include('Axioms/lift_machine_ieee^2.ax').
thf(tp_c_2Eiterate_2E_2E_2E,type,(c_2Eiterate_2E_2E_2E:$i)).
thf(mem_c_2Eiterate_2E_2E_2E,axiom,(mem@c_2Eiterate_2E_2E_2E@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@bool))))).
thf(tp_c_2Eiterate_2EFINREC,type,(c_2Eiterate_2EFINREC:(del>(del>$i)))).
thf(mem_c_2Eiterate_2EFINREC,axiom,( ! [A_27a:del] : ( ! [A_27b:del] : (mem@(c_2Eiterate_2EFINREC@A_27a@A_27b)@(arr@(arr@A_27a@(arr@A_27b@A_27b))@(arr@A_27b@(arr@(arr@A_27a@bool)@(arr@A_27b@(arr@ty_2Enum_2Enum@bool))))))))).
thf(tp_c_2Eiterate_2EITSET,type,(c_2Eiterate_2EITSET:(del>(del>$i)))).
thf(mem_c_2Eiterate_2EITSET,axiom,( ! [A_27a:del] : ( ! [A_27b:del] : (mem@(c_2Eiterate_2EITSET@A_27a@A_27b)@(arr@(arr@A_27b@(arr@A_27a@A_27a))@(arr@(arr@A_27b@bool)@(arr@A_27a@A_27a))))))).
thf(tp_c_2Eiterate_2ESum,type,(c_2Eiterate_2ESum:(del>$i))).
thf(mem_c_2Eiterate_2ESum,axiom,( ! [A_27a:del] : (mem@(c_2Eiterate_2ESum@A_27a)@(arr@(arr@A_27a@bool)@(arr@(arr@A_27a@ty_2Erealax_2Ereal)@ty_2Erealax_2Ereal))))).
thf(tp_c_2Eiterate_2Einf,type,(c_2Eiterate_2Einf:$i)).
thf(mem_c_2Eiterate_2Einf,axiom,(mem@c_2Eiterate_2Einf@(arr@(arr@ty_2Erealax_2Ereal@bool)@ty_2Erealax_2Ereal))).
thf(tp_c_2Eiterate_2Eiterate,type,(c_2Eiterate_2Eiterate:(del>(del>$i)))).
thf(mem_c_2Eiterate_2Eiterate,axiom,( ! [A_27a:del] : ( ! [A_27b:del] : (mem@(c_2Eiterate_2Eiterate@A_27a@A_27b)@(arr@(arr@A_27b@(arr@A_27b@A_27b))@(arr@(arr@A_27a@bool)@(arr@(arr@A_27a@A_27b)@A_27b))))))).
thf(tp_c_2Eiterate_2Emonoidal,type,(c_2Eiterate_2Emonoidal:(del>$i))).
thf(mem_c_2Eiterate_2Emonoidal,axiom,( ! [A_27a:del] : (mem@(c_2Eiterate_2Emonoidal@A_27a)@(arr@(arr@A_27a@(arr@A_27a@A_27a))@bool)))).
thf(tp_c_2Eiterate_2Eneutral,type,(c_2Eiterate_2Eneutral:(del>$i))).
thf(mem_c_2Eiterate_2Eneutral,axiom,( ! [A_27a:del] : (mem@(c_2Eiterate_2Eneutral@A_27a)@(arr@(arr@A_27a@(arr@A_27a@A_27a))@A_27a)))).
thf(tp_c_2Eiterate_2Ensum,type,(c_2Eiterate_2Ensum:(del>$i))).
thf(mem_c_2Eiterate_2Ensum,axiom,( ! [A_27a:del] : (mem@(c_2Eiterate_2Ensum@A_27a)@(arr@(arr@A_27a@bool)@(arr@(arr@A_27a@ty_2Enum_2Enum)@ty_2Enum_2Enum))))).
thf(tp_c_2Eiterate_2Epolynomial__function,type,(c_2Eiterate_2Epolynomial__function:$i)).
thf(mem_c_2Eiterate_2Epolynomial__function,axiom,(mem@c_2Eiterate_2Epolynomial__function@(arr@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@bool))).
thf(tp_c_2Eiterate_2Esupport,type,(c_2Eiterate_2Esupport:(del>(del>$i)))).
