%   ORIGINAL: h4/complex/COMPLEX__SUB__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/complex/complex__sub0: !z w. h4/complex/complex__sub z w = h4/complex/complex__add z (h4/complex/complex__neg w)
% Assm: h4/complex/COMPLEX__SCALAR__LMUL__NEG: !z k. h4/complex/complex__neg (h4/complex/complex__scalar__lmul k z) = h4/complex/complex__scalar__lmul (h4/realax/real__neg k) z
% Assm: h4/complex/COMPLEX__NEG__SCALAR__LMUL: !z k. h4/complex/complex__scalar__lmul k (h4/complex/complex__neg z) = h4/complex/complex__scalar__lmul (h4/realax/real__neg k) z
% Assm: h4/complex/COMPLEX__ADD__SCALAR__LMUL: !z w k. h4/complex/complex__scalar__lmul k (h4/complex/complex__add z w) = h4/complex/complex__add (h4/complex/complex__scalar__lmul k z) (h4/complex/complex__scalar__lmul k w)
% Goal: !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)
%   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_complexs_complexu_u_sub0]: !z w. h4/complex/complex__sub z w = h4/complex/complex__add z (h4/complex/complex__neg w)
% Assm [h4s_complexs_COMPLEXu_u_SCALARu_u_LMULu_u_NEG]: !z k. h4/complex/complex__neg (h4/complex/complex__scalar__lmul k z) = h4/complex/complex__scalar__lmul (h4/realax/real__neg k) z
% Assm [h4s_complexs_COMPLEXu_u_NEGu_u_SCALARu_u_LMUL]: !z k. h4/complex/complex__scalar__lmul k (h4/complex/complex__neg z) = h4/complex/complex__scalar__lmul (h4/realax/real__neg k) z
% Assm [h4s_complexs_COMPLEXu_u_ADDu_u_SCALARu_u_LMUL]: !z w k. h4/complex/complex__scalar__lmul k (h4/complex/complex__add z w) = h4/complex/complex__add (h4/complex/complex__scalar__lmul k z) (h4/complex/complex__scalar__lmul k w)
% Goal: !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)
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_Q258482,TV_Q258478]: ![V_f, V_g]: (![V_x]: s(TV_Q258478,happ(s(t_fun(TV_Q258482,TV_Q258478),V_f),s(TV_Q258482,V_x))) = s(TV_Q258478,happ(s(t_fun(TV_Q258482,TV_Q258478),V_g),s(TV_Q258482,V_x))) => s(t_fun(TV_Q258482,TV_Q258478),V_f) = s(t_fun(TV_Q258482,TV_Q258478),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_complexs_complexu_u_sub0, 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_add(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_neg(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_w)))))).
fof(ah4s_complexs_COMPLEXu_u_SCALARu_u_LMULu_u_NEG, axiom, ![V_z, V_k]: s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_complexs_complexu_u_neg(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,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_k))),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z)))).
fof(ah4s_complexs_COMPLEXu_u_NEGu_u_SCALARu_u_LMUL, 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),h4s_complexs_complexu_u_neg(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,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_k))),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),V_z)))).
fof(ah4s_complexs_COMPLEXu_u_ADDu_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_add(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_add(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(ch4s_complexs_COMPLEXu_u_SUBu_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),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)))))).
