%   ORIGINAL: h4/pred__set/CARD__INSERT
% 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/pred__set/CARD__DEF_c1: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
% Goal: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
%   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_predu_u_sets_CARDu_u_DEFu_c1]: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
% Goal: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q151533,TV_Q151529]: ![V_f, V_g]: (![V_x]: s(TV_Q151529,happ(s(t_fun(TV_Q151533,TV_Q151529),V_f),s(TV_Q151533,V_x))) = s(TV_Q151529,happ(s(t_fun(TV_Q151533,TV_Q151529),V_g),s(TV_Q151533,V_x))) => s(t_fun(TV_Q151533,TV_Q151529),V_f) = s(t_fun(TV_Q151533,TV_Q151529),V_g))).
fof(ah4s_predu_u_sets_CARDu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_x]: s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ch4s_predu_u_sets_CARDu_u_INSERT, conjecture, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_x]: s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