thf(mem_c_2Eiterate_2Esupport,axiom,( ! [A_27a:del] : ( ! [A_27b:del] : (mem@(c_2Eiterate_2Esupport@A_27a@A_27b)@(arr@(arr@A_27b@(arr@A_27b@A_27b))@(arr@(arr@A_27a@A_27b)@(arr@(arr@A_27a@bool)@(arr@A_27a@bool)))))))).
thf(conj_thm_2Eiterate_2EREAL__LT__BETWEEN,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2x))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))).
thf(conj_thm_2Eiterate_2ESIMP__REAL__ARCH,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (? [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Eiterate_2ELOWER__BOUND__FINITE__SET__REAL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) => (? [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(ap@V0f@V3x)))))))))))))).
thf(conj_thm_2Eiterate_2EUPPER__BOUND__FINITE__SET__REAL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) => (? [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@V3x))@(inj__ty_2Erealax_2Ereal @ V2a)))))))))))))).
thf(conj_thm_2Eiterate_2EREAL__WLOG__LT,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ((( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V1x)))) & (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))) <=> (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V2x)))))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ( ! [V5y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5y))) => (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5y)))))))) => ( ! [V6x:tp__ty_2Erealax_2Ereal] : ( ! [V7y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V6x))@(inj__ty_2Erealax_2Ereal @ V7y))))))))).
thf(ax_thm_2Eiterate_2EFINREC__def,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: (( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V1b:$i] : ((mem@V1b@A_27b) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3a:$i] : ((mem@V3a@A_27b) => ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V0f)@V1b)@V2s)@V3a)@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))) <=> ((V2s = (c_2Epred__set_2EEMPTY@A_27a)) & (V3a = V1b))))))))))) & ( ! [V4f:$i] : ((mem@V4f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V5b:$i] : ((mem@V5b@A_27b) => ( ! [V6s:$i] : ((mem@V6s@(arr@A_27a@bool)) => ( ! [V7a:$i] : ((mem@V7a@A_27b) => ( ! [V8n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V4f)@V5b)@V6s)@V7a)@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V8n)))) <=> (? [V9x:$i] : ((mem@V9x@A_27a) & (? [V10c:$i] : ((mem@V10c@A_27b) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V9x)@V6s)) & ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V4f)@V5b)@(ap@(ap@(c_2Epred__set_2EDELETE@A_27a)@V6s)@V9x))@V10c)@(inj__ty_2Enum_2Enum @ V8n))) & (V7a = (ap@(ap@V4f@V9x)@V10c)))))))))))))))))))))).
thf(conj_thm_2Eiterate_2EFINREC__def__compute,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: (( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V1b:$i] : ((mem@V1b@A_27b) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3a:$i] : ((mem@V3a@A_27b) => ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V0f)@V1b)@V2s)@V3a)@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))) <=> ((V2s = (c_2Epred__set_2EEMPTY@A_27a)) & (V3a = V1b))))))))))) & (( ! [V4f:$i] : ((mem@V4f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V5b:$i] : ((mem@V5b@A_27b) => ( ! [V6s:$i] : ((mem@V6s@(arr@A_27a@bool)) => ( ! [V7a:$i] : ((mem@V7a@A_27b) => ( ! [V8n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V4f)@V5b)@V6s)@V7a)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ V8n))))) <=> (? [V9x:$i] : ((mem@V9x@A_27a) & (? [V10c:$i] : ((mem@V10c@A_27b) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V9x)@V6s)) & ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V4f)@V5b)@(ap@(ap@(c_2Epred__set_2EDELETE@A_27a)@V6s)@V9x))@V10c)@(ap@(ap@c_2Earithmetic_2E_2D@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ V8n))))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) & (V7a = (ap@(ap@V4f@V9x)@V10c)))))))))))))))))) & ( ! [V11f:$i] : ((mem@V11f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V12b:$i] : ((mem@V12b@A_27b) => ( ! [V13s:$i] : ((mem@V13s@(arr@A_27a@bool)) => ( ! [V14a:$i] : ((mem@V14a@A_27b) => ( ! [V15n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V11f)@V12b)@V13s)@V14a)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ V15n))))) <=> (? [V16x:$i] : ((mem@V16x@A_27a) & (? [V17c:$i] : ((mem@V17c@A_27b) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V16x)@V13s)) & ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V11f)@V12b)@(ap@(ap@(c_2Epred__set_2EDELETE@A_27a)@V13s)@V16x))@V17c)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ V15n))))) & (V14a = (ap@(ap@V11f@V16x)@V17c))))))))))))))))))))))).
