenon_TY_s zenon_TZ_bi)) -> (lesseq zenon_TZ_dy zenon_TZ_bi) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H73 zenon_H1f zenon_H38 zenon_H65.
% 4.25/4.40  elim (classic ((~(zenon_TY_s = zenon_TZ_dy))/\(~(lesseq zenon_TY_s zenon_TZ_dy)))); [ zenon_intro zenon_H74 | zenon_intro zenon_H75 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H74). zenon_intro zenon_H77. zenon_intro zenon_H76.
% 4.25/4.40  exact (zenon_H76 zenon_H73).
% 4.25/4.40  cut ((lesseq zenon_TZ_dy zenon_TZ_bi) = (lesseq zenon_TY_s zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H38.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H65.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut ((zenon_TZ_dy = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H78].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H75); [ zenon_intro zenon_H7a | zenon_intro zenon_H79 ].
% 4.25/4.40  apply zenon_H7a. zenon_intro zenon_H7b.
% 4.25/4.40  elim (classic (zenon_TY_s = zenon_TY_s)); [ zenon_intro zenon_H49 | zenon_intro zenon_H4a ].
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s) = (zenon_TZ_dy = zenon_TY_s)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H78.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H49.
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H4a].
% 4.25/4.40  cut ((zenon_TY_s = zenon_TZ_dy)); [idtac | apply NNPP; zenon_intro zenon_H77].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H77 zenon_H7b).
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  apply zenon_H79. zenon_intro zenon_H73.
% 4.25/4.40  generalize (zenon_H1f zenon_TY_s). zenon_intro zenon_H40.
% 4.25/4.40  generalize (zenon_H40 zenon_TZ_dy). zenon_intro zenon_H7c.
% 4.25/4.40  generalize (zenon_H7c zenon_TZ_bi). zenon_intro zenon_H7d.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H7d); [ zenon_intro zenon_H76 | zenon_intro zenon_H7e ].
% 4.25/4.40  exact (zenon_H76 zenon_H73).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H7e); [ zenon_intro zenon_H72 | zenon_intro zenon_H44 ].
% 4.25/4.40  exact (zenon_H72 zenon_H65).
% 4.25/4.40  exact (zenon_H38 zenon_H44).
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  (* end of lemma zenon_L11_ *)
% 4.25/4.40  assert (zenon_L12_ : forall (zenon_TX_t : zenon_U) (zenon_TY_s : zenon_U), (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (~(zenon_TY_s = (max zenon_TX_t zenon_TY_s))) -> False).
% 4.25/4.40  do 2 intro. intros zenon_H17 zenon_Hf zenon_H10 zenon_H1b zenon_H1c.
% 4.25/4.40  generalize (zenon_H17 zenon_TX_t). zenon_intro zenon_H19.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H19); [ zenon_intro zenon_H11 | zenon_intro zenon_H1a ].
% 4.25/4.40  apply (zenon_L1_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  apply (zenon_L3_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  (* end of lemma zenon_L12_ *)
% 4.25/4.40  assert (zenon_L13_ : forall (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U) (zenon_TZ_ez : zenon_U), (~(lesseq zenon_TZ_ez (max zenon_TX_t zenon_TY_s))) -> (lesseq zenon_TZ_ez zenon_TY_s) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H7f zenon_H80 zenon_H17 zenon_Hf zenon_H10 zenon_H1b.
% 4.25/4.40  elim (classic (zenon_TY_s = (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_H48 | zenon_intro zenon_H1c ].
% 4.25/4.40  cut ((lesseq zenon_TZ_ez zenon_TY_s) = (lesseq zenon_TZ_ez (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H7f.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H80.
% 4.25/4.40  cut ((zenon_TY_s = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H1c].
% 4.25/4.40  cut ((zenon_TZ_ez = zenon_TZ_ez)); [idtac | apply NNPP; zenon_intro zenon_H82].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_H82. apply refl_equal.
% 4.25/4.40  exact (zenon_H1c zenon_H48).
% 4.25/4.40  apply (zenon_L12_ zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  (* end of lemma zenon_L13_ *)
% 4.25/4.40  assert (zenon_L14_ : forall (zenon_TX_t : zenon_U) (zenon_TY_s : zenon_U), (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (~(lesseq zenon_TX_t zenon_TY_s)) -> False).
% 4.25/4.40  do 2 intro. intros zenon_H17 zenon_Hf zenon_H10 zenon_H18.
% 4.25/4.40  generalize (zenon_H17 zenon_TX_t). zenon_intro zenon_H19.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H19); [ zenon_intro zenon_H11 | zenon_intro zenon_H1a ].
% 4.25/4.40  apply (zenon_L1_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  exact (zenon_H18 zenon_H1a).
% 4.25/4.40  (* end of lemma zenon_L14_ *)
% 4.25/4.40  assert (zenon_L15_ : forall (zenon_TZ_ez : zenon_U) (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (~(lesseq zenon_TX_t zenon_TZ_ez)) -> (lesseq zenon_TY_s zenon_TZ_ez) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H10 zenon_Hf zenon_H17 zenon_H1f zenon_H83 zenon_H84.
% 4.25/4.40  elim (classic ((~(zenon_TX_t = zenon_TY_s))/\(~(lesseq zenon_TX_t zenon_TY_s)))); [ zenon_intro zenon_H85 | zenon_intro zenon_H86 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H85). zenon_intro zenon_H87. zenon_intro zenon_H18.
% 4.25/4.40  apply (zenon_L14_ zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  cut ((lesseq zenon_TY_s zenon_TZ_ez) = (lesseq zenon_TX_t zenon_TZ_ez)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H83.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H84.
% 4.25/4.40  cut ((zenon_TZ_ez = zenon_TZ_ez)); [idtac | apply NNPP; zenon_intro zenon_H82].
% 4.25/4.40  cut ((zenon_TY_s = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H88].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H86); [ zenon_intro zenon_H8a | zenon_intro zenon_H89 ].
% 4.25/4.40  apply zenon_H8a. zenon_intro zenon_H8b.
% 4.25/4.40  elim (classic (zenon_TX_t = zenon_TX_t)); [ zenon_intro zenon_H2f | zenon_intro zenon_H30 ].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t) = (zenon_TY_s = zenon_TX_t)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H88.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H2f.
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H30].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H87].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H87 zenon_H8b).
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H89. zenon_intro zenon_H1a.
% 4.25/4.40  generalize (zenon_H1f zenon_TX_t). zenon_intro zenon_H32.
% 4.25/4.40  generalize (zenon_H32 zenon_TY_s). zenon_intro zenon_H8c.
% 4.25/4.40  generalize (zenon_H8c zenon_TZ_ez). zenon_intro zenon_H8d.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H8d); [ zenon_intro zenon_H18 | zenon_intro zenon_H8e ].
% 4.25/4.40  exact (zenon_H18 zenon_H1a).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H8e); [ zenon_intro zenon_H90 | zenon_intro zenon_H8f ].
% 4.25/4.40  exact (zenon_H90 zenon_H84).
% 4.25/4.40  exact (zenon_H83 zenon_H8f).
% 4.25/4.40  apply zenon_H82. apply refl_equal.
% 4.25/4.40  (* end of lemma zenon_L15_ *)
% 4.25/4.40  assert (zenon_L16_ : forall (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U) (zenon_TZ_ez : zenon_U), (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (forall Y : zenon_U, ((lesseq zenon_TZ_ez Y)\/(lesseq Y zenon_TZ_ez))) -> (~(lesseq zenon_TZ_ez (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (~(lesseq zenon_TX_t zenon_TZ_ez)) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H1f zenon_H91 zenon_H7f zenon_H1b zenon_H10 zenon_Hf zenon_H17 zenon_H83.
% 4.25/4.40  generalize (zenon_H91 zenon_TY_s). zenon_intro zenon_H92.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H92); [ zenon_intro zenon_H80 | zenon_intro zenon_H84 ].
% 4.25/4.40  apply (zenon_L13_ zenon_TY_s zenon_TX_t zenon_TZ_ez); trivial.
% 4.25/4.40  apply (zenon_L15_ zenon_TZ_ez zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  (* end of lemma zenon_L16_ *)
% 4.25/4.40  assert (zenon_L17_ : forall (zenon_TZ_ez : zenon_U) (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (~(exists Z : zenon_U, ((ub zenon_TX_t zenon_TY_s Z)/\(lesseq Z zenon_TZ_ez)))) -> (forall Y : zenon_U, ((lesseq zenon_TZ_ez Y)\/(lesseq Y zenon_TZ_ez))) -> (~(lesseq zenon_TZ_ez (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_ez) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H1f zenon_H93 zenon_H91 zenon_H7f zenon_H1b zenon_H10 zenon_Hf zenon_H17 zenon_H94.
% 4.25/4.40  apply zenon_H93. exists zenon_TZ_ez. apply NNPP. zenon_intro zenon_H95.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H95); [ zenon_intro zenon_H97 | zenon_intro zenon_H96 ].
% 4.25/4.40  generalize (ub zenon_TX_t). zenon_intro zenon_H5d.
% 4.25/4.40  generalize (zenon_H5d zenon_TY_s). zenon_intro zenon_H5e.
% 4.25/4.40  generalize (zenon_H5e zenon_TZ_ez). zenon_intro zenon_H98.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H98); [ zenon_intro zenon_H97; zenon_intro zenon_H9b | zenon_intro zenon_H9a; zenon_intro zenon_H99 ].
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H9b); [ zenon_intro zenon_H83 | zenon_intro zenon_H90 ].
% 4.25/4.40  apply (zenon_L16_ zenon_TY_s zenon_TX_t zenon_TZ_ez); trivial.
% 4.25/4.40  elim (classic ((~(zenon_TY_s = (max zenon_TX_t zenon_TY_s)))/\(~(lesseq zenon_TY_s (max zenon_TX_t zenon_TY_s))))); [ zenon_intro zenon_H39 | zenon_intro zenon_H3a ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H39). zenon_intro zenon_H1c. zenon_intro zenon_H3b.
% 4.25/4.40  apply (zenon_L12_ zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_ez) = (lesseq zenon_TY_s zenon_TZ_ez)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H90.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H94.
% 4.25/4.40  cut ((zenon_TZ_ez = zenon_TZ_ez)); [idtac | apply NNPP; zenon_intro zenon_H82].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H45].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H3a); [ zenon_intro zenon_H47 | zenon_intro zenon_H46 ].
% 4.25/4.40  apply zenon_H47. zenon_intro zenon_H48.
% 4.25/4.40  elim (classic (zenon_TY_s = zenon_TY_s)); [ zenon_intro zenon_H49 | zenon_intro zenon_H4a ].
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s) = ((max zenon_TX_t zenon_TY_s) = zenon_TY_s)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H45.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H49.
