# SZS status Theorem
# SZS status Theorem
# SZS output start CNFRefutation.
fof(1, axiom,p(s(t_bool,t)),file('i/f/probability/distribution__lebesgue__thm1', aHLu_TRUTH)).
fof(2, axiom,~(p(s(t_bool,f))),file('i/f/probability/distribution__lebesgue__thm1', aHLu_FALSITY)).
fof(3, axiom,![X1]:(s(t_bool,X1)=s(t_bool,t)|s(t_bool,X1)=s(t_bool,f)),file('i/f/probability/distribution__lebesgue__thm1', aHLu_BOOLu_CASES)).
fof(10, axiom,![X1]:(s(t_bool,X1)=s(t_bool,t)<=>p(s(t_bool,X1))),file('i/f/probability/distribution__lebesgue__thm1', ah4s_bools_EQu_u_CLAUSESu_c1)).
fof(13, axiom,![X7]:![X8]:![X6]:s(t_fun(X7,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_fun(t_fun(X7,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(X7,t_bool),X6),s(t_fun(t_fun(X7,t_bool),t_bool),X8)))))=s(t_fun(X7,t_bool),X6),file('i/f/probability/distribution__lebesgue__thm1', ah4s_measures_spaceu_u_def)).
fof(14, axiom,![X7]:![X8]:![X6]:s(t_fun(t_fun(X7,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_fun(t_fun(X7,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(X7,t_bool),X6),s(t_fun(t_fun(X7,t_bool),t_bool),X8)))))=s(t_fun(t_fun(X7,t_bool),t_bool),X8),file('i/f/probability/distribution__lebesgue__thm1', ah4s_measures_subsetsu_u_def)).
fof(15, axiom,![X14]:![X7]:![X4]:![X15]:![X16]:(p(s(t_bool,h4s_bools_in(s(t_fun(X7,X14),X4),s(t_fun(t_fun(X7,X14),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_fun(t_fun(X7,t_bool),t_bool)),X16),s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_fun(t_fun(X14,t_bool),t_bool)),X15))))))<=>(p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_fun(t_fun(X7,t_bool),t_bool)),X16))))&(p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_fun(t_fun(X14,t_bool),t_bool)),X15))))&(p(s(t_bool,h4s_bools_in(s(t_fun(X7,X14),X4),s(t_fun(t_fun(X7,X14),t_bool),h4s_utilu_u_probs_funset(s(t_fun(X7,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_fun(t_fun(X7,t_bool),t_bool)),X16))),s(t_fun(X14,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_fun(t_fun(X14,t_bool),t_bool)),X15))))))))&![X17]:(p(s(t_bool,h4s_bools_in(s(t_fun(X14,t_bool),X17),s(t_fun(t_fun(X14,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_fun(t_fun(X14,t_bool),t_bool)),X15))))))=>p(s(t_bool,h4s_bools_in(s(t_fun(X7,t_bool),h4s_predu_u_sets_inter(s(t_fun(X7,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(X7,X14),X4),s(t_fun(X14,t_bool),X17))),s(t_fun(X7,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_fun(t_fun(X7,t_bool),t_bool)),X16))))),s(t_fun(t_fun(X7,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_fun(t_fun(X7,t_bool),t_bool)),X16))))))))))),file('i/f/probability/distribution__lebesgue__thm1', ah4s_measures_INu_u_MEASURABLE)).
fof(16, axiom,![X7]:![X17]:![X18]:((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X18))))&p(s(t_bool,h4s_bools_in(s(t_fun(X7,t_bool),X17),s(t_fun(t_fun(X7,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(X7,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X18)))))))=>s(t_h4s_extreals_extreal,h4s_lebesgues_integral(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X18),s(t_fun(X7,t_h4s_extreals_extreal),h4s_measures_indicatoru_u_fn(s(t_fun(X7,t_bool),X17)))))=s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X18))),s(t_fun(X7,t_bool),X17)))))),file('i/f/probability/distribution__lebesgue__thm1', ah4s_lebesgues_integralu_u_indicator)).