thf(conj_thm_2Eiterate_2EFINREC__1__LEMMA,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V1b:$i] : ((mem@V1b@A_27b) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3a:$i] : ((mem@V3a@A_27b) => ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V0f)@V1b)@V2s)@V3a)@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> (? [V4x:$i] : ((mem@V4x@A_27a) & ((V2s = (ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V4x)@(c_2Epred__set_2EEMPTY@A_27a))) & (V3a = (ap@(ap@V0f@V4x)@V1b))))))))))))))))).
thf(conj_thm_2Eiterate_2EFINREC__SUC__LEMMA,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V1b:$i] : ((mem@V1b@A_27b) => (( ! [V2x:$i] : ((mem@V2x@A_27a) => ( ! [V3y:$i] : ((mem@V3y@A_27a) => ( ! [V4s:$i] : ((mem@V4s@A_27b) => ((~~ (V2x = V3y)) => ((ap@(ap@V0f@V2x)@(ap@(ap@V0f@V3y)@V4s)) = (ap@(ap@V0f@V3y)@(ap@(ap@V0f@V2x)@V4s)))))))))) => ( ! [V5n:tp__ty_2Enum_2Enum] : ( ! [V6s:$i] : ((mem@V6s@(arr@A_27a@bool)) => ( ! [V7z:$i] : ((mem@V7z@A_27b) => ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V0f)@V1b)@V6s)@V7z)@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V5n)))) => ( ! [V8x:$i] : ((mem@V8x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V8x)@V6s)) => (? [V9w:$i] : ((mem@V9w@A_27b) & ((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V0f)@V1b)@(ap@(ap@(c_2Epred__set_2EDELETE@A_27a)@V6s)@V8x))@V9w)@(inj__ty_2Enum_2Enum @ V5n))) & (V7z = (ap@(ap@V0f@V8x)@V9w)))))))))))))))))))))).
thf(conj_thm_2Eiterate_2EFINREC__UNIQUE__LEMMA,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V1b:$i] : ((mem@V1b@A_27b) => (( ! [V2x:$i] : ((mem@V2x@A_27a) => ( ! [V3y:$i] : ((mem@V3y@A_27a) => ( ! [V4s:$i] : ((mem@V4s@A_27b) => ((~~ (V2x = V3y)) => ((ap@(ap@V0f@V2x)@(ap@(ap@V0f@V3y)@V4s)) = (ap@(ap@V0f@V3y)@(ap@(ap@V0f@V2x)@V4s)))))))))) => ( ! [V5n1:tp__ty_2Enum_2Enum] : ( ! [V6n2:tp__ty_2Enum_2Enum] : ( ! [V7s:$i] : ((mem@V7s@(arr@A_27a@bool)) => ( ! [V8a1:$i] : ((mem@V8a1@A_27b) => ( ! [V9a2:$i] : ((mem@V9a2@A_27b) => (((p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V0f)@V1b)@V7s)@V8a1)@(inj__ty_2Enum_2Enum @ V5n1))) & (p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V0f)@V1b)@V7s)@V9a2)@(inj__ty_2Enum_2Enum @ V6n2)))) => ((V8a1 = V9a2) & (V5n1 = V6n2))))))))))))))))))).
thf(conj_thm_2Eiterate_2EFINREC__EXISTS__LEMMA,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V1b:$i] : ((mem@V1b@A_27b) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) => (? [V3a:$i] : ((mem@V3a@A_27b) & (? [V4n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(ap@(ap@(ap@(c_2Eiterate_2EFINREC@A_27a@A_27b)@V0f)@V1b)@V2s)@V3a)@(inj__ty_2Enum_2Enum @ V4n)))))))))))))))).
