%   ORIGINAL: h4/transc/LN__POW
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/transc/EXP__N: !x n. h4/transc/exp (h4/realax/real__mul (h4/real/real__of__num n) x) = h4/real/pow (h4/transc/exp x) n
% 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 n. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x ==> h4/transc/ln (h4/real/pow x n) = h4/realax/real__mul (h4/real/real__of__num n) (h4/transc/ln 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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_transcs_EXPu_u_N]: !x n. h4/transc/exp (h4/realax/real__mul (h4/real/real__of__num n) x) = h4/real/pow (h4/transc/exp x) n
% 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 n. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x ==> h4/transc/ln (h4/real/pow x n) = h4/realax/real__mul (h4/real/real__of__num n) (h4/transc/ln 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_Q192760,TV_Q192756]: ![V_f, V_g]: (![V_x]: s(TV_Q192756,happ(s(t_fun(TV_Q192760,TV_Q192756),V_f),s(TV_Q192760,V_x))) = s(TV_Q192756,happ(s(t_fun(TV_Q192760,TV_Q192756),V_g),s(TV_Q192760,V_x))) => s(t_fun(TV_Q192760,TV_Q192756),V_f) = s(t_fun(TV_Q192760,TV_Q192756),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_transcs_EXPu_u_N, axiom, ![V_x, V_n]: s(t_h4s_realaxs_real,h4s_transcs_exp(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,h4s_transcs_exp(s(t_h4s_realaxs_real,V_x))),s(t_h4s_nums_num,V_n)))).
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_LNu_u_POW, conjecture, ![V_x, V_n]: (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)))) => s(t_h4s_realaxs_real,h4s_transcs_ln(s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,V_n))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_transcs_ln(s(t_h4s_realaxs_real,V_x))))))).
