%   ORIGINAL: 'h4/thm/pred_set/partition_CARD_'
% 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: !R s. 'h4/const/pred_set/equiv_on' R s /\ 'h4/const/pred_set/FINITE' s ==> 'h4/const/pred_set/CARD' s = 'h4/const/pred_set/SUM_IMAGE' 'h4/const/pred_set/CARD' ('h4/const/pred_set/partition' R 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: !R s. 'h4/const/pred_set/equiv_on' R s /\ 'h4/const/pred_set/FINITE' s ==> happ 'h4/const/pred_set/CARD' s = 'h4/const/pred_set/SUM_IMAGE' 'h4/const/pred_set/CARD' ('h4/const/pred_set/partition' R 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_3f23852,V_3f23848]: ![F, G]: (![X]: s(V_3f23848,happ(s(fun(V_3f23852,V_3f23848),F),s(V_3f23852,X))) = s(V_3f23848,happ(s(fun(V_3f23852,V_3f23848),G),s(V_3f23852,X))) => s(fun(V_3f23852,V_3f23848),F) = s(fun(V_3f23852,V_3f23848),G))).
fof('h4/thm/pred_set/partition_CARD_', conjecture, ![A]: ![R, S]: ((p(s(bool,'h4/const/pred_set/equiv_on'(s(fun(A,fun(A,bool)),R),s(fun(A,bool),S)))) & p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S))))) => s('h4/type/num/num',happ(s(fun(fun(A,bool),'h4/type/num/num'),'h4/const/pred_set/CARD'),s(fun(A,bool),S))) = s('h4/type/num/num','h4/const/pred_set/SUM_IMAGE'(s(fun(fun(A,bool),'h4/type/num/num'),'h4/const/pred_set/CARD'),s(fun(fun(A,bool),bool),'h4/const/pred_set/partition'(s(fun(A,fun(A,bool)),R),s(fun(A,bool),S))))))).
