%   ORIGINAL: h4/arithmetic/ODD__MULT
% 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/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/arithmetic/ODD__EVEN: !n. h4/arithmetic/ODD n <=> ~h4/arithmetic/EVEN n
% Assm: h4/arithmetic/EVEN__MULT: !n m. h4/arithmetic/EVEN (h4/arithmetic/_2A m n) <=> h4/arithmetic/EVEN m \/ h4/arithmetic/EVEN n
% Goal: !n m. h4/arithmetic/ODD (h4/arithmetic/_2A m n) <=> h4/arithmetic/ODD m /\ h4/arithmetic/ODD n
%   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_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_arithmetics_ODDu_u_EVEN]: !n. h4/arithmetic/ODD n <=> ~h4/arithmetic/EVEN n
% Assm [h4s_arithmetics_EVENu_u_MULT]: !n m. h4/arithmetic/EVEN (h4/arithmetic/_2A m n) <=> h4/arithmetic/EVEN m \/ h4/arithmetic/EVEN n
% Goal: !n m. h4/arithmetic/ODD (h4/arithmetic/_2A m n) <=> h4/arithmetic/ODD m /\ h4/arithmetic/ODD n
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_Q308498,TV_Q308494]: ![V_f, V_g]: (![V_x]: s(TV_Q308494,happ(s(t_fun(TV_Q308498,TV_Q308494),V_f),s(TV_Q308498,V_x))) = s(TV_Q308494,happ(s(t_fun(TV_Q308498,TV_Q308494),V_g),s(TV_Q308498,V_x))) => s(t_fun(TV_Q308498,TV_Q308494),V_f) = s(t_fun(TV_Q308498,TV_Q308494),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_arithmetics_ODDu_u_EVEN, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))) <=> ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_EVENu_u_MULT, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))) <=> (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_m)))) | p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))))).
fof(ch4s_arithmetics_ODDu_u_MULT, conjecture, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))) <=> (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_m)))) & p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n))))))).
