%   ORIGINAL: h4/complex/COMPLEX__EXP__N2
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/real/REAL__ADD__RID: !x. h4/realax/real__add x (h4/real/real__of__num h4/num/0) = x
% Assm: h4/real/REAL__MUL__LZERO: !x. h4/realax/real__mul (h4/real/real__of__num h4/num/0) x = h4/real/real__of__num h4/num/0
% Assm: h4/real/REAL__SUB__RZERO: !x. h4/real/real__sub x (h4/real/real__of__num h4/num/0) = x
% Assm: h4/complex/RE__COMPLEX__OF__REAL: !x. h4/complex/RE (h4/complex/complex__of__real x) = x
% Assm: h4/complex/IM__COMPLEX__OF__REAL: !x. h4/complex/IM (h4/complex/complex__of__real x) = h4/real/real__of__num h4/num/0
% Assm: h4/complex/complex__of__num0: !n. h4/complex/complex__of__num n = h4/complex/complex__of__real (h4/real/real__of__num n)
% Assm: h4/complex/complex__mul0: !z w. h4/complex/complex__mul z w = h4/pair/_2C (h4/real/real__sub (h4/realax/real__mul (h4/complex/RE z) (h4/complex/RE w)) (h4/realax/real__mul (h4/complex/IM z) (h4/complex/IM w))) (h4/realax/real__add (h4/realax/real__mul (h4/complex/RE z) (h4/complex/IM w)) (h4/realax/real__mul (h4/complex/IM z) (h4/complex/RE w)))
% Assm: h4/complex/complex__scalar__lmul0: !z k. h4/complex/complex__scalar__lmul k z = h4/pair/_2C (h4/realax/real__mul k (h4/complex/RE z)) (h4/realax/real__mul k (h4/complex/IM z))
% Assm: h4/complex/COMPLEX__EXP__N: !z n. h4/complex/complex__exp (h4/complex/complex__scalar__lmul (h4/real/real__of__num n) z) = h4/complex/complex__pow (h4/complex/complex__exp z) n
% Goal: !z n. h4/complex/complex__exp (h4/complex/complex__mul (h4/complex/complex__of__num n) z) = h4/complex/complex__pow (h4/complex/complex__exp z) n
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> 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_reals_REALu_u_ADDu_u_RID]: !x. h4/realax/real__add x (h4/real/real__of__num h4/num/0) = x
% Assm [h4s_reals_REALu_u_MULu_u_LZERO]: !x. h4/realax/real__mul (h4/real/real__of__num h4/num/0) x = h4/real/real__of__num h4/num/0
% Assm [h4s_reals_REALu_u_SUBu_u_RZERO]: !x. h4/real/real__sub x (h4/real/real__of__num h4/num/0) = x
% Assm [h4s_complexs_REu_u_COMPLEXu_u_OFu_u_REAL]: !x. h4/complex/RE (h4/complex/complex__of__real x) = x
% Assm [h4s_complexs_IMu_u_COMPLEXu_u_OFu_u_REAL]: !x. h4/complex/IM (h4/complex/complex__of__real x) = h4/real/real__of__num h4/num/0
% Assm [h4s_complexs_complexu_u_ofu_u_num0]: !n. h4/complex/complex__of__num n = h4/complex/complex__of__real (h4/real/real__of__num n)
% Assm [h4s_complexs_complexu_u_mul0]: !z w. h4/complex/complex__mul z w = h4/pair/_2C (h4/real/real__sub (h4/realax/real__mul (h4/complex/RE z) (h4/complex/RE w)) (h4/realax/real__mul (h4/complex/IM z) (h4/complex/IM w))) (h4/realax/real__add (h4/realax/real__mul (h4/complex/RE z) (h4/complex/IM w)) (h4/realax/real__mul (h4/complex/IM z) (h4/complex/RE w)))
% Assm [h4s_complexs_complexu_u_scalaru_u_lmul0]: !z k. h4/complex/complex__scalar__lmul k z = h4/pair/_2C (h4/realax/real__mul k (h4/complex/RE z)) (h4/realax/real__mul k (h4/complex/IM z))
% Assm [h4s_complexs_COMPLEXu_u_EXPu_u_N]: !z n. h4/complex/complex__exp (h4/complex/complex__scalar__lmul (h4/real/real__of__num n) z) = h4/complex/complex__pow (h4/complex/complex__exp z) n
% Goal: !z n. h4/complex/complex__exp (h4/complex/complex__mul (h4/complex/complex__of__num n) z) = h4/complex/complex__pow (h4/complex/complex__exp z) n
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_Q261082,TV_Q261078]: ![V_f, V_g]: (![V_x]: s(TV_Q261078,happ(s(t_fun(TV_Q261082,TV_Q261078),V_f),s(TV_Q261082,V_x))) = s(TV_Q261078,happ(s(t_fun(TV_Q261082,TV_Q261078),V_g),s(TV_Q261082,V_x))) => s(t_fun(TV_Q261082,TV_Q261078),V_f) = s(t_fun(TV_Q261082,TV_Q261078),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_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_reals_REALu_u_ADDu_u_RID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),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)).
fof(ah4s_reals_REALu_u_MULu_u_LZERO, axiom, ![V_x]: 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,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_reals_REALu_u_SUBu_u_RZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),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)).
fof(ah4s_complexs_REu_u_COMPLEXu_u_OFu_u_REAL, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_complexs_re(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_ofu_u_real(s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_complexs_IMu_u_COMPLEXu_u_OFu_u_REAL, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_complexs_im(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_ofu_u_real(s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_complexs_complexu_u_ofu_u_num0, axiom, ![V_n]: s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_ofu_u_real(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n)))))).
fof(ah4s_complexs_complexu_u_mul0, axiom, ![V_z, V_w]: s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_mul(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_w))) = s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_complexs_re(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))),s(t_h4s_realaxs_real,h4s_complexs_re(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_w))))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_complexs_im(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))),s(t_h4s_realaxs_real,h4s_complexs_im(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_w))))))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_complexs_re(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))),s(t_h4s_realaxs_real,h4s_complexs_im(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_w))))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_complexs_im(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))),s(t_h4s_realaxs_real,h4s_complexs_re(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_w)))))))))).
fof(ah4s_complexs_complexu_u_scalaru_u_lmul0, axiom, ![V_z, V_k]: s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_scalaru_u_lmul(s(t_h4s_realaxs_real,V_k),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))) = s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_k),s(t_h4s_realaxs_real,h4s_complexs_re(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_k),s(t_h4s_realaxs_real,h4s_complexs_im(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z)))))))).
fof(ah4s_complexs_COMPLEXu_u_EXPu_u_N, axiom, ![V_z, V_n]: s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_exp(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_scalaru_u_lmul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))))) = s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_pow(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_exp(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))),s(t_h4s_nums_num,V_n)))).
fof(ch4s_complexs_COMPLEXu_u_EXPu_u_N2, conjecture, ![V_z, V_n]: s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_exp(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_mul(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))))) = s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_pow(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_exp(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z))),s(t_h4s_nums_num,V_n)))).
