%   ORIGINAL: 'h4/thm/pred_set/BIJ_INSERT_'
% 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: !t s f e. 'h4/const/pred_set/BIJ' f ('h4/const/pred_set/INSERT' e s) t <=> ~'h4/const/bool/IN' e s /\ 'h4/const/bool/IN' (f e) t /\ 'h4/const/pred_set/BIJ' f s ('h4/const/pred_set/DELETE' t (f e)) \/ 'h4/const/bool/IN' e s /\ 'h4/const/pred_set/BIJ' f s t
%   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: !t s f e. 'h4/const/pred_set/BIJ' f ('h4/const/pred_set/INSERT' e s) t <=> ~'h4/const/bool/IN' e s /\ 'h4/const/bool/IN' (happ f e) t /\ 'h4/const/pred_set/BIJ' f s ('h4/const/pred_set/DELETE' t (happ f e)) \/ 'h4/const/bool/IN' e s /\ 'h4/const/pred_set/BIJ' f s t
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_3f18995,V_3f18991]: ![F, G]: (![X]: s(V_3f18991,happ(s(fun(V_3f18995,V_3f18991),F),s(V_3f18995,X))) = s(V_3f18991,happ(s(fun(V_3f18995,V_3f18991),G),s(V_3f18995,X))) => s(fun(V_3f18995,V_3f18991),F) = s(fun(V_3f18995,V_3f18991),G))).
fof('h4/thm/pred_set/BIJ_INSERT_', conjecture, ![A,B]: ![T, S, F, E]: (p(s(bool,'h4/const/pred_set/BIJ'(s(fun(A,B),F),s(fun(A,bool),'h4/const/pred_set/INSERT'(s(A,E),s(fun(A,bool),S))),s(fun(B,bool),T)))) <=> ((~ (p(s(bool,'h4/const/bool/IN'(s(A,E),s(fun(A,bool),S))))) & (p(s(bool,'h4/const/bool/IN'(s(B,happ(s(fun(A,B),F),s(A,E))),s(fun(B,bool),T)))) & p(s(bool,'h4/const/pred_set/BIJ'(s(fun(A,B),F),s(fun(A,bool),S),s(fun(B,bool),'h4/const/pred_set/DELETE'(s(fun(B,bool),T),s(B,happ(s(fun(A,B),F),s(A,E)))))))))) | (p(s(bool,'h4/const/bool/IN'(s(A,E),s(fun(A,bool),S)))) & p(s(bool,'h4/const/pred_set/BIJ'(s(fun(A,B),F),s(fun(A,bool),S),s(fun(B,bool),T)))))))).