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H4a].
% 4.25/4.40  cut ((zenon_TY_s = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H1c].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H1c zenon_H48).
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  apply zenon_H46. zenon_intro zenon_H4b.
% 4.25/4.40  generalize (zenon_H1f zenon_TY_s). zenon_intro zenon_H40.
% 4.25/4.40  generalize (zenon_H40 (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H4c.
% 4.25/4.40  generalize (zenon_H4c zenon_TZ_ez). zenon_intro zenon_H9c.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H9c); [ zenon_intro zenon_H3b | zenon_intro zenon_H9d ].
% 4.25/4.40  exact (zenon_H3b zenon_H4b).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H9d); [ zenon_intro zenon_H9e | zenon_intro zenon_H84 ].
% 4.25/4.40  exact (zenon_H9e zenon_H94).
% 4.25/4.40  exact (zenon_H90 zenon_H84).
% 4.25/4.40  apply zenon_H82. apply refl_equal.
% 4.25/4.40  exact (zenon_H97 zenon_H9a).
% 4.25/4.40  generalize (lesseq_ref zenon_TZ_ez). zenon_intro zenon_H9f.
% 4.25/4.40  exact (zenon_H96 zenon_H9f).
% 4.25/4.40  (* end of lemma zenon_L17_ *)
% 4.25/4.40  assert (zenon_L18_ : forall (zenon_TZ_ez : zenon_U) (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (~(exists Z : zenon_U, ((ub zenon_TX_t zenon_TY_s Z)/\(lesseq Z zenon_TZ_ez)))) -> (forall Y : zenon_U, ((lesseq zenon_TZ_ez Y)\/(lesseq Y zenon_TZ_ez))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_ez) -> ((max zenon_TZ_ez zenon_TZ_ez) = zenon_TZ_ez) -> (~(lesseq (max zenon_TZ_ez zenon_TZ_ez) (max zenon_TX_t zenon_TY_s))) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H1f zenon_H93 zenon_H91 zenon_H1b zenon_H10 zenon_Hf zenon_H17 zenon_H94 zenon_Ha0 zenon_Ha1.
% 4.25/4.40  elim (classic (lesseq (max zenon_TZ_ez zenon_TZ_ez) (max zenon_TZ_ez zenon_TZ_ez))); [ zenon_intro zenon_Ha2 | zenon_intro zenon_Ha3 ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TZ_ez zenon_TZ_ez) zenon_TZ_ez)); [ zenon_intro zenon_Ha4 | zenon_intro zenon_Ha5 ].
% 4.25/4.40  elim (classic (lesseq zenon_TZ_ez (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_Ha6 | zenon_intro zenon_H7f ].
% 4.25/4.40  generalize (zenon_H1f (max zenon_TZ_ez zenon_TZ_ez)). zenon_intro zenon_Ha7.
% 4.25/4.40  generalize (zenon_Ha7 zenon_TZ_ez). zenon_intro zenon_Ha8.
% 4.25/4.40  generalize (zenon_Ha8 (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_Ha9.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_Ha9); [ zenon_intro zenon_Ha5 | zenon_intro zenon_Haa ].
% 4.25/4.40  exact (zenon_Ha5 zenon_Ha4).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_Haa); [ zenon_intro zenon_H7f | zenon_intro zenon_Hab ].
% 4.25/4.40  exact (zenon_H7f zenon_Ha6).
% 4.25/4.40  exact (zenon_Ha1 zenon_Hab).
% 4.25/4.40  apply (zenon_L17_ zenon_TZ_ez zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  cut ((lesseq (max zenon_TZ_ez zenon_TZ_ez) (max zenon_TZ_ez zenon_TZ_ez)) = (lesseq (max zenon_TZ_ez zenon_TZ_ez) zenon_TZ_ez)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Ha5.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Ha2.
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TZ_ez) = zenon_TZ_ez)); [idtac | apply NNPP; zenon_intro zenon_Hac].
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TZ_ez) = (max zenon_TZ_ez zenon_TZ_ez))); [idtac | apply NNPP; zenon_intro zenon_Had].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_Had. apply refl_equal.
% 4.25/4.40  exact (zenon_Hac zenon_Ha0).
% 4.25/4.40  generalize (lesseq_ref (max zenon_TZ_ez zenon_TZ_ez)). zenon_intro zenon_Ha2.
% 4.25/4.40  exact (zenon_Ha3 zenon_Ha2).
% 4.25/4.40  (* end of lemma zenon_L18_ *)
% 4.25/4.40  assert (zenon_L19_ : forall (zenon_TZ_ez : zenon_U) (zenon_TX_t : zenon_U) (zenon_TY_s : zenon_U), (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> ((max zenon_TZ_ez zenon_TY_s) = zenon_TY_s) -> (~(lesseq (max zenon_TZ_ez zenon_TY_s) (max zenon_TX_t zenon_TY_s))) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H17 zenon_Hf zenon_H10 zenon_H1b zenon_Hae zenon_Haf.
% 4.25/4.40  elim (classic (lesseq (max zenon_TZ_ez zenon_TY_s) (max zenon_TZ_ez zenon_TY_s))); [ zenon_intro zenon_Hb0 | zenon_intro zenon_Hb1 ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TZ_ez zenon_TY_s) zenon_TY_s)); [ zenon_intro zenon_Hb2 | zenon_intro zenon_Hb3 ].
% 4.25/4.40  cut ((lesseq (max zenon_TZ_ez zenon_TY_s) zenon_TY_s) = (lesseq (max zenon_TZ_ez zenon_TY_s) (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Haf.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hb2.
% 4.25/4.40  cut ((zenon_TY_s = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H1c].
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = (max zenon_TZ_ez zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_Hb4].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_Hb4. apply refl_equal.
% 4.25/4.40  apply (zenon_L12_ zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  cut ((lesseq (max zenon_TZ_ez zenon_TY_s) (max zenon_TZ_ez zenon_TY_s)) = (lesseq (max zenon_TZ_ez zenon_TY_s) zenon_TY_s)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hb3.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hb0.
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_Hb5].
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = (max zenon_TZ_ez zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_Hb4].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_Hb4. apply refl_equal.
% 4.25/4.40  exact (zenon_Hb5 zenon_Hae).
% 4.25/4.40  generalize (lesseq_ref (max zenon_TZ_ez zenon_TY_s)). zenon_intro zenon_Hb0.
% 4.25/4.40  exact (zenon_Hb1 zenon_Hb0).
% 4.25/4.40  (* end of lemma zenon_L19_ *)
% 4.25/4.40  assert (zenon_L20_ : forall (zenon_TX_t : zenon_U) (zenon_TY_s : zenon_U) (zenon_TZ_ez : zenon_U), (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> ((max zenon_TZ_ez zenon_TY_s) = zenon_TY_s) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> ((max zenon_TX_t zenon_TX_t) = zenon_TX_t) -> (~(lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TZ_ez zenon_TY_s))) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H1f zenon_Hae zenon_H10 zenon_Hf zenon_H17 zenon_Hb6 zenon_Hb7.
% 4.25/4.40  elim (classic (lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TX_t zenon_TX_t))); [ zenon_intro zenon_Hb8 | zenon_intro zenon_Hb9 ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TX_t zenon_TX_t) zenon_TX_t)); [ zenon_intro zenon_Hba | zenon_intro zenon_Hbb ].
% 4.25/4.40  elim (classic (lesseq zenon_TX_t (max zenon_TZ_ez zenon_TY_s))); [ zenon_intro zenon_Hbc | zenon_intro zenon_Hbd ].
% 4.25/4.40  generalize (zenon_H1f (max zenon_TX_t zenon_TX_t)). zenon_intro zenon_Hbe.
% 4.25/4.40  generalize (zenon_Hbe zenon_TX_t). zenon_intro zenon_Hbf.
% 4.25/4.40  generalize (zenon_Hbf (max zenon_TZ_ez zenon_TY_s)). zenon_intro zenon_Hc0.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_Hc0); [ zenon_intro zenon_Hbb | zenon_intro zenon_Hc1 ].
% 4.25/4.40  exact (zenon_Hbb zenon_Hba).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_Hc1); [ zenon_intro zenon_Hbd | zenon_intro zenon_Hc2 ].
% 4.25/4.40  exact (zenon_Hbd zenon_Hbc).
% 4.25/4.40  exact (zenon_Hb7 zenon_Hc2).
% 4.25/4.40  elim (classic (zenon_TY_s = (max zenon_TZ_ez zenon_TY_s))); [ zenon_intro zenon_Hc3 | zenon_intro zenon_Hc4 ].
% 4.25/4.40  elim (classic (lesseq zenon_TX_t zenon_TY_s)); [ zenon_intro zenon_H1a | zenon_intro zenon_H18 ].
% 4.25/4.40  cut ((lesseq zenon_TX_t zenon_TY_s) = (lesseq zenon_TX_t (max zenon_TZ_ez zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hbd.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H1a.
% 4.25/4.40  cut ((zenon_TY_s = (max zenon_TZ_ez zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_Hc4].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H30].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  exact (zenon_Hc4 zenon_Hc3).
% 4.25/4.40  apply (zenon_L14_ zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  elim (classic ((max zenon_TZ_ez zenon_TY_s) = (max zenon_TZ_ez zenon_TY_s))); [ zenon_intro zenon_Hc5 | zenon_intro zenon_Hb4 ].
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = (max zenon_TZ_ez zenon_TY_s)) = (zenon_TY_s = (max zenon_TZ_ez zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hc4.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hc5.
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = (max zenon_TZ_ez zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_Hb4].
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_Hb5].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_Hb5 zenon_Hae).
% 4.25/4.40  apply zenon_Hb4. apply refl_equal.
% 4.25/4.40  apply zenon_Hb4. apply refl_equal.
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TX_t zenon_TX_t)) = (lesseq (max zenon_TX_t zenon_TX_t) zenon_TX_t)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hbb.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hb8.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_Hc6].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t))); [idtac | apply NNPP; zenon_intro zenon_Hc7].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_Hc7. apply refl_equal.
% 4.25/4.40  exact (zenon_Hc6 zenon_Hb6).
% 4.25/4.40  generalize (lesseq_ref (max zenon_TX_t zenon_TX_t)). zenon_intro zenon_Hb8.
% 4.25/4.40  exact (zenon_Hb9 zenon_Hb8).
