%   ORIGINAL: h4/realax/TREAL__NEG__WELLDEF
% 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/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/hreal/HREAL__ADD__SYM: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal__add Y X
% Assm: h4/realax/treal__neg0: !y x. h4/realax/treal__neg (h4/pair/_2C x y) = h4/pair/_2C y x
% Assm: h4/realax/treal__eq0: !y2 y1 x2 x1. h4/realax/treal__eq (h4/pair/_2C x1 y1) (h4/pair/_2C x2 y2) <=> h4/hreal/hreal__add x1 y2 = h4/hreal/hreal__add x2 y1
% Goal: !x2 x1. h4/realax/treal__eq x1 x2 ==> h4/realax/treal__eq (h4/realax/treal__neg x1) (h4/realax/treal__neg x2)
%   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_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_hreals_HREALu_u_ADDu_u_SYM]: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal__add Y X
% Assm [h4s_realaxs_trealu_u_neg0]: !y x. h4/realax/treal__neg (h4/pair/_2C x y) = h4/pair/_2C y x
% Assm [h4s_realaxs_trealu_u_eq0]: !y2 y1 x2 x1. h4/realax/treal__eq (h4/pair/_2C x1 y1) (h4/pair/_2C x2 y2) <=> h4/hreal/hreal__add x1 y2 = h4/hreal/hreal__add x2 y1
% Goal: !x2 x1. h4/realax/treal__eq x1 x2 ==> h4/realax/treal__eq (h4/realax/treal__neg x1) (h4/realax/treal__neg x2)
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_Q329016,TV_Q329012]: ![V_f, V_g]: (![V_x]: s(TV_Q329012,happ(s(t_fun(TV_Q329016,TV_Q329012),V_f),s(TV_Q329016,V_x))) = s(TV_Q329012,happ(s(t_fun(TV_Q329016,TV_Q329012),V_g),s(TV_Q329016,V_x))) => s(t_fun(TV_Q329016,TV_Q329012),V_f) = s(t_fun(TV_Q329016,TV_Q329012),V_g))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
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_realaxs_trealu_u_neg0, axiom, ![V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_neg(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_y),s(t_h4s_hreals_hreal,V_x)))).
fof(ah4s_realaxs_trealu_u_eq0, axiom, ![V_y2, V_y1, V_x2, V_x1]: (p(s(t_bool,h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y2)))))) <=> s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y2))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y1))))).
fof(ch4s_realaxs_TREALu_u_NEGu_u_WELLDEF, conjecture, ![V_x2, V_x1]: (p(s(t_bool,h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x1),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x2)))) => p(s(t_bool,h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_neg(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_neg(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x2)))))))).
