%   ORIGINAL: h4/integral/INTEGRABLE__DINT
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/integral/integrable__def: !f b a. h4/integral/integrable (h4/pair/_2C a b) f <=> (?i. h4/transc/Dint (h4/pair/_2C a b) f i)
% Assm: h4/integral/integral__def: !f b a. h4/integral/integral (h4/pair/_2C a b) f = h4/min/_40 (\i. h4/transc/Dint (h4/pair/_2C a b) f i)
% Goal: !f b a. h4/integral/integrable (h4/pair/_2C a b) f ==> h4/transc/Dint (h4/pair/_2C a b) f (h4/integral/integral (h4/pair/_2C a b) f)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_integrals_integrableu_u_def]: !f b a. h4/integral/integrable (h4/pair/_2C a b) f <=> (?i. h4/transc/Dint (h4/pair/_2C a b) f i)
% Assm [h4s_integrals_integralu_u_def]: !_0. (!a b f i. happ (happ (happ (happ _0 a) b) f) i <=> h4/transc/Dint (h4/pair/_2C a b) f i) ==> (!f b a. h4/integral/integral (h4/pair/_2C a b) f = h4/min/_40 (happ (happ (happ _0 a) b) f))
% Goal: !f b a. h4/integral/integrable (h4/pair/_2C a b) f ==> h4/transc/Dint (h4/pair/_2C a b) f (h4/integral/integral (h4/pair/_2C a b) f)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q271583,TV_Q271579]: ![V_f, V_g]: (![V_x]: s(TV_Q271579,happ(s(t_fun(TV_Q271583,TV_Q271579),V_f),s(TV_Q271583,V_x))) = s(TV_Q271579,happ(s(t_fun(TV_Q271583,TV_Q271579),V_g),s(TV_Q271583,V_x))) => s(t_fun(TV_Q271583,TV_Q271579),V_f) = s(t_fun(TV_Q271583,TV_Q271579),V_g))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_integrals_integrableu_u_def, axiom, ![V_f, V_b, V_a]: (p(s(t_bool,h4s_integrals_integrable(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f)))) <=> ?[V_i]: p(s(t_bool,h4s_transcs_dint(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_i)))))).
fof(ah4s_integrals_integralu_u_def, axiom, ![V_uu_0]: (![V_a, V_b, V_f, V_i]: s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_bool))),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_bool)))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_i))) = s(t_bool,h4s_transcs_dint(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_i))) => ![V_f, V_b, V_a]: s(t_h4s_realaxs_real,h4s_integrals_integral(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))) = s(t_h4s_realaxs_real,h4s_mins_u_40(s(t_fun(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_bool))),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_bool)))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))))))).
fof(ch4s_integrals_INTEGRABLEu_u_DINT, conjecture, ![V_f, V_b, V_a]: (p(s(t_bool,h4s_integrals_integrable(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f)))) => p(s(t_bool,h4s_transcs_dint(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,h4s_integrals_integral(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f)))))))).