% 4.25/4.40  (* end of lemma zenon_L20_ *)
% 4.25/4.40  assert (zenon_L21_ : forall (zenon_TZ_ez : zenon_U) (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (~(exists Z : zenon_U, ((ub zenon_TX_t zenon_TY_s Z)/\(lesseq Z zenon_TZ_ez)))) -> (forall Y : zenon_U, ((lesseq zenon_TZ_ez Y)\/(lesseq Y zenon_TZ_ez))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_ez) -> ((max zenon_TX_t zenon_TX_t) = zenon_TX_t) -> (~(lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TX_t zenon_TY_s))) -> (~(lesseq zenon_TX_t (max zenon_TX_t zenon_TY_s))) -> (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> ((max zenon_TZ_ez zenon_TZ_ez) = zenon_TZ_ez) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H93 zenon_H91 zenon_H1b zenon_H10 zenon_Hf zenon_H17 zenon_H94 zenon_Hb6 zenon_Hc8 zenon_H25 zenon_H1f zenon_Ha0.
% 4.25/4.40  elim (classic (zenon_TZ_ez = (max zenon_TZ_ez zenon_TZ_ez))); [ zenon_intro zenon_Hc9 | zenon_intro zenon_Hca ].
% 4.25/4.40  elim (classic (lesseq zenon_TX_t zenon_TZ_ez)); [ zenon_intro zenon_H8f | zenon_intro zenon_H83 ].
% 4.25/4.40  elim (classic (lesseq zenon_TX_t (max zenon_TZ_ez zenon_TZ_ez))); [ zenon_intro zenon_Hcb | zenon_intro zenon_Hcc ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TZ_ez zenon_TZ_ez) (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_Hab | zenon_intro zenon_Ha1 ].
% 4.25/4.40  generalize (zenon_H1f zenon_TX_t). zenon_intro zenon_H32.
% 4.25/4.40  generalize (zenon_H32 (max zenon_TZ_ez zenon_TZ_ez)). zenon_intro zenon_Hcd.
% 4.25/4.40  generalize (zenon_Hcd (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_Hce.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_Hce); [ zenon_intro zenon_Hcc | zenon_intro zenon_Hcf ].
% 4.25/4.40  exact (zenon_Hcc zenon_Hcb).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_Hcf); [ zenon_intro zenon_Ha1 | zenon_intro zenon_H31 ].
% 4.25/4.40  exact (zenon_Ha1 zenon_Hab).
% 4.25/4.40  exact (zenon_H25 zenon_H31).
% 4.25/4.40  apply (zenon_L18_ zenon_TZ_ez zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  cut ((lesseq zenon_TX_t zenon_TZ_ez) = (lesseq zenon_TX_t (max zenon_TZ_ez zenon_TZ_ez))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hcc.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H8f.
% 4.25/4.40  cut ((zenon_TZ_ez = (max zenon_TZ_ez zenon_TZ_ez))); [idtac | apply NNPP; zenon_intro zenon_Hca].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H30].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  exact (zenon_Hca zenon_Hc9).
% 4.25/4.40  generalize (max_2 zenon_TZ_ez). zenon_intro zenon_Hd0.
% 4.25/4.40  generalize (zenon_H91 zenon_TY_s). zenon_intro zenon_H92.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H92); [ zenon_intro zenon_H80 | zenon_intro zenon_H84 ].
% 4.25/4.40  generalize (zenon_Hd0 zenon_TY_s). zenon_intro zenon_Hd1.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_Hd1); [ zenon_intro zenon_Hae | zenon_intro zenon_Hd2 ].
% 4.25/4.40  elim (classic (zenon_TY_s = (max zenon_TZ_ez zenon_TY_s))); [ zenon_intro zenon_Hc3 | zenon_intro zenon_Hc4 ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TX_t zenon_TX_t) zenon_TY_s)); [ zenon_intro zenon_Hd3 | zenon_intro zenon_Hd4 ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TZ_ez zenon_TY_s))); [ zenon_intro zenon_Hc2 | zenon_intro zenon_Hb7 ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TZ_ez zenon_TY_s) (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_Hd5 | zenon_intro zenon_Haf ].
% 4.25/4.40  generalize (zenon_H1f (max zenon_TX_t zenon_TX_t)). zenon_intro zenon_Hbe.
% 4.25/4.40  generalize (zenon_Hbe (max zenon_TZ_ez zenon_TY_s)). zenon_intro zenon_Hd6.
% 4.25/4.40  generalize (zenon_Hd6 (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_Hd7.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_Hd7); [ zenon_intro zenon_Hb7 | zenon_intro zenon_Hd8 ].
% 4.25/4.40  exact (zenon_Hb7 zenon_Hc2).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_Hd8); [ zenon_intro zenon_Haf | zenon_intro zenon_Hd9 ].
% 4.25/4.40  exact (zenon_Haf zenon_Hd5).
% 4.25/4.40  exact (zenon_Hc8 zenon_Hd9).
% 4.25/4.40  apply (zenon_L19_ zenon_TZ_ez zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TX_t) zenon_TY_s) = (lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TZ_ez zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hb7.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hd3.
% 4.25/4.40  cut ((zenon_TY_s = (max zenon_TZ_ez zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_Hc4].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t))); [idtac | apply NNPP; zenon_intro zenon_Hc7].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_Hc7. apply refl_equal.
% 4.25/4.40  exact (zenon_Hc4 zenon_Hc3).
% 4.25/4.40  elim (classic (lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TZ_ez zenon_TY_s))); [ zenon_intro zenon_Hc2 | zenon_intro zenon_Hb7 ].
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TZ_ez zenon_TY_s)) = (lesseq (max zenon_TX_t zenon_TX_t) zenon_TY_s)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hd4.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hc2.
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_Hb5].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t))); [idtac | apply NNPP; zenon_intro zenon_Hc7].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_Hc7. apply refl_equal.
% 4.25/4.40  exact (zenon_Hb5 zenon_Hae).
% 4.25/4.40  apply (zenon_L20_ zenon_TX_t zenon_TY_s zenon_TZ_ez); trivial.
% 4.25/4.40  elim (classic ((max zenon_TZ_ez zenon_TY_s) = (max zenon_TZ_ez zenon_TY_s))); [ zenon_intro zenon_Hc5 | zenon_intro zenon_Hb4 ].
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = (max zenon_TZ_ez zenon_TY_s)) = (zenon_TY_s = (max zenon_TZ_ez zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hc4.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hc5.
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = (max zenon_TZ_ez zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_Hb4].
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TY_s) = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_Hb5].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_Hb5 zenon_Hae).
% 4.25/4.40  apply zenon_Hb4. apply refl_equal.
% 4.25/4.40  apply zenon_Hb4. apply refl_equal.
% 4.25/4.40  exact (zenon_Hd2 zenon_H80).
% 4.25/4.40  apply (zenon_L15_ zenon_TZ_ez zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  elim (classic ((max zenon_TZ_ez zenon_TZ_ez) = (max zenon_TZ_ez zenon_TZ_ez))); [ zenon_intro zenon_Hda | zenon_intro zenon_Had ].
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TZ_ez) = (max zenon_TZ_ez zenon_TZ_ez)) = (zenon_TZ_ez = (max zenon_TZ_ez zenon_TZ_ez))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hca.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hda.
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TZ_ez) = (max zenon_TZ_ez zenon_TZ_ez))); [idtac | apply NNPP; zenon_intro zenon_Had].
% 4.25/4.40  cut (((max zenon_TZ_ez zenon_TZ_ez) = zenon_TZ_ez)); [idtac | apply NNPP; zenon_intro zenon_Hac].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_Hac zenon_Ha0).
% 4.25/4.40  apply zenon_Had. apply refl_equal.
% 4.25/4.40  apply zenon_Had. apply refl_equal.
% 4.25/4.40  (* end of lemma zenon_L21_ *)
% 4.25/4.40  assert (zenon_L22_ : forall (zenon_TZ_ez : zenon_U) (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (~(lesseq zenon_TX_t (max zenon_TX_t zenon_TY_s))) -> (~(zenon_TX_t = (max zenon_TX_t zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (forall Y : zenon_U, ((lesseq zenon_TZ_ez Y)\/(lesseq Y zenon_TZ_ez))) -> ((max zenon_TZ_ez zenon_TZ_ez) = zenon_TZ_ez) -> (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_ez) -> (~(exists Z : zenon_U, ((ub zenon_TX_t zenon_TY_s Z)/\(lesseq Z zenon_TZ_ez)))) -> (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> ((max zenon_TX_t zenon_TX_t) = zenon_TX_t) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H25 zenon_H10 zenon_Hf zenon_H17 zenon_H1b zenon_H91 zenon_Ha0 zenon_H94 zenon_H93 zenon_H1f zenon_Hb6.
% 4.25/4.40  elim (classic (zenon_TX_t = (max zenon_TX_t zenon_TX_t))); [ zenon_intro zenon_Hdb | zenon_intro zenon_Hdc ].
% 4.25/4.40  elim (classic (lesseq zenon_TX_t zenon_TX_t)); [ zenon_intro zenon_H63 | zenon_intro zenon_H5b ].
% 4.25/4.40  elim (classic (lesseq zenon_TX_t (max zenon_TX_t zenon_TX_t))); [ zenon_intro zenon_Hdd | zenon_intro zenon_Hde ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_Hd9 | zenon_intro zenon_Hc8 ].
% 4.25/4.40  generalize (zenon_H1f zenon_TX_t). zenon_intro zenon_H32.
% 4.25/4.40  generalize (zenon_H32 (max zenon_TX_t zenon_TX_t)). zenon_intro zenon_Hdf.
% 4.25/4.40  generalize (zenon_Hdf (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_He0.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_He0); [ zenon_intro zenon_Hde | zenon_intro zenon_He1 ].
% 4.25/4.40  exact (zenon_Hde zenon_Hdd).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_He1); [ zenon_intro zenon_Hc8 | zenon_intro zenon_H31 ].
% 4.25/4.40  exact (zenon_Hc8 zenon_Hd9).
% 4.25/4.40  exact (zenon_H25 zenon_H31).
% 4.25/4.40  apply (zenon_L21_ zenon_TZ_ez zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  cut ((lesseq zenon_TX_t zenon_TX_t) = (lesseq zenon_TX_t (max zenon_TX_t zenon_TX_t))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hde.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H63.
% 4.25/4.40  cut ((zenon_TX_t = (max zenon_TX_t zenon_TX_t))); [idtac | apply NNPP; zenon_intro zenon_Hdc].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H30].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  exact (zenon_Hdc zenon_Hdb).
% 4.25/4.40  generalize (lesseq_ref zenon_TX_t). zenon_intro zenon_H63.
% 4.25/4.40  exact (zenon_H5b zenon_H63).
% 4.25/4.40  elim (classic ((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t))); [ zenon_intro zenon_He2 | zenon_intro zenon_Hc7 ].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t)) = (zenon_TX_t = (max zenon_TX_t zenon_TX_t))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hdc.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_He2.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t))); [idtac | apply NNPP; zenon_intro zenon_Hc7].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_Hc6].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_Hc6 zenon_Hb6).
% 4.25/4.40  apply zenon_Hc7. apply refl_equal.