thf(conj_thm_2Eiterate_2EFINREC__FUN__LEMMA,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1R:$i] : ((mem@V1R@(arr@A_27a@(arr@A_27b@(arr@A_27c@bool)))) => ((( ! [V2s:$i] : ((mem@V2s@A_27a) => ((p@(ap@V0P@V2s)) => (? [V3a:$i] : ((mem@V3a@A_27b) & (? [V4n:$i] : ((mem@V4n@A_27c) & (p@(ap@(ap@(ap@V1R@V2s)@V3a)@V4n))))))))) & ( ! [V5n1:$i] : ((mem@V5n1@A_27c) => ( ! [V6n2:$i] : ((mem@V6n2@A_27c) => ( ! [V7s:$i] : ((mem@V7s@A_27a) => ( ! [V8a1:$i] : ((mem@V8a1@A_27b) => ( ! [V9a2:$i] : ((mem@V9a2@A_27b) => (((p@(ap@(ap@(ap@V1R@V7s)@V8a1)@V5n1)) & (p@(ap@(ap@(ap@V1R@V7s)@V9a2)@V6n2))) => ((V8a1 = V9a2) & (V5n1 = V6n2)))))))))))))) => (? [V10f:$i] : ((mem@V10f@(arr@A_27a@A_27b)) & ( ! [V11s:$i] : ((mem@V11s@A_27a) => ( ! [V12a:$i] : ((mem@V12a@A_27b) => ((p@(ap@V0P@V11s)) => ((? [V13n:$i] : ((mem@V13n@A_27c) & (p@(ap@(ap@(ap@V1R@V11s)@V12a)@V13n)))) <=> ((ap@V10f@V11s) = V12a)))))))))))))))))).
thf(conj_thm_2Eiterate_2EFINREC__FUN,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V1b:$i] : ((mem@V1b@A_27b) => (( ! [V2x:$i] : ((mem@V2x@A_27a) => ( ! [V3y:$i] : ((mem@V3y@A_27a) => ( ! [V4s:$i] : ((mem@V4s@A_27b) => ((~~ (V2x = V3y)) => ((ap@(ap@V0f@V2x)@(ap@(ap@V0f@V3y)@V4s)) = (ap@(ap@V0f@V3y)@(ap@(ap@V0f@V2x)@V4s)))))))))) => (? [V5g:$i] : ((mem@V5g@(arr@(arr@A_27a@bool)@A_27b)) & (((ap@V5g@(c_2Epred__set_2EEMPTY@A_27a)) = V1b) & ( ! [V6s:$i] : ((mem@V6s@(arr@A_27a@bool)) => ( ! [V7x:$i] : ((mem@V7x@A_27a) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V6s)) & (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V7x)@V6s))) => ((ap@V5g@V6s) = (ap@(ap@V0f@V7x)@(ap@V5g@(ap@(ap@(c_2Epred__set_2EDELETE@A_27a)@V6s)@V7x)))))))))))))))))))).
thf(conj_thm_2Eiterate_2ESET__RECURSION__LEMMA,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V1b:$i] : ((mem@V1b@A_27b) => (( ! [V2x:$i] : ((mem@V2x@A_27a) => ( ! [V3y:$i] : ((mem@V3y@A_27a) => ( ! [V4s:$i] : ((mem@V4s@A_27b) => ((~~ (V2x = V3y)) => ((ap@(ap@V0f@V2x)@(ap@(ap@V0f@V3y)@V4s)) = (ap@(ap@V0f@V3y)@(ap@(ap@V0f@V2x)@V4s)))))))))) => (? [V5g:$i] : ((mem@V5g@(arr@(arr@A_27a@bool)@A_27b)) & (((ap@V5g@(c_2Epred__set_2EEMPTY@A_27a)) = V1b) & ( ! [V6x:$i] : ((mem@V6x@A_27a) => ( ! [V7s:$i] : ((mem@V7s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V7s)) => ((ap@V5g@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V6x)@V7s)) = (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27b)@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6x)@V7s))@(ap@V5g@V7s))@(ap@(ap@V0f@V6x)@(ap@V5g@V7s)))))))))))))))))))).
