%   ORIGINAL: 'h4/thm/quotient_sum/SUM_REL_ind_'
% 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. (!R1 R2 a1 a2. P R1 R2 ('h4/const/sum/INL' a1) ('h4/const/sum/INL' a2)) /\ (!R1 R2 b1 b2. P R1 R2 ('h4/const/sum/INR' b1) ('h4/const/sum/INR' b2)) /\ (!R1 R2 a1 b2. P R1 R2 ('h4/const/sum/INL' a1) ('h4/const/sum/INR' b2)) /\ (!R1 R2 b1 a2. P R1 R2 ('h4/const/sum/INR' b1) ('h4/const/sum/INL' a2)) ==> (!v v1 v2 v3. P v v1 v2 v3)
%   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. (!R1 R2 a1 a2. happ (happ (happ (happ P R1) R2) ('h4/const/sum/INL' a1)) ('h4/const/sum/INL' a2)) /\ (!R1 R2 b1 b2. happ (happ (happ (happ P R1) R2) ('h4/const/sum/INR' b1)) ('h4/const/sum/INR' b2)) /\ (!R1 R2 a1 b2. happ (happ (happ (happ P R1) R2) ('h4/const/sum/INL' a1)) ('h4/const/sum/INR' b2)) /\ (!R1 R2 b1 a2. happ (happ (happ (happ P R1) R2) ('h4/const/sum/INR' b1)) ('h4/const/sum/INL' a2)) ==> (!v v1 v2 v3. happ (happ (happ (happ P v) v1) v2) v3)
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_3f14169,V_3f14165]: ![F, G]: (![X]: s(V_3f14165,happ(s(fun(V_3f14169,V_3f14165),F),s(V_3f14169,X))) = s(V_3f14165,happ(s(fun(V_3f14169,V_3f14165),G),s(V_3f14169,X))) => s(fun(V_3f14169,V_3f14165),F) = s(fun(V_3f14169,V_3f14165),G))).
fof('h4/thm/quotient_sum/SUM_REL_ind_', conjecture, ![A,B]: ![P]: ((![R1, R2, A1, A2]: p(s(bool,happ(s(fun('h4/type/sum/sum'(A,B),bool),happ(s(fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)),happ(s(fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool))),happ(s(fun(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)))),P),s(fun(A,fun(A,bool)),R1))),s(fun(B,fun(B,bool)),R2))),s('h4/type/sum/sum'(A,B),'h4/const/sum/INL'(s(A,A1))))),s('h4/type/sum/sum'(A,B),'h4/const/sum/INL'(s(A,A2)))))) & (![R1, R2, B1, B2]: p(s(bool,happ(s(fun('h4/type/sum/sum'(A,B),bool),happ(s(fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)),happ(s(fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool))),happ(s(fun(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)))),P),s(fun(A,fun(A,bool)),R1))),s(fun(B,fun(B,bool)),R2))),s('h4/type/sum/sum'(A,B),'h4/const/sum/INR'(s(B,B1))))),s('h4/type/sum/sum'(A,B),'h4/const/sum/INR'(s(B,B2)))))) & (![R1, R2, A1, B2]: p(s(bool,happ(s(fun('h4/type/sum/sum'(A,B),bool),happ(s(fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)),happ(s(fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool))),happ(s(fun(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)))),P),s(fun(A,fun(A,bool)),R1))),s(fun(B,fun(B,bool)),R2))),s('h4/type/sum/sum'(A,B),'h4/const/sum/INL'(s(A,A1))))),s('h4/type/sum/sum'(A,B),'h4/const/sum/INR'(s(B,B2)))))) & ![R1, R2, B1, A2]: p(s(bool,happ(s(fun('h4/type/sum/sum'(A,B),bool),happ(s(fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)),happ(s(fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool))),happ(s(fun(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)))),P),s(fun(A,fun(A,bool)),R1))),s(fun(B,fun(B,bool)),R2))),s('h4/type/sum/sum'(A,B),'h4/const/sum/INR'(s(B,B1))))),s('h4/type/sum/sum'(A,B),'h4/const/sum/INL'(s(A,A2))))))))) => ![V, V1, V2, V3]: p(s(bool,happ(s(fun('h4/type/sum/sum'(A,B),bool),happ(s(fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)),happ(s(fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool))),happ(s(fun(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),fun('h4/type/sum/sum'(A,B),fun('h4/type/sum/sum'(A,B),bool)))),P),s(fun(A,fun(A,bool)),V))),s(fun(B,fun(B,bool)),V1))),s('h4/type/sum/sum'(A,B),V2))),s('h4/type/sum/sum'(A,B),V3)))))).