% 4.25/4.40  apply zenon_Hc7. apply refl_equal.
% 4.25/4.40  (* end of lemma zenon_L22_ *)
% 4.25/4.40  assert (zenon_L23_ : forall (zenon_TZ_ez : zenon_U) (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (~(exists Z : zenon_U, ((ub zenon_TX_t zenon_TY_s Z)/\(lesseq Z zenon_TZ_ez)))) -> ((max zenon_TZ_ez zenon_TZ_ez) = zenon_TZ_ez) -> (forall Y : zenon_U, ((lesseq zenon_TZ_ez Y)\/(lesseq Y zenon_TZ_ez))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> ((max zenon_TX_t zenon_TX_t) = zenon_TX_t) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (~(lesseq zenon_TX_t zenon_TZ_ez)) -> (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_ez) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H93 zenon_Ha0 zenon_H91 zenon_H1b zenon_Hb6 zenon_H17 zenon_Hf zenon_H1f zenon_H83 zenon_H94.
% 4.25/4.40  elim (classic ((~(zenon_TX_t = (max zenon_TX_t zenon_TY_s)))/\(~(lesseq zenon_TX_t (max zenon_TX_t zenon_TY_s))))); [ zenon_intro zenon_H23 | zenon_intro zenon_H24 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H23). zenon_intro zenon_H10. zenon_intro zenon_H25.
% 4.25/4.40  apply (zenon_L22_ zenon_TZ_ez zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_ez) = (lesseq zenon_TX_t zenon_TZ_ez)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H83.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H94.
% 4.25/4.40  cut ((zenon_TZ_ez = zenon_TZ_ez)); [idtac | apply NNPP; zenon_intro zenon_H82].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H2b].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H24); [ zenon_intro zenon_H2d | zenon_intro zenon_H2c ].
% 4.25/4.40  apply zenon_H2d. zenon_intro zenon_H2e.
% 4.25/4.40  elim (classic (zenon_TX_t = zenon_TX_t)); [ zenon_intro zenon_H2f | zenon_intro zenon_H30 ].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t) = ((max zenon_TX_t zenon_TY_s) = zenon_TX_t)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H2b.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H2f.
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H30].
% 4.25/4.40  cut ((zenon_TX_t = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H10].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H10 zenon_H2e).
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H2c. zenon_intro zenon_H31.
% 4.25/4.40  generalize (zenon_H1f zenon_TX_t). zenon_intro zenon_H32.
% 4.25/4.40  generalize (zenon_H32 (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H33.
% 4.25/4.40  generalize (zenon_H33 zenon_TZ_ez). zenon_intro zenon_He3.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_He3); [ zenon_intro zenon_H25 | zenon_intro zenon_He4 ].
% 4.25/4.40  exact (zenon_H25 zenon_H31).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_He4); [ zenon_intro zenon_H9e | zenon_intro zenon_H8f ].
% 4.25/4.40  exact (zenon_H9e zenon_H94).
% 4.25/4.40  exact (zenon_H83 zenon_H8f).
% 4.25/4.40  apply zenon_H82. apply refl_equal.
% 4.25/4.40  (* end of lemma zenon_L23_ *)
% 4.25/4.40  assert (zenon_L24_ : forall (zenon_TZ_ez : zenon_U) (zenon_TY_s : zenon_U) (zenon_TX_t : zenon_U), (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z)))))) -> (~(exists Z : zenon_U, ((ub zenon_TX_t zenon_TY_s Z)/\(lesseq Z zenon_TZ_ez)))) -> (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_ez) -> ((max zenon_TZ_ez zenon_TZ_ez) = zenon_TZ_ez) -> (forall Y : zenon_U, ((lesseq zenon_TZ_ez Y)\/(lesseq Y zenon_TZ_ez))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = zenon_TX_t)\/(~(lesseq Y zenon_TX_t)))) -> ((max zenon_TX_t zenon_TX_t) = zenon_TX_t) -> (~(lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TX_t zenon_TY_s))) -> False).
% 4.25/4.40  do 3 intro. intros zenon_H1f zenon_H93 zenon_H94 zenon_Ha0 zenon_H91 zenon_H1b zenon_H17 zenon_Hf zenon_Hb6 zenon_Hc8.
% 4.25/4.40  elim (classic (lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TX_t zenon_TX_t))); [ zenon_intro zenon_Hb8 | zenon_intro zenon_Hb9 ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TX_t zenon_TX_t) zenon_TX_t)); [ zenon_intro zenon_Hba | zenon_intro zenon_Hbb ].
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TX_t) zenon_TX_t) = (lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hc8.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hba.
% 4.25/4.40  cut ((zenon_TX_t = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H10].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t))); [idtac | apply NNPP; zenon_intro zenon_Hc7].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_Hc7. apply refl_equal.
% 4.25/4.40  elim (classic (lesseq zenon_TX_t (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_H31 | zenon_intro zenon_H25 ].
% 4.25/4.40  generalize (zenon_H1f (max zenon_TX_t zenon_TX_t)). zenon_intro zenon_Hbe.
% 4.25/4.40  generalize (zenon_Hbe zenon_TX_t). zenon_intro zenon_Hbf.
% 4.25/4.40  generalize (zenon_Hbf (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_He5.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_He5); [ zenon_intro zenon_Hbb | zenon_intro zenon_He6 ].
% 4.25/4.40  exact (zenon_Hbb zenon_Hba).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_He6); [ zenon_intro zenon_H25 | zenon_intro zenon_Hd9 ].
% 4.25/4.40  exact (zenon_H25 zenon_H31).
% 4.25/4.40  exact (zenon_Hc8 zenon_Hd9).
% 4.25/4.40  apply (zenon_L21_ zenon_TZ_ez zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TX_t zenon_TX_t)) = (lesseq (max zenon_TX_t zenon_TX_t) zenon_TX_t)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hbb.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_Hb8.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_Hc6].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t))); [idtac | apply NNPP; zenon_intro zenon_Hc7].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_Hc7. apply refl_equal.
% 4.25/4.40  exact (zenon_Hc6 zenon_Hb6).
% 4.25/4.40  generalize (lesseq_ref (max zenon_TX_t zenon_TX_t)). zenon_intro zenon_Hb8.
% 4.25/4.40  exact (zenon_Hb9 zenon_Hb8).
% 4.25/4.40  (* end of lemma zenon_L24_ *)
% 4.25/4.40  assert (zenon_L25_ : forall (zenon_TX_t : zenon_U) (zenon_TY_s : zenon_U), (forall Y : zenon_U, ((lesseq zenon_TY_s Y)\/(lesseq Y zenon_TY_s))) -> (~(lesseq zenon_TY_s zenon_TX_t)) -> (forall Y : zenon_U, (((max zenon_TX_t Y) = Y)\/(~(lesseq zenon_TX_t Y)))) -> (~(zenon_TY_s = (max zenon_TX_t zenon_TY_s))) -> False).
% 4.25/4.40  do 2 intro. intros zenon_H17 zenon_H15 zenon_H1b zenon_H1c.
% 4.25/4.40  generalize (zenon_H17 zenon_TX_t). zenon_intro zenon_H19.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H19); [ zenon_intro zenon_H11 | zenon_intro zenon_H1a ].
% 4.25/4.40  exact (zenon_H15 zenon_H11).
% 4.25/4.40  apply (zenon_L3_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  (* end of lemma zenon_L25_ *)
% 4.25/4.40  apply NNPP. intro zenon_G.
% 4.25/4.40  elim (classic (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((lesseq x y)->((lesseq y z)->(lesseq x z))))))); [ zenon_intro zenon_H1f | zenon_intro zenon_He7 ].
% 4.25/4.40  apply (zenon_notallex_s (fun X : zenon_U => (forall Y : zenon_U, (forall Z : zenon_U, ((minsol_model_ub X Y Z)<->(minsol_model_max X Y Z))))) zenon_G); [ zenon_intro zenon_He8; idtac ].
% 4.25/4.40  elim zenon_He8. zenon_intro zenon_TX_t. zenon_intro zenon_He9.
% 4.25/4.40  apply (zenon_notallex_s (fun Y : zenon_U => (forall Z : zenon_U, ((minsol_model_ub zenon_TX_t Y Z)<->(minsol_model_max zenon_TX_t Y Z)))) zenon_He9); [ zenon_intro zenon_Hea; idtac ].
% 4.25/4.40  elim zenon_Hea. zenon_intro zenon_TY_s. zenon_intro zenon_Heb.
% 4.25/4.40  apply (zenon_notallex_s (fun Z : zenon_U => ((minsol_model_ub zenon_TX_t zenon_TY_s Z)<->(minsol_model_max zenon_TX_t zenon_TY_s Z))) zenon_Heb); [ zenon_intro zenon_Hec; idtac ].
% 4.25/4.40  elim zenon_Hec. zenon_intro zenon_TZ_bi. zenon_intro zenon_Hed.
% 4.25/4.40  apply (zenon_notequiv_s _ _ zenon_Hed); [ zenon_intro zenon_Hf1; zenon_intro zenon_Hf0 | zenon_intro zenon_Hef; zenon_intro zenon_Hee ].
% 4.25/4.40  generalize (minsol_model_ub zenon_TX_t). zenon_intro zenon_Hf2.
% 4.25/4.40  generalize (zenon_Hf2 zenon_TY_s). zenon_intro zenon_Hf3.
% 4.25/4.40  generalize (zenon_Hf3 zenon_TZ_bi). zenon_intro zenon_Hf4.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_Hf4); [ zenon_intro zenon_Hf1; zenon_intro zenon_Hf6 | zenon_intro zenon_Hef; zenon_intro zenon_Hf5 ].
% 4.25/4.40  generalize (minsol_model_max zenon_TX_t). zenon_intro zenon_Hf7.
% 4.25/4.40  generalize (zenon_Hf7 zenon_TY_s). zenon_intro zenon_Hf8.
% 4.25/4.40  generalize (zenon_Hf8 zenon_TZ_bi). zenon_intro zenon_Hf9.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_Hf9); [ zenon_intro zenon_Hee; zenon_intro zenon_Hfb | zenon_intro zenon_Hf0; zenon_intro zenon_Hfa ].
% 4.25/4.40  exact (zenon_Hee zenon_Hf0).
% 4.25/4.40  apply (zenon_and_s _ _ zenon_Hfa). zenon_intro zenon_Hfd. zenon_intro zenon_Hfc.
% 4.25/4.40  generalize (model_max_2 zenon_TX_t). zenon_intro zenon_Hfe.
% 4.25/4.40  generalize (zenon_Hfe zenon_TY_s). zenon_intro zenon_Hff.