thf(ax_thm_2Eiterate_2EITSET__def,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27b@(arr@A_27a@A_27a))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27b@bool)) => ( ! [V2b:$i] : ((mem@V2b@A_27a) => ((ap@(ap@(ap@(c_2Eiterate_2EITSET@A_27a@A_27b)@V0f)@V1s)@V2b) = (ap@(ap@(c_2Emin_2E_40@(arr@(arr@A_27b@bool)@A_27a))@(lam@(arr@(arr@A_27b@bool)@A_27a)@(^ [V3g:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Emin_2E_3D@A_27a)@(ap@V3g@(c_2Epred__set_2EEMPTY@A_27b)))@V2b))@(ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V4x:$i] : (ap@(c_2Ebool_2E_21@(arr@A_27b@bool))@(lam@(arr@A_27b@bool)@(^ [V5s:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(c_2Epred__set_2EFINITE@A_27b)@V5s))@(ap@(ap@(c_2Emin_2E_3D@A_27a)@(ap@V3g@(ap@(ap@(c_2Epred__set_2EINSERT@A_27b)@V4x)@V5s)))@(ap@(ap@(ap@(c_2Ebool_2ECOND@A_27a)@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V4x)@V5s))@(ap@V3g@V5s))@(ap@(ap@V0f@V4x)@(ap@V3g@V5s)))))))))))))))@V1s))))))))))).
thf(conj_thm_2Eiterate_2EFINITE__RECURSION,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@A_27b))) => ( ! [V1b:$i] : ((mem@V1b@A_27b) => (( ! [V2x:$i] : ((mem@V2x@A_27a) => ( ! [V3y:$i] : ((mem@V3y@A_27a) => ( ! [V4s:$i] : ((mem@V4s@A_27b) => ((~~ (V2x = V3y)) => ((ap@(ap@V0f@V2x)@(ap@(ap@V0f@V3y)@V4s)) = (ap@(ap@V0f@V3y)@(ap@(ap@V0f@V2x)@V4s)))))))))) => (((ap@(ap@(ap@(c_2Eiterate_2EITSET@A_27b@A_27a)@V0f)@(c_2Epred__set_2EEMPTY@A_27a))@V1b) = V1b) & ( ! [V5x:$i] : ((mem@V5x@A_27a) => ( ! [V6s:$i] : ((mem@V6s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V6s)) => ((ap@(ap@(ap@(c_2Eiterate_2EITSET@A_27b@A_27a)@V0f)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V5x)@V6s))@V1b) = (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27b)@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V6s))@(ap@(ap@(ap@(c_2Eiterate_2EITSET@A_27b@A_27a)@V0f)@V6s)@V1b))@(ap@(ap@V0f@V5x)@(ap@(ap@(ap@(c_2Eiterate_2EITSET@A_27b@A_27a)@V0f)@V6s)@V1b)))))))))))))))))).
thf(conj_thm_2Eiterate_2ECARD__UNION__EQ,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2u)) & (((ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@A_27a)) & ((ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V0s)@V1t) = V2u))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s))@(ap@(c_2Epred__set_2ECARD@A_27a)@V1t))) = (surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@A_27a)@V2u)))))))))))).
thf(conj_thm_2Eiterate_2ESUBSET__RESTRICT,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1P:$i] : ((mem@V1P@(arr@A_27a@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V2x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V0s))@(ap@V1P@V2x)))))))@V0s)))))))).
thf(conj_thm_2Eiterate_2EFINITE__RESTRICT,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1P:$i] : ((mem@V1P@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) => (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V2x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V0s))@(ap@V1P@V2x))))))))))))))).
thf(conj_thm_2Eiterate_2EFINITE__REAL__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@V1x))))))))) & (( ! [V2a:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@V3x))))))))) & (( ! [V4b:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Erealax_2Ereal__lt@V5x)@(inj__ty_2Erealax_2Ereal @ V4b)))))))))) & (( ! [V6b:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ereal_2Ereal__lte@V7x)@(inj__ty_2Erealax_2Ereal @ V6b)))))))))) & (( ! [V8a:tp__ty_2Erealax_2Ereal] : ( ! [V9b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V10x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V10x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V8a))@V10x))@(ap@(ap@c_2Erealax_2Ereal__lt@V10x)@(inj__ty_2Erealax_2Ereal @ V9b))))))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V9b))@(inj__ty_2Erealax_2Ereal @ V8a)))))) & (( ! [V11a:tp__ty_2Erealax_2Ereal] : ( ! [V12b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V13x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V13x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V11a))@V13x))@(ap@(ap@c_2Erealax_2Ereal__lt@V13x)@(inj__ty_2Erealax_2Ereal @ V12b))))))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V12b))@(inj__ty_2Erealax_2Ereal @ V11a)))))) & (( ! [V14a:tp__ty_2Erealax_2Ereal] : ( ! [V15b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V16x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V16x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V14a))@V16x))@(ap@(ap@c_2Ereal_2Ereal__lte@V16x)@(inj__ty_2Erealax_2Ereal @ V15b))))))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V15b))@(inj__ty_2Erealax_2Ereal @ V14a)))))) & ( ! [V17a:tp__ty_2Erealax_2Ereal] : ( ! [V18b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V19x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V19x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V17a))@V19x))@(ap@(ap@c_2Ereal_2Ereal__lte@V19x)@(inj__ty_2Erealax_2Ereal @ V18b))))))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V18b))@(inj__ty_2Erealax_2Ereal @ V17a)))))))))))))).
