%   ORIGINAL: 'h4/thm/pred_set/FINITE_ISO_NUM_'
% 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. 'h4/const/pred_set/FINITE' s ==> (?f. (!n m. 'h4/const/prim_rec/<' n ('h4/const/pred_set/CARD' s) /\ 'h4/const/prim_rec/<' m ('h4/const/pred_set/CARD' s) ==> f n = f m ==> n = m) /\ s = 'h4/const/pred_set/GSPEC' (\n. 'h4/const/pair/,' (f n) ('h4/const/prim_rec/<' n ('h4/const/pred_set/CARD' 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: !_0. (!f s n. happ (happ (happ _0 f) s) n = 'h4/const/pair/,' (happ f n) ('h4/const/prim_rec/<' n ('h4/const/pred_set/CARD' s))) ==> (!s. 'h4/const/pred_set/FINITE' s ==> (?f. (!n m. 'h4/const/prim_rec/<' n ('h4/const/pred_set/CARD' s) /\ 'h4/const/prim_rec/<' m ('h4/const/pred_set/CARD' s) ==> happ f n = happ f m ==> n = m) /\ s = 'h4/const/pred_set/GSPEC' (happ (happ _0 f) 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_3f20321,V_3f20317]: ![F, G]: (![X]: s(V_3f20317,happ(s(fun(V_3f20321,V_3f20317),F),s(V_3f20321,X))) = s(V_3f20317,happ(s(fun(V_3f20321,V_3f20317),G),s(V_3f20321,X))) => s(fun(V_3f20321,V_3f20317),F) = s(fun(V_3f20321,V_3f20317),G))).
fof('h4/thm/pred_set/FINITE_ISO_NUM_', conjecture, ![A]: ![V__0]: (![F, S, N]: s('h4/type/pair/prod'(A,bool),happ(s(fun('h4/type/num/num','h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,bool),fun('h4/type/num/num','h4/type/pair/prod'(A,bool))),happ(s(fun(fun('h4/type/num/num',A),fun(fun(A,bool),fun('h4/type/num/num','h4/type/pair/prod'(A,bool)))),V__0),s(fun('h4/type/num/num',A),F))),s(fun(A,bool),S))),s('h4/type/num/num',N))) = s('h4/type/pair/prod'(A,bool),'h4/const/pair/,'(s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N))),s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/pred_set/CARD'(s(fun(A,bool),S))))))) => ![S]: (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) => ?[F]: (![N, M]: ((p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/pred_set/CARD'(s(fun(A,bool),S)))))) & p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',M),s('h4/type/num/num','h4/const/pred_set/CARD'(s(fun(A,bool),S))))))) => (s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N))) = s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',M))) => s('h4/type/num/num',N) = s('h4/type/num/num',M))) & s(fun(A,bool),S) = s(fun(A,bool),'h4/const/pred_set/GSPEC'(s(fun('h4/type/num/num','h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,bool),fun('h4/type/num/num','h4/type/pair/prod'(A,bool))),happ(s(fun(fun('h4/type/num/num',A),fun(fun(A,bool),fun('h4/type/num/num','h4/type/pair/prod'(A,bool)))),V__0),s(fun('h4/type/num/num',A),F))),s(fun(A,bool),S))))))))).
