%   ORIGINAL: h4/poly/POLY__CMUL__CLAUSES_c1
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/poly/poly__cmul__def_c0: !c. h4/poly/_23_23 c h4/list/NIL = h4/list/NIL
% Assm: h4/poly/poly__cmul__def_c1: !t h c. h4/poly/_23_23 c (h4/list/CONS h t) = h4/list/CONS (h4/realax/real__mul c h) (h4/poly/_23_23 c t)
% Goal: !t h c. h4/poly/_23_23 c (h4/list/CONS h t) = h4/list/CONS (h4/realax/real__mul c h) (h4/poly/_23_23 c t)
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_polys_polyu_u_cmulu_u_defu_c0]: !c. h4/poly/_23_23 c h4/list/NIL = h4/list/NIL
% Assm [h4s_polys_polyu_u_cmulu_u_defu_c1]: !t h c. h4/poly/_23_23 c (h4/list/CONS h t) = h4/list/CONS (h4/realax/real__mul c h) (h4/poly/_23_23 c t)
% Goal: !t h c. h4/poly/_23_23 c (h4/list/CONS h t) = h4/list/CONS (h4/realax/real__mul c h) (h4/poly/_23_23 c t)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q213811,TV_Q213807]: ![V_f, V_g]: (![V_x]: s(TV_Q213807,happ(s(t_fun(TV_Q213811,TV_Q213807),V_f),s(TV_Q213811,V_x))) = s(TV_Q213807,happ(s(t_fun(TV_Q213811,TV_Q213807),V_g),s(TV_Q213811,V_x))) => s(t_fun(TV_Q213811,TV_Q213807),V_f) = s(t_fun(TV_Q213811,TV_Q213807),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_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,t0)))).
fof(ah4s_polys_polyu_u_cmulu_u_defu_c0, axiom, ![V_c]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil)).
fof(ah4s_polys_polyu_u_cmulu_u_defu_c1, axiom, ![V_t, V_h, V_c]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_h),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,V_h))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t)))))).
fof(ch4s_polys_POLYu_u_CMULu_u_CLAUSESu_c1, conjecture, ![V_t, V_h, V_c]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_h),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,V_h))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t)))))).