% 4.25/4.40  generalize (zenon_Hff zenon_TZ_bi). zenon_intro zenon_H100.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H100); [ zenon_intro zenon_H101; zenon_intro zenon_H37 | zenon_intro zenon_Hfd; zenon_intro zenon_H21 ].
% 4.25/4.40  exact (zenon_H101 zenon_Hfd).
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_Hf6); [ zenon_intro zenon_H103 | zenon_intro zenon_H102 ].
% 4.25/4.40  generalize (model_ub_2 zenon_TX_t). zenon_intro zenon_H54.
% 4.25/4.40  generalize (zenon_H54 zenon_TY_s). zenon_intro zenon_H55.
% 4.25/4.40  generalize (zenon_H55 zenon_TZ_bi). zenon_intro zenon_H104.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H104); [ zenon_intro zenon_H103; zenon_intro zenon_H107 | zenon_intro zenon_H106; zenon_intro zenon_H105 ].
% 4.25/4.40  apply zenon_H107. exists zenon_TZ_bi. apply NNPP. zenon_intro zenon_H108.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H108); [ zenon_intro zenon_H10a | zenon_intro zenon_H109 ].
% 4.25/4.40  generalize (ub zenon_TX_t). zenon_intro zenon_H5d.
% 4.25/4.40  generalize (zenon_H5d zenon_TY_s). zenon_intro zenon_H5e.
% 4.25/4.40  generalize (zenon_H5e zenon_TZ_bi). zenon_intro zenon_H10b.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H10b); [ zenon_intro zenon_H10a; zenon_intro zenon_H10e | zenon_intro zenon_H10d; zenon_intro zenon_H10c ].
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H10e); [ zenon_intro zenon_H20 | zenon_intro zenon_H38 ].
% 4.25/4.40  elim (classic ((~(zenon_TX_t = (max zenon_TX_t zenon_TY_s)))/\(~(lesseq zenon_TX_t (max zenon_TX_t zenon_TY_s))))); [ zenon_intro zenon_H23 | zenon_intro zenon_H24 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H23). zenon_intro zenon_H10. zenon_intro zenon_H25.
% 4.25/4.40  generalize (summation_monotone zenon_TX_t). zenon_intro zenon_H10f.
% 4.25/4.40  generalize (lesseq_total zenon_TY_s). zenon_intro zenon_H17.
% 4.25/4.40  generalize (zenon_H10f zenon_TY_s). zenon_intro zenon_H110.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H110); [ zenon_intro zenon_H18; zenon_intro zenon_H112 | zenon_intro zenon_H1a; zenon_intro zenon_H111 ].
% 4.25/4.40  apply (zenon_L2_ zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  elim (classic (lesseq zenon_TY_s zenon_TZ_bi)); [ zenon_intro zenon_H44 | zenon_intro zenon_H38 ].
% 4.25/4.40  generalize (zenon_H1f zenon_TX_t). zenon_intro zenon_H32.
% 4.25/4.40  generalize (zenon_H32 zenon_TY_s). zenon_intro zenon_H8c.
% 4.25/4.40  generalize (zenon_H8c zenon_TZ_bi). zenon_intro zenon_H113.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H113); [ zenon_intro zenon_H18 | zenon_intro zenon_H114 ].
% 4.25/4.40  exact (zenon_H18 zenon_H1a).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H114); [ zenon_intro zenon_H38 | zenon_intro zenon_H36 ].
% 4.25/4.40  exact (zenon_H38 zenon_H44).
% 4.25/4.40  exact (zenon_H20 zenon_H36).
% 4.25/4.40  apply (zenon_L7_ zenon_TX_t zenon_TZ_bi zenon_TY_s); trivial.
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_bi) = (lesseq zenon_TX_t zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H20.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H21.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H2b].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H24); [ zenon_intro zenon_H2d | zenon_intro zenon_H2c ].
% 4.25/4.40  apply zenon_H2d. zenon_intro zenon_H2e.
% 4.25/4.40  elim (classic (zenon_TX_t = zenon_TX_t)); [ zenon_intro zenon_H2f | zenon_intro zenon_H30 ].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t) = ((max zenon_TX_t zenon_TY_s) = zenon_TX_t)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H2b.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H2f.
% 4.25/4.40  cut ((zenon_TX_t = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H30].
% 4.25/4.40  cut ((zenon_TX_t = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H10].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H10 zenon_H2e).
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H30. apply refl_equal.
% 4.25/4.40  apply zenon_H2c. zenon_intro zenon_H31.
% 4.25/4.40  generalize (zenon_H1f zenon_TX_t). zenon_intro zenon_H32.
% 4.25/4.40  generalize (zenon_H32 (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H33.
% 4.25/4.40  generalize (zenon_H33 zenon_TZ_bi). zenon_intro zenon_H34.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H34); [ zenon_intro zenon_H25 | zenon_intro zenon_H35 ].
% 4.25/4.40  exact (zenon_H25 zenon_H31).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H35); [ zenon_intro zenon_H37 | zenon_intro zenon_H36 ].
% 4.25/4.40  exact (zenon_H37 zenon_H21).
% 4.25/4.40  exact (zenon_H20 zenon_H36).
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  apply (zenon_L7_ zenon_TX_t zenon_TZ_bi zenon_TY_s); trivial.
% 4.25/4.40  exact (zenon_H10a zenon_H10d).
% 4.25/4.40  generalize (lesseq_ref zenon_TZ_bi). zenon_intro zenon_H115.
% 4.25/4.40  exact (zenon_H109 zenon_H115).
% 4.25/4.40  exact (zenon_H103 zenon_H106).
% 4.25/4.40  apply (zenon_notallex_s (fun Z : zenon_U => ((model_ub zenon_TX_t zenon_TY_s Z)->(lesseq zenon_TZ_bi Z))) zenon_H102); [ zenon_intro zenon_H116; idtac ].
% 4.25/4.40  elim zenon_H116. zenon_intro zenon_TZ_kt. zenon_intro zenon_H118.
% 4.25/4.40  apply (zenon_notimply_s _ _ zenon_H118). zenon_intro zenon_H11a. zenon_intro zenon_H119.
% 4.25/4.40  generalize (model_ub_2 zenon_TX_t). zenon_intro zenon_H54.
% 4.25/4.40  generalize (zenon_H54 zenon_TY_s). zenon_intro zenon_H55.
% 4.25/4.40  generalize (zenon_H55 zenon_TZ_kt). zenon_intro zenon_H11b.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H11b); [ zenon_intro zenon_H11e; zenon_intro zenon_H11d | zenon_intro zenon_H11a; zenon_intro zenon_H11c ].
% 4.25/4.40  exact (zenon_H11e zenon_H11a).
% 4.25/4.40  elim zenon_H11c. zenon_intro zenon_TZ_lb. zenon_intro zenon_H120.
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H120). zenon_intro zenon_H122. zenon_intro zenon_H121.
% 4.25/4.40  generalize (ub zenon_TX_t). zenon_intro zenon_H5d.
% 4.25/4.40  generalize (zenon_H5d zenon_TY_s). zenon_intro zenon_H5e.
% 4.25/4.40  generalize (zenon_H5e zenon_TZ_lb). zenon_intro zenon_H123.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H123); [ zenon_intro zenon_H126; zenon_intro zenon_H125 | zenon_intro zenon_H122; zenon_intro zenon_H124 ].
% 4.25/4.40  exact (zenon_H126 zenon_H122).
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H124). zenon_intro zenon_H128. zenon_intro zenon_H127.
% 4.25/4.40  generalize (lesseq_total zenon_TX_t). zenon_intro zenon_H129.
% 4.25/4.40  generalize (zenon_Hfc zenon_TZ_kt). zenon_intro zenon_H12a.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H12a); [ zenon_intro zenon_H12c | zenon_intro zenon_H12b ].
% 4.25/4.40  generalize (model_max_2 zenon_TX_t). zenon_intro zenon_Hfe.
% 4.25/4.40  generalize (zenon_Hfe zenon_TY_s). zenon_intro zenon_Hff.
% 4.25/4.40  generalize (zenon_Hff zenon_TZ_kt). zenon_intro zenon_H12d.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H12d); [ zenon_intro zenon_H12c; zenon_intro zenon_H130 | zenon_intro zenon_H12f; zenon_intro zenon_H12e ].
% 4.25/4.40  elim (classic ((~((max zenon_TX_t zenon_TY_s) = zenon_TZ_lb))/\(~(lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_lb)))); [ zenon_intro zenon_H131 | zenon_intro zenon_H132 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H131). zenon_intro zenon_H134. zenon_intro zenon_H133.
% 4.25/4.40  elim (classic ((~((max zenon_TX_t zenon_TY_s) = zenon_TY_s))/\(~(lesseq (max zenon_TX_t zenon_TY_s) zenon_TY_s)))); [ zenon_intro zenon_H135 | zenon_intro zenon_H136 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H135). zenon_intro zenon_H45. zenon_intro zenon_H137.
% 4.25/4.40  elim (classic ((~((max zenon_TX_t zenon_TY_s) = zenon_TX_t))/\(~(lesseq (max zenon_TX_t zenon_TY_s) zenon_TX_t)))); [ zenon_intro zenon_H138 | zenon_intro zenon_H139 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H138). zenon_intro zenon_H2b. zenon_intro zenon_H13a.
% 4.25/4.40  generalize (max_2 zenon_TX_t). zenon_intro zenon_H1b.
% 4.25/4.40  generalize (zenon_H129 zenon_TY_s). zenon_intro zenon_H13b.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H13b); [ zenon_intro zenon_H1a | zenon_intro zenon_H11 ].
% 4.25/4.40  apply (zenon_L8_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  generalize (max_1 zenon_TX_t). zenon_intro zenon_Hf.
% 4.25/4.40  generalize (zenon_Hf zenon_TY_s). zenon_intro zenon_H14.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H14); [ zenon_intro zenon_H16 | zenon_intro zenon_H15 ].
% 4.25/4.40  exact (zenon_H2b zenon_H16).
% 4.25/4.40  exact (zenon_H15 zenon_H11).
% 4.25/4.40  cut ((lesseq zenon_TX_t zenon_TZ_lb) = (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_lb)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H133.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H128.
% 4.25/4.40  cut ((zenon_TZ_lb = zenon_TZ_lb)); [idtac | apply NNPP; zenon_intro zenon_H13c].
% 4.25/4.40  cut ((zenon_TX_t = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H10].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H139); [ zenon_intro zenon_H13e | zenon_intro zenon_H13d ].
% 4.25/4.40  apply zenon_H13e. zenon_intro zenon_H16.
% 4.25/4.40  elim (classic ((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_H13f | zenon_intro zenon_H140 ].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s)) = (zenon_TX_t = (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H10.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H13f.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H140].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_H2b].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H2b zenon_H16).
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H13d. zenon_intro zenon_H141.
