%   ORIGINAL: h4/transc/EXP__LN
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/transc/EXP__POS__LT: !x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/transc/exp x)
% Assm: h4/transc/EXP__INJ: !y x. h4/transc/exp x = h4/transc/exp y <=> x = y
% Assm: h4/transc/EXP__TOTAL: !y. h4/realax/real__lt (h4/real/real__of__num h4/num/0) y ==> (?x. h4/transc/exp x = y)
% Assm: h4/transc/LN__EXP: !x. h4/transc/ln (h4/transc/exp x) = x
% Goal: !x. h4/transc/exp (h4/transc/ln x) = x <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) x
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_transcs_EXPu_u_POSu_u_LT]: !x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/transc/exp x)
% Assm [h4s_transcs_EXPu_u_INJ]: !y x. h4/transc/exp x = h4/transc/exp y <=> x = y
% Assm [h4s_transcs_EXPu_u_TOTAL]: !y. h4/realax/real__lt (h4/real/real__of__num h4/num/0) y ==> (?x. h4/transc/exp x = y)
% Assm [h4s_transcs_LNu_u_EXP]: !x. h4/transc/ln (h4/transc/exp x) = x
% Goal: !x. h4/transc/exp (h4/transc/ln x) = x <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) x
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_Q192560,TV_Q192556]: ![V_f, V_g]: (![V_x]: s(TV_Q192556,happ(s(t_fun(TV_Q192560,TV_Q192556),V_f),s(TV_Q192560,V_x))) = s(TV_Q192556,happ(s(t_fun(TV_Q192560,TV_Q192556),V_g),s(TV_Q192560,V_x))) => s(t_fun(TV_Q192560,TV_Q192556),V_f) = s(t_fun(TV_Q192560,TV_Q192556),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
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_transcs_EXPu_u_POSu_u_LT, axiom, ![V_x]: p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_transcs_exp(s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_transcs_EXPu_u_INJ, axiom, ![V_y, V_x]: (s(t_h4s_realaxs_real,h4s_transcs_exp(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_transcs_exp(s(t_h4s_realaxs_real,V_y))) <=> s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y))).
fof(ah4s_transcs_EXPu_u_TOTAL, axiom, ![V_y]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_y)))) => ?[V_x]: s(t_h4s_realaxs_real,h4s_transcs_exp(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,V_y))).
fof(ah4s_transcs_LNu_u_EXP, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_transcs_ln(s(t_h4s_realaxs_real,h4s_transcs_exp(s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,V_x)).
fof(ch4s_transcs_EXPu_u_LN, conjecture, ![V_x]: (s(t_h4s_realaxs_real,h4s_transcs_exp(s(t_h4s_realaxs_real,h4s_transcs_ln(s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,V_x) <=> p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x)))))).
