%   ORIGINAL: 'h4/thm/arithmetic/MODEQ_INTRO_CONG_'
% 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 e1 e0. 'h4/const/prim_rec/<' 'h4/const/num/0' n ==> 'h4/const/arithmetic/MODEQ' n e0 e1 ==> 'h4/const/arithmetic/MOD' e0 n = 'h4/const/arithmetic/MOD' e1 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 e1 e0. 'h4/const/prim_rec/<' 'h4/const/num/0' n ==> 'h4/const/arithmetic/MODEQ' n e0 e1 ==> 'h4/const/arithmetic/MOD' e0 n = 'h4/const/arithmetic/MOD' e1 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_3f85969,V_3f85965]: ![F, G]: (![X]: s(V_3f85965,happ(s(fun(V_3f85969,V_3f85965),F),s(V_3f85969,X))) = s(V_3f85965,happ(s(fun(V_3f85969,V_3f85965),G),s(V_3f85969,X))) => s(fun(V_3f85969,V_3f85965),F) = s(fun(V_3f85969,V_3f85965),G))).
fof('h4/thm/arithmetic/MODEQ_INTRO_CONG_', conjecture, ![N, E1, E0]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num',N)))) => (p(s(bool,'h4/const/arithmetic/MODEQ'(s('h4/type/num/num',N),s('h4/type/num/num',E0),s('h4/type/num/num',E1)))) => s('h4/type/num/num','h4/const/arithmetic/MOD'(s('h4/type/num/num',E0),s('h4/type/num/num',N))) = s('h4/type/num/num','h4/const/arithmetic/MOD'(s('h4/type/num/num',E1),s('h4/type/num/num',N)))))).