% 4.25/4.40  generalize (zenon_H1f (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H142.
% 4.25/4.40  generalize (zenon_H142 zenon_TX_t). zenon_intro zenon_H143.
% 4.25/4.40  generalize (zenon_H143 zenon_TZ_lb). zenon_intro zenon_H144.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H144); [ zenon_intro zenon_H13a | zenon_intro zenon_H145 ].
% 4.25/4.40  exact (zenon_H13a zenon_H141).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H145); [ zenon_intro zenon_H147 | zenon_intro zenon_H146 ].
% 4.25/4.40  exact (zenon_H147 zenon_H128).
% 4.25/4.40  exact (zenon_H133 zenon_H146).
% 4.25/4.40  apply zenon_H13c. apply refl_equal.
% 4.25/4.40  cut ((lesseq zenon_TY_s zenon_TZ_lb) = (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_lb)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H133.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H127.
% 4.25/4.40  cut ((zenon_TZ_lb = zenon_TZ_lb)); [idtac | apply NNPP; zenon_intro zenon_H13c].
% 4.25/4.40  cut ((zenon_TY_s = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H1c].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H136); [ zenon_intro zenon_H149 | zenon_intro zenon_H148 ].
% 4.25/4.40  apply zenon_H149. zenon_intro zenon_H1e.
% 4.25/4.40  elim (classic ((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_H13f | zenon_intro zenon_H140 ].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s)) = (zenon_TY_s = (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H1c.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H13f.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H140].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H45].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H45 zenon_H1e).
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H148. zenon_intro zenon_H14a.
% 4.25/4.40  generalize (zenon_H1f (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H142.
% 4.25/4.40  generalize (zenon_H142 zenon_TY_s). zenon_intro zenon_H14b.
% 4.25/4.40  generalize (zenon_H14b zenon_TZ_lb). zenon_intro zenon_H14c.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H14c); [ zenon_intro zenon_H137 | zenon_intro zenon_H14d ].
% 4.25/4.40  exact (zenon_H137 zenon_H14a).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H14d); [ zenon_intro zenon_H14e | zenon_intro zenon_H146 ].
% 4.25/4.40  exact (zenon_H14e zenon_H127).
% 4.25/4.40  exact (zenon_H133 zenon_H146).
% 4.25/4.40  apply zenon_H13c. apply refl_equal.
% 4.25/4.40  cut ((lesseq zenon_TZ_lb zenon_TZ_kt) = (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_kt)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H130.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H121.
% 4.25/4.40  cut ((zenon_TZ_kt = zenon_TZ_kt)); [idtac | apply NNPP; zenon_intro zenon_H14f].
% 4.25/4.40  cut ((zenon_TZ_lb = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H150].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H132); [ zenon_intro zenon_H152 | zenon_intro zenon_H151 ].
% 4.25/4.40  apply zenon_H152. zenon_intro zenon_H153.
% 4.25/4.40  elim (classic ((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_H13f | zenon_intro zenon_H140 ].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s)) = (zenon_TZ_lb = (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H150.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H13f.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H140].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TZ_lb)); [idtac | apply NNPP; zenon_intro zenon_H134].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H134 zenon_H153).
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H151. zenon_intro zenon_H146.
% 4.25/4.40  generalize (zenon_H1f (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H142.
% 4.25/4.40  generalize (zenon_H142 zenon_TZ_lb). zenon_intro zenon_H154.
% 4.25/4.40  generalize (zenon_H154 zenon_TZ_kt). zenon_intro zenon_H155.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H155); [ zenon_intro zenon_H133 | zenon_intro zenon_H156 ].
% 4.25/4.40  exact (zenon_H133 zenon_H146).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H156); [ zenon_intro zenon_H157 | zenon_intro zenon_H12e ].
% 4.25/4.40  exact (zenon_H157 zenon_H121).
% 4.25/4.40  exact (zenon_H130 zenon_H12e).
% 4.25/4.40  apply zenon_H14f. apply refl_equal.
% 4.25/4.40  exact (zenon_H12c zenon_H12f).
% 4.25/4.40  exact (zenon_H119 zenon_H12b).
% 4.25/4.40  exact (zenon_Hf1 zenon_Hef).
% 4.25/4.40  generalize (minsol_model_ub zenon_TX_t). zenon_intro zenon_Hf2.
% 4.25/4.40  generalize (zenon_Hf2 zenon_TY_s). zenon_intro zenon_Hf3.
% 4.25/4.40  generalize (zenon_Hf3 zenon_TZ_bi). zenon_intro zenon_Hf4.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_Hf4); [ zenon_intro zenon_Hf1; zenon_intro zenon_Hf6 | zenon_intro zenon_Hef; zenon_intro zenon_Hf5 ].
% 4.25/4.40  exact (zenon_Hf1 zenon_Hef).
% 4.25/4.40  apply (zenon_and_s _ _ zenon_Hf5). zenon_intro zenon_H106. zenon_intro zenon_H4f.
% 4.25/4.40  generalize (model_ub_2 zenon_TX_t). zenon_intro zenon_H54.
% 4.25/4.40  generalize (zenon_H54 zenon_TY_s). zenon_intro zenon_H55.
% 4.25/4.40  generalize (zenon_H55 zenon_TZ_bi). zenon_intro zenon_H104.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H104); [ zenon_intro zenon_H103; zenon_intro zenon_H107 | zenon_intro zenon_H106; zenon_intro zenon_H105 ].
% 4.25/4.40  exact (zenon_H103 zenon_H106).
% 4.25/4.40  elim zenon_H105. zenon_intro zenon_TZ_dy. zenon_intro zenon_H158.
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H158). zenon_intro zenon_H159. zenon_intro zenon_H65.
% 4.25/4.40  generalize (ub zenon_TX_t). zenon_intro zenon_H5d.
% 4.25/4.40  generalize (zenon_H5d zenon_TY_s). zenon_intro zenon_H5e.
% 4.25/4.40  generalize (zenon_H5e zenon_TZ_dy). zenon_intro zenon_H15a.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H15a); [ zenon_intro zenon_H15d; zenon_intro zenon_H15c | zenon_intro zenon_H159; zenon_intro zenon_H15b ].
% 4.25/4.40  exact (zenon_H15d zenon_H159).
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H15b). zenon_intro zenon_H64. zenon_intro zenon_H73.
% 4.25/4.40  generalize (minsol_model_max zenon_TX_t). zenon_intro zenon_Hf7.
% 4.25/4.40  generalize (zenon_Hf7 zenon_TY_s). zenon_intro zenon_Hf8.
% 4.25/4.40  generalize (zenon_Hf8 zenon_TZ_bi). zenon_intro zenon_Hf9.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_Hf9); [ zenon_intro zenon_Hee; zenon_intro zenon_Hfb | zenon_intro zenon_Hf0; zenon_intro zenon_Hfa ].
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_Hfb); [ zenon_intro zenon_H101 | zenon_intro zenon_H15e ].
% 4.25/4.40  generalize (model_max_2 zenon_TX_t). zenon_intro zenon_Hfe.
% 4.25/4.40  generalize (zenon_Hfe zenon_TY_s). zenon_intro zenon_Hff.
% 4.25/4.40  generalize (zenon_Hff zenon_TZ_bi). zenon_intro zenon_H100.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H100); [ zenon_intro zenon_H101; zenon_intro zenon_H37 | zenon_intro zenon_Hfd; zenon_intro zenon_H21 ].
% 4.25/4.40  elim (classic ((~((max zenon_TX_t zenon_TY_s) = zenon_TZ_dy))/\(~(lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_dy)))); [ zenon_intro zenon_H15f | zenon_intro zenon_H160 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H15f). zenon_intro zenon_H162. zenon_intro zenon_H161.
% 4.25/4.40  elim (classic ((~((max zenon_TX_t zenon_TY_s) = zenon_TY_s))/\(~(lesseq (max zenon_TX_t zenon_TY_s) zenon_TY_s)))); [ zenon_intro zenon_H135 | zenon_intro zenon_H136 ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H135). zenon_intro zenon_H45. zenon_intro zenon_H137.
% 4.25/4.40  generalize (max_1 zenon_TZ_bi). zenon_intro zenon_H163.
% 4.25/4.40  generalize (lesseq_total zenon_TY_s). zenon_intro zenon_H17.
% 4.25/4.40  generalize (lesseq_antisymmetric zenon_TZ_bi). zenon_intro zenon_H164.
% 4.25/4.40  generalize (max_2 zenon_TX_t). zenon_intro zenon_H1b.
% 4.25/4.40  generalize (zenon_H163 zenon_TZ_bi). zenon_intro zenon_H165.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H165); [ zenon_intro zenon_H166 | zenon_intro zenon_H109 ].
% 4.25/4.40  elim (classic (lesseq zenon_TZ_bi zenon_TZ_bi)); [ zenon_intro zenon_H115 | zenon_intro zenon_H109 ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TZ_bi zenon_TZ_bi) zenon_TZ_bi)); [ zenon_intro zenon_H167 | zenon_intro zenon_H168 ].
% 4.25/4.40  cut ((lesseq (max zenon_TZ_bi zenon_TZ_bi) zenon_TZ_bi) = (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H37.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H167.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut (((max zenon_TZ_bi zenon_TZ_bi) = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H169].
% 4.25/4.40  congruence.
% 4.25/4.40  elim (classic ((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_H13f | zenon_intro zenon_H140 ].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s)) = ((max zenon_TZ_bi zenon_TZ_bi) = (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H169.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H13f.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H140].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TZ_bi zenon_TZ_bi))); [idtac | apply NNPP; zenon_intro zenon_H16a].
% 4.25/4.40  congruence.
% 4.25/4.40  cut ((zenon_TY_s = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H16b].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H16c].
% 4.25/4.40  congruence.
% 4.25/4.40  generalize (zenon_H17 zenon_TX_t). zenon_intro zenon_H19.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H19); [ zenon_intro zenon_H11 | zenon_intro zenon_H1a ].
% 4.25/4.40  generalize (zenon_H164 zenon_TX_t). zenon_intro zenon_H16d.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H16d); [ zenon_intro zenon_H16f | zenon_intro zenon_H16e ].
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H16f); [ zenon_intro zenon_H50 | zenon_intro zenon_H20 ].
% 4.25/4.40  apply (zenon_L9_ zenon_TZ_bi zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  apply (zenon_L10_ zenon_TZ_bi zenon_TZ_dy zenon_TX_t); trivial.
% 4.25/4.40  apply zenon_H16c. apply sym_equal. exact zenon_H16e.
% 4.25/4.40  apply (zenon_L8_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  generalize (zenon_H17 zenon_TX_t). zenon_intro zenon_H19.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H19); [ zenon_intro zenon_H11 | zenon_intro zenon_H1a ].
