%   ORIGINAL: 'h4/thm/pred_set/CROSS_SUBSET_'
% 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: !P Q P0 Q0. 'h4/const/pred_set/SUBSET' ('h4/const/pred_set/CROSS' P0 Q0) ('h4/const/pred_set/CROSS' P Q) <=> P0 = 'h4/const/pred_set/EMPTY' \/ Q0 = 'h4/const/pred_set/EMPTY' \/ 'h4/const/pred_set/SUBSET' P0 P /\ 'h4/const/pred_set/SUBSET' Q0 Q
%   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: !P Q P0 Q0. 'h4/const/pred_set/SUBSET' ('h4/const/pred_set/CROSS' P0 Q0) ('h4/const/pred_set/CROSS' P Q) <=> P0 = 'h4/const/pred_set/EMPTY' \/ Q0 = 'h4/const/pred_set/EMPTY' \/ 'h4/const/pred_set/SUBSET' P0 P /\ 'h4/const/pred_set/SUBSET' Q0 Q
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_3f21296,V_3f21292]: ![F, G]: (![X]: s(V_3f21292,happ(s(fun(V_3f21296,V_3f21292),F),s(V_3f21296,X))) = s(V_3f21292,happ(s(fun(V_3f21296,V_3f21292),G),s(V_3f21296,X))) => s(fun(V_3f21296,V_3f21292),F) = s(fun(V_3f21296,V_3f21292),G))).
fof('h4/thm/pred_set/CROSS_SUBSET_', conjecture, ![A,B]: ![P, Q, P0, Q0]: (p(s(bool,'h4/const/pred_set/SUBSET'(s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/pred_set/CROSS'(s(fun(A,bool),P0),s(fun(B,bool),Q0))),s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/pred_set/CROSS'(s(fun(A,bool),P),s(fun(B,bool),Q)))))) <=> (s(fun(A,bool),P0) = s(fun(A,bool),'h4/const/pred_set/EMPTY') | (s(fun(B,bool),Q0) = s(fun(B,bool),'h4/const/pred_set/EMPTY') | (p(s(bool,'h4/const/pred_set/SUBSET'(s(fun(A,bool),P0),s(fun(A,bool),P)))) & p(s(bool,'h4/const/pred_set/SUBSET'(s(fun(B,bool),Q0),s(fun(B,bool),Q))))))))).
