%   ORIGINAL: 'h4/thm/pred_set/partition_elements_interrelate_'
% 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: !s R. 'h4/const/pred_set/equiv_on' R s ==> (!t. 'h4/const/bool/IN' t ('h4/const/pred_set/partition' R s) ==> (!x y. 'h4/const/bool/IN' x t /\ 'h4/const/bool/IN' y t ==> R x y))
%   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: !s R. 'h4/const/pred_set/equiv_on' R s ==> (!t. 'h4/const/bool/IN' t ('h4/const/pred_set/partition' R s) ==> (!x y. 'h4/const/bool/IN' x t /\ 'h4/const/bool/IN' y t ==> happ (happ R x) y))
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_3f23777,V_3f23773]: ![F, G]: (![X]: s(V_3f23773,happ(s(fun(V_3f23777,V_3f23773),F),s(V_3f23777,X))) = s(V_3f23773,happ(s(fun(V_3f23777,V_3f23773),G),s(V_3f23777,X))) => s(fun(V_3f23777,V_3f23773),F) = s(fun(V_3f23777,V_3f23773),G))).
fof('h4/thm/pred_set/partition_elements_interrelate_', conjecture, ![A]: ![S, R]: (p(s(bool,'h4/const/pred_set/equiv_on'(s(fun(A,fun(A,bool)),R),s(fun(A,bool),S)))) => ![T]: (p(s(bool,'h4/const/bool/IN'(s(fun(A,bool),T),s(fun(fun(A,bool),bool),'h4/const/pred_set/partition'(s(fun(A,fun(A,bool)),R),s(fun(A,bool),S)))))) => ![X, Y]: ((p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),T)))) & p(s(bool,'h4/const/bool/IN'(s(A,Y),s(fun(A,bool),T))))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))),s(A,Y)))))))).