% 4.25/4.40  generalize (zenon_H164 zenon_TX_t). zenon_intro zenon_H16d.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H16d); [ zenon_intro zenon_H16f | zenon_intro zenon_H16e ].
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H16f); [ zenon_intro zenon_H50 | zenon_intro zenon_H20 ].
% 4.25/4.40  apply (zenon_L9_ zenon_TZ_bi zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  apply (zenon_L10_ zenon_TZ_bi zenon_TZ_dy zenon_TX_t); trivial.
% 4.25/4.40  generalize (zenon_H164 zenon_TY_s). zenon_intro zenon_H170.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H170); [ zenon_intro zenon_H172 | zenon_intro zenon_H171 ].
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H172); [ zenon_intro zenon_H173 | zenon_intro zenon_H38 ].
% 4.25/4.40  generalize (zenon_H17 zenon_TZ_bi). zenon_intro zenon_H174.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H174); [ zenon_intro zenon_H44 | zenon_intro zenon_H175 ].
% 4.25/4.40  generalize (zenon_H163 zenon_TY_s). zenon_intro zenon_H176.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H176); [ zenon_intro zenon_H177 | zenon_intro zenon_H38 ].
% 4.25/4.40  elim (classic (lesseq zenon_TZ_bi zenon_TZ_bi)); [ zenon_intro zenon_H115 | zenon_intro zenon_H109 ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TZ_bi zenon_TY_s) zenon_TZ_bi)); [ zenon_intro zenon_H178 | zenon_intro zenon_H179 ].
% 4.25/4.40  cut ((lesseq (max zenon_TZ_bi zenon_TY_s) zenon_TZ_bi) = (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H37.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H178.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut (((max zenon_TZ_bi zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H17a].
% 4.25/4.40  congruence.
% 4.25/4.40  elim (classic ((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_H13f | zenon_intro zenon_H140 ].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s)) = ((max zenon_TZ_bi zenon_TY_s) = (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H17a.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H13f.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H140].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TZ_bi zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H17b].
% 4.25/4.40  congruence.
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H4a].
% 4.25/4.40  cut ((zenon_TX_t = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H16c].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_H16c. apply sym_equal. exact zenon_H16e.
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  cut ((lesseq zenon_TZ_bi zenon_TZ_bi) = (lesseq (max zenon_TZ_bi zenon_TY_s) zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H179.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H115.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut ((zenon_TZ_bi = (max zenon_TZ_bi zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H17c].
% 4.25/4.40  congruence.
% 4.25/4.40  elim (classic ((max zenon_TZ_bi zenon_TY_s) = (max zenon_TZ_bi zenon_TY_s))); [ zenon_intro zenon_H17d | zenon_intro zenon_H17e ].
% 4.25/4.40  cut (((max zenon_TZ_bi zenon_TY_s) = (max zenon_TZ_bi zenon_TY_s)) = (zenon_TZ_bi = (max zenon_TZ_bi zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H17c.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H17d.
% 4.25/4.40  cut (((max zenon_TZ_bi zenon_TY_s) = (max zenon_TZ_bi zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H17e].
% 4.25/4.40  cut (((max zenon_TZ_bi zenon_TY_s) = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H17f].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H17f zenon_H177).
% 4.25/4.40  apply zenon_H17e. apply refl_equal.
% 4.25/4.40  apply zenon_H17e. apply refl_equal.
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  generalize (lesseq_ref zenon_TZ_bi). zenon_intro zenon_H115.
% 4.25/4.40  exact (zenon_H109 zenon_H115).
% 4.25/4.40  exact (zenon_H38 zenon_H44).
% 4.25/4.40  exact (zenon_H173 zenon_H175).
% 4.25/4.40  apply (zenon_L11_ zenon_TZ_bi zenon_TZ_dy zenon_TY_s); trivial.
% 4.25/4.40  apply zenon_H16b. apply sym_equal. exact zenon_H171.
% 4.25/4.40  apply (zenon_L8_ zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  cut ((lesseq zenon_TZ_bi zenon_TZ_bi) = (lesseq (max zenon_TZ_bi zenon_TZ_bi) zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H168.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H115.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut ((zenon_TZ_bi = (max zenon_TZ_bi zenon_TZ_bi))); [idtac | apply NNPP; zenon_intro zenon_H180].
% 4.25/4.40  congruence.
% 4.25/4.40  elim (classic ((max zenon_TZ_bi zenon_TZ_bi) = (max zenon_TZ_bi zenon_TZ_bi))); [ zenon_intro zenon_H181 | zenon_intro zenon_H182 ].
% 4.25/4.40  cut (((max zenon_TZ_bi zenon_TZ_bi) = (max zenon_TZ_bi zenon_TZ_bi)) = (zenon_TZ_bi = (max zenon_TZ_bi zenon_TZ_bi))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H180.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H181.
% 4.25/4.40  cut (((max zenon_TZ_bi zenon_TZ_bi) = (max zenon_TZ_bi zenon_TZ_bi))); [idtac | apply NNPP; zenon_intro zenon_H182].
% 4.25/4.40  cut (((max zenon_TZ_bi zenon_TZ_bi) = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H183].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H183 zenon_H166).
% 4.25/4.40  apply zenon_H182. apply refl_equal.
% 4.25/4.40  apply zenon_H182. apply refl_equal.
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  generalize (lesseq_ref zenon_TZ_bi). zenon_intro zenon_H115.
% 4.25/4.40  exact (zenon_H109 zenon_H115).
% 4.25/4.40  generalize (lesseq_ref zenon_TZ_bi). zenon_intro zenon_H115.
% 4.25/4.40  exact (zenon_H109 zenon_H115).
% 4.25/4.40  cut ((lesseq zenon_TY_s zenon_TZ_dy) = (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_dy)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H161.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H73.
% 4.25/4.40  cut ((zenon_TZ_dy = zenon_TZ_dy)); [idtac | apply NNPP; zenon_intro zenon_H184].
% 4.25/4.40  cut ((zenon_TY_s = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H1c].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H136); [ zenon_intro zenon_H149 | zenon_intro zenon_H148 ].
% 4.25/4.40  apply zenon_H149. zenon_intro zenon_H1e.
% 4.25/4.40  elim (classic ((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_H13f | zenon_intro zenon_H140 ].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s)) = (zenon_TY_s = (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H1c.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H13f.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H140].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H45].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H45 zenon_H1e).
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H148. zenon_intro zenon_H14a.
% 4.25/4.40  generalize (zenon_H1f (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H142.
% 4.25/4.40  generalize (zenon_H142 zenon_TY_s). zenon_intro zenon_H14b.
% 4.25/4.40  generalize (zenon_H14b zenon_TZ_dy). zenon_intro zenon_H185.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H185); [ zenon_intro zenon_H137 | zenon_intro zenon_H186 ].
% 4.25/4.40  exact (zenon_H137 zenon_H14a).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H186); [ zenon_intro zenon_H76 | zenon_intro zenon_H187 ].
% 4.25/4.40  exact (zenon_H76 zenon_H73).
% 4.25/4.40  exact (zenon_H161 zenon_H187).
% 4.25/4.40  apply zenon_H184. apply refl_equal.
% 4.25/4.40  cut ((lesseq zenon_TZ_dy zenon_TZ_bi) = (lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_bi)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H37.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H65.
% 4.25/4.40  cut ((zenon_TZ_bi = zenon_TZ_bi)); [idtac | apply NNPP; zenon_intro zenon_H2a].
% 4.25/4.40  cut ((zenon_TZ_dy = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H188].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H160); [ zenon_intro zenon_H18a | zenon_intro zenon_H189 ].
% 4.25/4.40  apply zenon_H18a. zenon_intro zenon_H18b.
% 4.25/4.40  elim (classic ((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_H13f | zenon_intro zenon_H140 ].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s)) = (zenon_TZ_dy = (max zenon_TX_t zenon_TY_s))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H188.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H13f.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H140].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TZ_dy)); [idtac | apply NNPP; zenon_intro zenon_H162].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H162 zenon_H18b).
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H140. apply refl_equal.
% 4.25/4.40  apply zenon_H189. zenon_intro zenon_H187.
% 4.25/4.40  generalize (zenon_H1f (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H142.
% 4.25/4.40  generalize (zenon_H142 zenon_TZ_dy). zenon_intro zenon_H18c.
% 4.25/4.40  generalize (zenon_H18c zenon_TZ_bi). zenon_intro zenon_H18d.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H18d); [ zenon_intro zenon_H161 | zenon_intro zenon_H18e ].
% 4.25/4.40  exact (zenon_H161 zenon_H187).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H18e); [ zenon_intro zenon_H72 | zenon_intro zenon_H21 ].
% 4.25/4.40  exact (zenon_H72 zenon_H65).
% 4.25/4.40  exact (zenon_H37 zenon_H21).
% 4.25/4.40  apply zenon_H2a. apply refl_equal.
% 4.25/4.40  exact (zenon_H101 zenon_Hfd).
% 4.25/4.40  apply (zenon_notallex_s (fun Z : zenon_U => ((model_max zenon_TX_t zenon_TY_s Z)->(lesseq zenon_TZ_bi Z))) zenon_H15e); [ zenon_intro zenon_H18f; idtac ].
% 4.25/4.40  elim zenon_H18f. zenon_intro zenon_TZ_ez. zenon_intro zenon_H190.
% 4.25/4.40  apply (zenon_notimply_s _ _ zenon_H190). zenon_intro zenon_H192. zenon_intro zenon_H191.
% 4.25/4.40  generalize (model_max_2 zenon_TX_t). zenon_intro zenon_Hfe.
% 4.25/4.40  generalize (zenon_Hfe zenon_TY_s). zenon_intro zenon_Hff.
% 4.25/4.40  generalize (zenon_Hff zenon_TZ_ez). zenon_intro zenon_H193.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H193); [ zenon_intro zenon_H194; zenon_intro zenon_H9e | zenon_intro zenon_H192; zenon_intro zenon_H94 ].
% 4.25/4.40  exact (zenon_H194 zenon_H192).
% 4.25/4.40  generalize (max_1 zenon_TZ_ez). zenon_intro zenon_H195.
% 4.25/4.40  generalize (lesseq_total zenon_TZ_ez). zenon_intro zenon_H91.
% 4.25/4.40  generalize (max_2 zenon_TX_t). zenon_intro zenon_H1b.
% 4.25/4.40  generalize (zenon_H4f zenon_TZ_ez). zenon_intro zenon_H196.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H196); [ zenon_intro zenon_H198 | zenon_intro zenon_H197 ].
% 4.25/4.40  generalize (model_ub_2 zenon_TX_t). zenon_intro zenon_H54.
% 4.25/4.40  generalize (zenon_H54 zenon_TY_s). zenon_intro zenon_H55.
