%   ORIGINAL: h4/complex/COMPLEX__EQ__SCALAR__LMUL
% 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/complex/COMPLEX__SUB__0: !z w. h4/complex/complex__sub z w = h4/complex/complex__of__num h4/num/0 <=> z = w
% Assm: h4/complex/COMPLEX__SUB__SCALAR__LMUL: !z w k. h4/complex/complex__scalar__lmul k (h4/complex/complex__sub z w) = h4/complex/complex__sub (h4/complex/complex__scalar__lmul k z) (h4/complex/complex__scalar__lmul k w)
% Assm: h4/complex/COMPLEX__SCALAR__LMUL__ENTIRE: !z k. h4/complex/complex__scalar__lmul k z = h4/complex/complex__of__num h4/num/0 <=> k = h4/real/real__of__num h4/num/0 \/ z = h4/complex/complex__of__num h4/num/0
% Goal: !z w k. h4/complex/complex__scalar__lmul k z = h4/complex/complex__scalar__lmul k w <=> k = h4/real/real__of__num h4/num/0 \/ z = w
%   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_complexs_COMPLEXu_u_SUBu_u_0]: !z w. h4/complex/complex__sub z w = h4/complex/complex__of__num h4/num/0 <=> z = w
% Assm [h4s_complexs_COMPLEXu_u_SUBu_u_SCALARu_u_LMUL]: !z w k. h4/complex/complex__scalar__lmul k (h4/complex/complex__sub z w) = h4/complex/complex__sub (h4/complex/complex__scalar__lmul k z) (h4/complex/complex__scalar__lmul k w)
% Assm [h4s_complexs_COMPLEXu_u_SCALARu_u_LMULu_u_ENTIRE]: !z k. h4/complex/complex__scalar__lmul k z = h4/complex/complex__of__num h4/num/0 <=> k = h4/real/real__of__num h4/num/0 \/ z = h4/complex/complex__of__num h4/num/0
% Goal: !z w k. h4/complex/complex__scalar__lmul k z = h4/complex/complex__scalar__lmul k w <=> k = h4/real/real__of__num h4/num/0 \/ z = w
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_Q258732,TV_Q258728]: ![V_f, V_g]: (![V_x]: s(TV_Q258728,happ(s(t_fun(TV_Q258732,TV_Q258728),V_f),s(TV_Q258732,V_x))) = s(TV_Q258728,happ(s(t_fun(TV_Q258732,TV_Q258728),V_g),s(TV_Q258732,V_x))) => s(t_fun(TV_Q258732,TV_Q258728),V_f) = s(t_fun(TV_Q258732,TV_Q258728),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_complexs_COMPLEXu_u_SUBu_u_0, axiom, ![V_z, V_w]: (s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_sub(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_complexs_complexu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) <=> 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))).
fof(ah4s_complexs_COMPLEXu_u_SUBu_u_SCALARu_u_LMUL, axiom, ![V_z, V_w, 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),h4s_complexs_complexu_u_sub(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_complexs_complexu_u_sub(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_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_w)))))).
fof(ah4s_complexs_COMPLEXu_u_SCALARu_u_LMULu_u_ENTIRE, 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_complexs_complexu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) <=> (s(t_h4s_realaxs_real,V_k) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) | 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_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ch4s_complexs_COMPLEXu_u_EQu_u_SCALARu_u_LMUL, conjecture, ![V_z, V_w, 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_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_w))) <=> (s(t_h4s_realaxs_real,V_k) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) | 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)))).
