%   ORIGINAL: h4/hreal/HRAT__RDISTRIB
% 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/hrat/HRAT__MUL__SYM: !i h. h4/hrat/hrat__mul h i = h4/hrat/hrat__mul i h
% Assm: h4/hrat/HRAT__LDISTRIB: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__mul h i) (h4/hrat/hrat__mul h j)
% Goal: !z y x. h4/hrat/hrat__mul (h4/hrat/hrat__add x y) z = h4/hrat/hrat__add (h4/hrat/hrat__mul x z) (h4/hrat/hrat__mul y z)
%   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_hrats_HRATu_u_MULu_u_SYM]: !i h. h4/hrat/hrat__mul h i = h4/hrat/hrat__mul i h
% Assm [h4s_hrats_HRATu_u_LDISTRIB]: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__mul h i) (h4/hrat/hrat__mul h j)
% Goal: !z y x. h4/hrat/hrat__mul (h4/hrat/hrat__add x y) z = h4/hrat/hrat__add (h4/hrat/hrat__mul x z) (h4/hrat/hrat__mul y z)
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_Q253756,TV_Q253752]: ![V_f, V_g]: (![V_x]: s(TV_Q253752,happ(s(t_fun(TV_Q253756,TV_Q253752),V_f),s(TV_Q253756,V_x))) = s(TV_Q253752,happ(s(t_fun(TV_Q253756,TV_Q253752),V_g),s(TV_Q253756,V_x))) => s(t_fun(TV_Q253756,TV_Q253752),V_f) = s(t_fun(TV_Q253756,TV_Q253752),V_g))).
fof(ah4s_hrats_HRATu_u_MULu_u_SYM, axiom, ![V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_h)))).
fof(ah4s_hrats_HRATu_u_LDISTRIB, axiom, ![V_j, V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_j))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_j)))))).
fof(ch4s_hreals_HRATu_u_RDISTRIB, conjecture, ![V_z, V_y, V_x]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))),s(t_h4s_hrats_hrat,V_z))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_z)))))).
