%   ORIGINAL: 'h4/thm/arithmetic/MODEQ_TRANS_'
% 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 x y z. 'h4/const/arithmetic/MODEQ' n x y /\ 'h4/const/arithmetic/MODEQ' n y z ==> 'h4/const/arithmetic/MODEQ' n x z
%   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 x y z. 'h4/const/arithmetic/MODEQ' n x y /\ 'h4/const/arithmetic/MODEQ' n y z ==> 'h4/const/arithmetic/MODEQ' n x z
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_3f86094,V_3f86090]: ![F, G]: (![X]: s(V_3f86090,happ(s(fun(V_3f86094,V_3f86090),F),s(V_3f86094,X))) = s(V_3f86090,happ(s(fun(V_3f86094,V_3f86090),G),s(V_3f86094,X))) => s(fun(V_3f86094,V_3f86090),F) = s(fun(V_3f86094,V_3f86090),G))).
fof('h4/thm/arithmetic/MODEQ_TRANS_', conjecture, ![N, X, Y, Z]: ((p(s(bool,'h4/const/arithmetic/MODEQ'(s('h4/type/num/num',N),s('h4/type/num/num',X),s('h4/type/num/num',Y)))) & p(s(bool,'h4/const/arithmetic/MODEQ'(s('h4/type/num/num',N),s('h4/type/num/num',Y),s('h4/type/num/num',Z))))) => p(s(bool,'h4/const/arithmetic/MODEQ'(s('h4/type/num/num',N),s('h4/type/num/num',X),s('h4/type/num/num',Z)))))).
