%   ORIGINAL: 'h4/thm/arithmetic/MODEQ_NUMERAL_2'
% 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: !n m. 'h4/const/arithmetic/<=' ('h4/const/arithmetic/NUMERAL' n) ('h4/const/arithmetic/NUMERAL' m) ==> 'h4/const/arithmetic/MODEQ' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' n)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' m)) ('h4/const/arithmetic/MOD' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' m)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' n)))
%   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: !n m. 'h4/const/arithmetic/<=' ('h4/const/arithmetic/NUMERAL' n) ('h4/const/arithmetic/NUMERAL' m) ==> 'h4/const/arithmetic/MODEQ' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' n)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' m)) ('h4/const/arithmetic/MOD' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' m)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' n)))
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_3f86169,V_3f86165]: ![F, G]: (![X]: s(V_3f86165,happ(s(fun(V_3f86169,V_3f86165),F),s(V_3f86169,X))) = s(V_3f86165,happ(s(fun(V_3f86169,V_3f86165),G),s(V_3f86169,X))) => s(fun(V_3f86169,V_3f86165),F) = s(fun(V_3f86169,V_3f86165),G))).
fof('h4/thm/arithmetic/MODEQ_NUMERAL_2', conjecture, ![N, M]: (p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num',N))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num',M)))))) => p(s(bool,'h4/const/arithmetic/MODEQ'(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',N))))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',M))))),s('h4/type/num/num','h4/const/arithmetic/MOD'(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',M))))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',N)))))))))))).