% 4.25/4.40  generalize (zenon_H55 zenon_TZ_ez). zenon_intro zenon_H199.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H199); [ zenon_intro zenon_H198; zenon_intro zenon_H93 | zenon_intro zenon_H19b; zenon_intro zenon_H19a ].
% 4.25/4.40  generalize (lesseq_total zenon_TY_s). zenon_intro zenon_H17.
% 4.25/4.40  generalize (max_1 zenon_TX_t). zenon_intro zenon_Hf.
% 4.25/4.40  generalize (zenon_H195 zenon_TZ_ez). zenon_intro zenon_H19c.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H19c); [ zenon_intro zenon_Ha0 | zenon_intro zenon_H96 ].
% 4.25/4.40  generalize (zenon_H1b zenon_TX_t). zenon_intro zenon_H19d.
% 4.25/4.40  apply (zenon_or_s _ _ zenon_H19d); [ zenon_intro zenon_Hb6 | zenon_intro zenon_H5b ].
% 4.25/4.40  apply zenon_H93. exists zenon_TZ_ez. apply NNPP. zenon_intro zenon_H95.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H95); [ zenon_intro zenon_H97 | zenon_intro zenon_H96 ].
% 4.25/4.40  generalize (ub zenon_TX_t). zenon_intro zenon_H5d.
% 4.25/4.40  generalize (zenon_H5d zenon_TY_s). zenon_intro zenon_H5e.
% 4.25/4.40  generalize (zenon_H5e zenon_TZ_ez). zenon_intro zenon_H98.
% 4.25/4.40  apply (zenon_equiv_s _ _ zenon_H98); [ zenon_intro zenon_H97; zenon_intro zenon_H9b | zenon_intro zenon_H9a; zenon_intro zenon_H99 ].
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H9b); [ zenon_intro zenon_H83 | zenon_intro zenon_H90 ].
% 4.25/4.40  apply (zenon_L23_ zenon_TZ_ez zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  elim (classic ((~(zenon_TY_s = (max zenon_TX_t zenon_TY_s)))/\(~(lesseq zenon_TY_s (max zenon_TX_t zenon_TY_s))))); [ zenon_intro zenon_H39 | zenon_intro zenon_H3a ].
% 4.25/4.40  apply (zenon_and_s _ _ zenon_H39). zenon_intro zenon_H1c. zenon_intro zenon_H3b.
% 4.25/4.40  elim (classic (zenon_TX_t = (max zenon_TX_t zenon_TX_t))); [ zenon_intro zenon_Hdb | zenon_intro zenon_Hdc ].
% 4.25/4.40  elim (classic (lesseq zenon_TY_s zenon_TX_t)); [ zenon_intro zenon_H11 | zenon_intro zenon_H15 ].
% 4.25/4.40  elim (classic (lesseq zenon_TY_s (max zenon_TX_t zenon_TX_t))); [ zenon_intro zenon_H19e | zenon_intro zenon_H19f ].
% 4.25/4.40  elim (classic (lesseq (max zenon_TX_t zenon_TX_t) (max zenon_TX_t zenon_TY_s))); [ zenon_intro zenon_Hd9 | zenon_intro zenon_Hc8 ].
% 4.25/4.40  generalize (zenon_H1f zenon_TY_s). zenon_intro zenon_H40.
% 4.25/4.40  generalize (zenon_H40 (max zenon_TX_t zenon_TX_t)). zenon_intro zenon_H1a0.
% 4.25/4.40  generalize (zenon_H1a0 (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H1a1.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H1a1); [ zenon_intro zenon_H19f | zenon_intro zenon_H1a2 ].
% 4.25/4.40  exact (zenon_H19f zenon_H19e).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H1a2); [ zenon_intro zenon_Hc8 | zenon_intro zenon_H4b ].
% 4.25/4.40  exact (zenon_Hc8 zenon_Hd9).
% 4.25/4.40  exact (zenon_H3b zenon_H4b).
% 4.25/4.40  apply (zenon_L24_ zenon_TZ_ez zenon_TY_s zenon_TX_t); trivial.
% 4.25/4.40  cut ((lesseq zenon_TY_s zenon_TX_t) = (lesseq zenon_TY_s (max zenon_TX_t zenon_TX_t))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H19f.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H11.
% 4.25/4.40  cut ((zenon_TX_t = (max zenon_TX_t zenon_TX_t))); [idtac | apply NNPP; zenon_intro zenon_Hdc].
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H4a].
% 4.25/4.40  congruence.
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  exact (zenon_Hdc zenon_Hdb).
% 4.25/4.40  apply (zenon_L25_ zenon_TX_t zenon_TY_s); trivial.
% 4.25/4.40  elim (classic ((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t))); [ zenon_intro zenon_He2 | zenon_intro zenon_Hc7 ].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t)) = (zenon_TX_t = (max zenon_TX_t zenon_TX_t))).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_Hdc.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_He2.
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = (max zenon_TX_t zenon_TX_t))); [idtac | apply NNPP; zenon_intro zenon_Hc7].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TX_t) = zenon_TX_t)); [idtac | apply NNPP; zenon_intro zenon_Hc6].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_Hc6 zenon_Hb6).
% 4.25/4.40  apply zenon_Hc7. apply refl_equal.
% 4.25/4.40  apply zenon_Hc7. apply refl_equal.
% 4.25/4.40  cut ((lesseq (max zenon_TX_t zenon_TY_s) zenon_TZ_ez) = (lesseq zenon_TY_s zenon_TZ_ez)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H90.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H94.
% 4.25/4.40  cut ((zenon_TZ_ez = zenon_TZ_ez)); [idtac | apply NNPP; zenon_intro zenon_H82].
% 4.25/4.40  cut (((max zenon_TX_t zenon_TY_s) = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H45].
% 4.25/4.40  congruence.
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H3a); [ zenon_intro zenon_H47 | zenon_intro zenon_H46 ].
% 4.25/4.40  apply zenon_H47. zenon_intro zenon_H48.
% 4.25/4.40  elim (classic (zenon_TY_s = zenon_TY_s)); [ zenon_intro zenon_H49 | zenon_intro zenon_H4a ].
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s) = ((max zenon_TX_t zenon_TY_s) = zenon_TY_s)).
% 4.25/4.40  intro zenon_D_pnotp.
% 4.25/4.40  apply zenon_H45.
% 4.25/4.40  rewrite <- zenon_D_pnotp.
% 4.25/4.40  exact zenon_H49.
% 4.25/4.40  cut ((zenon_TY_s = zenon_TY_s)); [idtac | apply NNPP; zenon_intro zenon_H4a].
% 4.25/4.40  cut ((zenon_TY_s = (max zenon_TX_t zenon_TY_s))); [idtac | apply NNPP; zenon_intro zenon_H1c].
% 4.25/4.40  congruence.
% 4.25/4.40  exact (zenon_H1c zenon_H48).
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  apply zenon_H4a. apply refl_equal.
% 4.25/4.40  apply zenon_H46. zenon_intro zenon_H4b.
% 4.25/4.40  generalize (zenon_H1f zenon_TY_s). zenon_intro zenon_H40.
% 4.25/4.40  generalize (zenon_H40 (max zenon_TX_t zenon_TY_s)). zenon_intro zenon_H4c.
% 4.25/4.40  generalize (zenon_H4c zenon_TZ_ez). zenon_intro zenon_H9c.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H9c); [ zenon_intro zenon_H3b | zenon_intro zenon_H9d ].
% 4.25/4.40  exact (zenon_H3b zenon_H4b).
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H9d); [ zenon_intro zenon_H9e | zenon_intro zenon_H84 ].
% 4.25/4.40  exact (zenon_H9e zenon_H94).
% 4.25/4.40  exact (zenon_H90 zenon_H84).
% 4.25/4.40  apply zenon_H82. apply refl_equal.
% 4.25/4.40  exact (zenon_H97 zenon_H9a).
% 4.25/4.40  generalize (lesseq_ref zenon_TZ_ez). zenon_intro zenon_H9f.
% 4.25/4.40  exact (zenon_H96 zenon_H9f).
% 4.25/4.40  generalize (lesseq_ref zenon_TX_t). zenon_intro zenon_H63.
% 4.25/4.40  exact (zenon_H5b zenon_H63).
% 4.25/4.40  generalize (lesseq_ref zenon_TZ_ez). zenon_intro zenon_H9f.
% 4.25/4.40  exact (zenon_H96 zenon_H9f).
% 4.25/4.40  exact (zenon_H198 zenon_H19b).
% 4.25/4.40  exact (zenon_H191 zenon_H197).
% 4.25/4.40  exact (zenon_Hee zenon_Hf0).
% 4.25/4.40  apply zenon_He7. zenon_intro zenon_Tx_qd. apply NNPP. zenon_intro zenon_H1a4.
% 4.25/4.40  apply zenon_H1a4. zenon_intro zenon_Ty_qf. apply NNPP. zenon_intro zenon_H1a6.
% 4.25/4.40  apply zenon_H1a6. zenon_intro zenon_Tz_qh. apply NNPP. zenon_intro zenon_H1a8.
% 4.25/4.40  apply (zenon_notimply_s _ _ zenon_H1a8). zenon_intro zenon_H1aa. zenon_intro zenon_H1a9.
% 4.25/4.40  apply (zenon_notimply_s _ _ zenon_H1a9). zenon_intro zenon_H1ac. zenon_intro zenon_H1ab.
% 4.25/4.40  generalize (lesseq_trans zenon_Tx_qd). zenon_intro zenon_H1ad.
% 4.25/4.40  generalize (zenon_H1ad zenon_Ty_qf). zenon_intro zenon_H1ae.
% 4.25/4.40  generalize (zenon_H1ae zenon_Tz_qh). zenon_intro zenon_H1af.
% 4.25/4.40  apply (zenon_imply_s _ _ zenon_H1af); [ zenon_intro zenon_H1b1 | zenon_intro zenon_H1b0 ].
% 4.25/4.40  apply (zenon_notand_s _ _ zenon_H1b1); [ zenon_intro zenon_H1b3 | zenon_intro zenon_H1b2 ].
% 4.25/4.40  exact (zenon_H1b3 zenon_H1aa).
% 4.25/4.40  exact (zenon_H1b2 zenon_H1ac).
% 4.25/4.40  exact (zenon_H1ab zenon_H1b0).
% 4.25/4.40  Qed.
% 4.25/4.40  % SZS output end Proof
% 4.25/4.40  (* END-PROOF *)
% 4.25/4.40  nodes searched: 325053
% 4.25/4.40  max branch formulas: 3553
% 4.25/4.40  proof nodes created: 7664
% 4.25/4.40  formulas created: 216661
% 4.25/4.40  
%------------------------------------------------------------------------------