%   ORIGINAL: 'h4/thm/pred_set/MIN_SET_UNION_'
% 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: !A B. 'h4/const/pred_set/FINITE' A /\ 'h4/const/pred_set/FINITE' B /\ ~(A = 'h4/const/pred_set/EMPTY') /\ ~(B = 'h4/const/pred_set/EMPTY') ==> 'h4/const/pred_set/MIN_SET' ('h4/const/pred_set/UNION' A B) = 'h4/const/arithmetic/MIN' ('h4/const/pred_set/MIN_SET' A) ('h4/const/pred_set/MIN_SET' B)
%   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: !A B. 'h4/const/pred_set/FINITE' A /\ 'h4/const/pred_set/FINITE' B /\ ~(A = 'h4/const/pred_set/EMPTY') /\ ~(B = 'h4/const/pred_set/EMPTY') ==> 'h4/const/pred_set/MIN_SET' ('h4/const/pred_set/UNION' A B) = 'h4/const/arithmetic/MIN' ('h4/const/pred_set/MIN_SET' A) ('h4/const/pred_set/MIN_SET' B)
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_3f23148,V_3f23144]: ![F, G]: (![X]: s(V_3f23144,happ(s(fun(V_3f23148,V_3f23144),F),s(V_3f23148,X))) = s(V_3f23144,happ(s(fun(V_3f23148,V_3f23144),G),s(V_3f23148,X))) => s(fun(V_3f23148,V_3f23144),F) = s(fun(V_3f23148,V_3f23144),G))).
fof('h4/thm/pred_set/MIN_SET_UNION_', conjecture, ![A, B]: ((p(s(bool,'h4/const/pred_set/FINITE'(s(fun('h4/type/num/num',bool),A)))) & (p(s(bool,'h4/const/pred_set/FINITE'(s(fun('h4/type/num/num',bool),B)))) & (~ (s(fun('h4/type/num/num',bool),A) = s(fun('h4/type/num/num',bool),'h4/const/pred_set/EMPTY')) & ~ (s(fun('h4/type/num/num',bool),B) = s(fun('h4/type/num/num',bool),'h4/const/pred_set/EMPTY'))))) => s('h4/type/num/num','h4/const/pred_set/MIN_SET'(s(fun('h4/type/num/num',bool),'h4/const/pred_set/UNION'(s(fun('h4/type/num/num',bool),A),s(fun('h4/type/num/num',bool),B))))) = s('h4/type/num/num','h4/const/arithmetic/MIN'(s('h4/type/num/num','h4/const/pred_set/MIN_SET'(s(fun('h4/type/num/num',bool),A))),s('h4/type/num/num','h4/const/pred_set/MIN_SET'(s(fun('h4/type/num/num',bool),B))))))).
