%   ORIGINAL: 'h4/thm/pred_set/FINITE_BIGUNION_'
% 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: !P. 'h4/const/pred_set/FINITE' P /\ (!s. 'h4/const/bool/IN' s P ==> 'h4/const/pred_set/FINITE' s) ==> 'h4/const/pred_set/FINITE' ('h4/const/pred_set/BIGUNION' P)
%   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: !P. 'h4/const/pred_set/FINITE' P /\ (!s. 'h4/const/bool/IN' s P ==> 'h4/const/pred_set/FINITE' s) ==> 'h4/const/pred_set/FINITE' ('h4/const/pred_set/BIGUNION' P)
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_3f20696,V_3f20692]: ![F, G]: (![X]: s(V_3f20692,happ(s(fun(V_3f20696,V_3f20692),F),s(V_3f20696,X))) = s(V_3f20692,happ(s(fun(V_3f20696,V_3f20692),G),s(V_3f20696,X))) => s(fun(V_3f20696,V_3f20692),F) = s(fun(V_3f20696,V_3f20692),G))).
fof('h4/thm/pred_set/FINITE_BIGUNION_', conjecture, ![A]: ![P]: ((p(s(bool,'h4/const/pred_set/FINITE'(s(fun(fun(A,bool),bool),P)))) & ![S]: (p(s(bool,'h4/const/bool/IN'(s(fun(A,bool),S),s(fun(fun(A,bool),bool),P)))) => p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))))) => p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),'h4/const/pred_set/BIGUNION'(s(fun(fun(A,bool),bool),P)))))))).
