%   ORIGINAL: 'h4/thm/nets/DORDER_LEMMA_'
% 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: !g. 'h4/const/nets/dorder' g ==> (!P Q. (?n. g n n /\ (!m. g m n ==> P m)) /\ (?n. g n n /\ (!m. g m n ==> Q m)) ==> (?n. g n n /\ (!m. g m n ==> P m /\ Q m)))
%   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: !g. 'h4/const/nets/dorder' g ==> (!P Q. (?n. happ (happ g n) n /\ (!m. happ (happ g m) n ==> happ P m)) /\ (?n. happ (happ g n) n /\ (!m. happ (happ g m) n ==> happ Q m)) ==> (?n. happ (happ g n) n /\ (!m. happ (happ g m) n ==> happ P m /\ happ Q m)))
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_3f63736,V_3f63732]: ![F, G]: (![X]: s(V_3f63732,happ(s(fun(V_3f63736,V_3f63732),F),s(V_3f63736,X))) = s(V_3f63732,happ(s(fun(V_3f63736,V_3f63732),G),s(V_3f63736,X))) => s(fun(V_3f63736,V_3f63732),F) = s(fun(V_3f63736,V_3f63732),G))).
fof('h4/thm/nets/DORDER_LEMMA_', conjecture, ![A]: ![G]: (p(s(bool,'h4/const/nets/dorder'(s(fun(A,fun(A,bool)),G)))) => ![P, Q]: ((?[N]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),G),s(A,N))),s(A,N)))) & ![M]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),G),s(A,M))),s(A,N)))) => p(s(bool,happ(s(fun(A,bool),P),s(A,M)))))) & ?[N]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),G),s(A,N))),s(A,N)))) & ![M]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),G),s(A,M))),s(A,N)))) => p(s(bool,happ(s(fun(A,bool),Q),s(A,M))))))) => ?[N]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),G),s(A,N))),s(A,N)))) & ![M]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),G),s(A,M))),s(A,N)))) => (p(s(bool,happ(s(fun(A,bool),P),s(A,M)))) & p(s(bool,happ(s(fun(A,bool),Q),s(A,M)))))))))).