fof(17, axiom,![X7]:s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(X7,t_bool)),h4s_probabilitys_pu_u_space)=s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(X7,t_bool)),h4s_measures_mu_u_space),file('i/f/probability/distribution__lebesgue__thm1', ah4s_probabilitys_pu_u_spaceu_u_def)).
fof(18, axiom,![X7]:s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(X7,t_bool),t_bool)),h4s_probabilitys_events)=s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(X7,t_bool),t_bool)),h4s_measures_measurableu_u_sets),file('i/f/probability/distribution__lebesgue__thm1', ah4s_probabilitys_eventsu_u_def)).
fof(19, axiom,![X7]:s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob)=s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),file('i/f/probability/distribution__lebesgue__thm1', ah4s_probabilitys_probu_u_def)).
fof(20, axiom,![X7]:![X19]:(p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19))))<=>(p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19))))&s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19))),s(t_fun(X7,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(X7,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19)))))=s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),file('i/f/probability/distribution__lebesgue__thm1', ah4s_probabilitys_probu_u_spaceu_u_def)).
fof(21, axiom,![X7]:![X14]:![X17]:![X19]:![X20]:(p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(X7,X14),X20),s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19),s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_fun(t_fun(X14,t_bool),t_bool)),X17))))<=>(p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19))))&p(s(t_bool,h4s_bools_in(s(t_fun(X7,X14),X20),s(t_fun(t_fun(X7,X14),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_fun(t_fun(X7,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(X7,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(X7,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19))),s(t_fun(t_fun(X7,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(X7,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19))))),s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_fun(t_fun(X14,t_bool),t_bool)),X17)))))))),file('i/f/probability/distribution__lebesgue__thm1', ah4s_probabilitys_randomu_u_variableu_u_def)).
fof(22, axiom,![X7]:![X14]:![X19]:![X20]:![X6]:s(t_h4s_realaxs_real,h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X14,t_bool),t_bool),t_fun(t_fun(X14,t_bool),t_h4s_realaxs_real))),X19),s(t_fun(X14,X7),X20),s(t_fun(X7,t_bool),X6)))=s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(X14,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(X14,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X14,t_bool),t_bool),t_fun(t_fun(X14,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(X14,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X14,t_bool),t_bool),t_fun(t_fun(X14,t_bool),t_h4s_realaxs_real))),X19))),s(t_fun(X14,t_bool),h4s_predu_u_sets_inter(s(t_fun(X14,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(X14,X7),X20),s(t_fun(X7,t_bool),X6))),s(t_fun(X14,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(X14,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X14,t_bool),t_bool),t_fun(t_fun(X14,t_bool),t_h4s_realaxs_real))),t_fun(X14,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X14,t_bool),t_bool),t_fun(t_fun(X14,t_bool),t_h4s_realaxs_real))),X19))))))),file('i/f/probability/distribution__lebesgue__thm1', ah4s_probabilitys_distributionu_u_def)).
fof(23, conjecture,![X14]:![X7]:![X17]:![X19]:![X20]:![X21]:((p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(X7,X14),X20),s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19),s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_fun(t_fun(X14,t_bool),t_bool)),X17))))&p(s(t_bool,h4s_bools_in(s(t_fun(X14,t_bool),X21),s(t_fun(t_fun(X14,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(X14,t_bool),t_fun(t_fun(X14,t_bool),t_bool)),X17)))))))=>s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19),s(t_fun(X7,X14),X20),s(t_fun(X14,t_bool),X21)))))=s(t_h4s_extreals_extreal,h4s_lebesgues_integral(s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19),s(t_fun(X7,t_h4s_extreals_extreal),h4s_measures_indicatoru_u_fn(s(t_fun(X7,t_bool),h4s_predu_u_sets_inter(s(t_fun(X7,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(X7,X14),X20),s(t_fun(X14,t_bool),X21))),s(t_fun(X7,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),t_fun(X7,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(X7,t_bool),t_h4s_pairs_prod(t_fun(t_fun(X7,t_bool),t_bool),t_fun(t_fun(X7,t_bool),t_h4s_realaxs_real))),X19)))))))))),file('i/f/probability/distribution__lebesgue__thm1', ch4s_probabilitys_distributionu_u_lebesgueu_u_thm1)).
# SZS output end CNFRefutation
