%   ORIGINAL: 'h4/thm/pred_set/POW_EQNS_e1'
% 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
% Goal: !e s. 'h4/const/pred_set/POW' ('h4/const/pred_set/INSERT' e s) = 'h4/const/bool/LET' (\ps. 'h4/const/pred_set/UNION' ('h4/const/pred_set/IMAGE' ('h4/const/pred_set/INSERT' e) ps) ps) ('h4/const/pred_set/POW' s)
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_0. (!e ps. happ (happ _0 e) ps = 'h4/const/pred_set/UNION' ('h4/const/pred_set/IMAGE' ('h4/const/pred_set/INSERT' e) ps) ps) ==> (!e s. 'h4/const/pred_set/POW' (happ ('h4/const/pred_set/INSERT' e) s) = 'h4/const/bool/LET' (happ _0 e) ('h4/const/pred_set/POW' s))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f23398,V_3f23394]: ![F, G]: (![X]: s(V_3f23394,happ(s(fun(V_3f23398,V_3f23394),F),s(V_3f23398,X))) = s(V_3f23394,happ(s(fun(V_3f23398,V_3f23394),G),s(V_3f23398,X))) => s(fun(V_3f23398,V_3f23394),F) = s(fun(V_3f23398,V_3f23394),G))).
fof('h4/thm/pred_set/POW_EQNS_e1', conjecture, ![A]: ![V__0]: (![E, Ps]: s(fun(fun(A,bool),bool),happ(s(fun(fun(fun(A,bool),bool),fun(fun(A,bool),bool)),happ(s(fun(A,fun(fun(fun(A,bool),bool),fun(fun(A,bool),bool))),V__0),s(A,E))),s(fun(fun(A,bool),bool),Ps))) = s(fun(fun(A,bool),bool),'h4/const/pred_set/UNION'(s(fun(fun(A,bool),bool),'h4/const/pred_set/IMAGE'(s(fun(fun(A,bool),fun(A,bool)),'h4/const/pred_set/INSERT'(s(A,E))),s(fun(fun(A,bool),bool),Ps))),s(fun(fun(A,bool),bool),Ps))) => ![E, S]: s(fun(fun(A,bool),bool),'h4/const/pred_set/POW'(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),'h4/const/pred_set/INSERT'(s(A,E))),s(fun(A,bool),S))))) = s(fun(fun(A,bool),bool),'h4/const/bool/LET'(s(fun(fun(fun(A,bool),bool),fun(fun(A,bool),bool)),happ(s(fun(A,fun(fun(fun(A,bool),bool),fun(fun(A,bool),bool))),V__0),s(A,E))),s(fun(fun(A,bool),bool),'h4/const/pred_set/POW'(s(fun(A,bool),S))))))).
