%   ORIGINAL: h4/realax/HREAL__LT__ADD2
% 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/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/hreal/HREAL__ADD__SYM: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal__add Y X
% Assm: h4/hreal/HREAL__ADD__ASSOC: !Z Y X. h4/hreal/hreal__add X (h4/hreal/hreal__add Y Z) = h4/hreal/hreal__add (h4/hreal/hreal__add X Y) Z
% Assm: h4/hreal/HREAL__LT: !Y X. h4/hreal/hreal__lt X Y <=> (?D. Y = h4/hreal/hreal__add X D)
% Goal: !y2 y1 x2 x1. h4/hreal/hreal__lt x1 y1 /\ h4/hreal/hreal__lt x2 y2 ==> h4/hreal/hreal__lt (h4/hreal/hreal__add x1 x2) (h4/hreal/hreal__add y1 y2)
%   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_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_hreals_HREALu_u_ADDu_u_SYM]: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal__add Y X
% Assm [h4s_hreals_HREALu_u_ADDu_u_ASSOC]: !Z Y X. h4/hreal/hreal__add X (h4/hreal/hreal__add Y Z) = h4/hreal/hreal__add (h4/hreal/hreal__add X Y) Z
% Assm [h4s_hreals_HREALu_u_LT]: !Y X. h4/hreal/hreal__lt X Y <=> (?D. Y = h4/hreal/hreal__add X D)
% Goal: !y2 y1 x2 x1. h4/hreal/hreal__lt x1 y1 /\ h4/hreal/hreal__lt x2 y2 ==> h4/hreal/hreal__lt (h4/hreal/hreal__add x1 x2) (h4/hreal/hreal__add y1 y2)
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_Q328339,TV_Q328335]: ![V_f, V_g]: (![V_x]: s(TV_Q328335,happ(s(t_fun(TV_Q328339,TV_Q328335),V_f),s(TV_Q328339,V_x))) = s(TV_Q328335,happ(s(t_fun(TV_Q328339,TV_Q328335),V_g),s(TV_Q328339,V_x))) => s(t_fun(TV_Q328339,TV_Q328335),V_f) = s(t_fun(TV_Q328339,TV_Q328335),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_hreals_HREALu_u_ADDu_u_SYM, axiom, ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X)))).
fof(ah4s_hreals_HREALu_u_ADDu_u_ASSOC, axiom, ![V_Z, V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_Z))))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hreals_hreal,V_Z)))).
fof(ah4s_hreals_HREALu_u_LT, axiom, ![V_Y, V_X]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) <=> ?[V_D]: s(t_h4s_hreals_hreal,V_Y) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_D))))).
fof(ch4s_realaxs_HREALu_u_LTu_u_ADD2, conjecture, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y1)))) & p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y2))))) => p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_x2))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_y1),s(t_h4s_hreals_hreal,V_y2)))))))).