thf(conj_thm_2Eiterate_2Ereal__INFINITE,lemma,(~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Eiterate_2ESET__PROVE__CASES,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@(arr@A_27a@bool)@bool)) => (((p@(ap@V0P@(c_2Epred__set_2EEMPTY@A_27a))) & ( ! [V1a:$i] : ((mem@V1a@A_27a) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V1a)@V2s))) => (p@(ap@V0P@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1a)@V2s))))))))) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27a@bool)) => (p@(ap@V0P@V3s))))))))).
thf(conj_thm_2Eiterate_2ECHOOSE__SUBSET__STRONG,lemma,( ! [A_27a:del]: ( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0n))@(ap@(c_2Epred__set_2ECARD@A_27a)@V1s)))) => (? [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V2t)@V1s)) & (p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@A_27a)@V2t)@(inj__ty_2Enum_2Enum @ V0n)))))))))))).
thf(conj_thm_2Eiterate_2ECHOOSE__SUBSET,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) => ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1n))@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s))) => (? [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V2t)@V0s)) & (p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@A_27a)@V2t)@(inj__ty_2Enum_2Enum @ V1n))))))))))))).
thf(conj_thm_2Eiterate_2EREAL__COMPLETE,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => (((? [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V1x)))) & (? [V2M:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V3x))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2M))))))) => (? [V4M:tp__ty_2Erealax_2Ereal] : (( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V4M))))) & ( ! [V6M_27:tp__ty_2Erealax_2Ereal] : (( ! [V7x:tp__ty_2Erealax_2Ereal] : ((p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V7x))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V7x))@(inj__ty_2Erealax_2Ereal @ V6M_27))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4M))@(inj__ty_2Erealax_2Ereal @ V6M_27))))))))))).
thf(conj_thm_2Eiterate_2Esup__alt,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@V0s)) = (surj__ty_2Erealax_2Ereal @ (ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1a:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@c_2Ereal_2Ereal__lte@V2x)@V1a))))))@(ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3b:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V0s))@(ap@(ap@c_2Ereal_2Ereal__lte@V4x)@V3b))))))@(ap@(ap@c_2Ereal_2Ereal__lte@V1a)@V3b)))))))))))))).
thf(conj_thm_2Eiterate_2ESUP__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2b:tp__ty_2Erealax_2Ereal] : (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2b))))) <=> ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1t)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V2b))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@V0s)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@V1t))))))))).
thf(conj_thm_2Eiterate_2ESUP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (? [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) => (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@c_2Ereal_2Esup@V0s))))) & ( ! [V4b:tp__ty_2Erealax_2Ereal] : (( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V4b))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Esup@V0s))@(inj__ty_2Erealax_2Ereal @ V4b)))))))))).
thf(conj_thm_2Eiterate_2ESUP__FINITE__LEMMA,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@V0s)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Eiterate_2ESUP__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Esup@V0s))@V0s)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal_2Esup@V0s)))))))))).
thf(conj_thm_2Eiterate_2EREAL__LE__SUP__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Ereal_2Esup@V0s))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Eiterate_2EREAL__SUP__LE__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Esup@V0s))@(inj__ty_2Erealax_2Ereal @ V1a))) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1a))))))))))).
thf(conj_thm_2Eiterate_2EREAL__LT__SUP__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Ereal_2Esup@V0s))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Eiterate_2EREAL__SUP__LT__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Esup@V0s))@(inj__ty_2Erealax_2Ereal @ V1a))) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1a))))))))))).
thf(conj_thm_2Eiterate_2EREAL__SUP__LE__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (? [V2b:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2b))))))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Esup@V0s))@(inj__ty_2Erealax_2Ereal @ V1y))) <=> ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V1y))))))))))).
thf(conj_thm_2Eiterate_2EREAL__SUP__UNIQUE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))) & ( ! [V3b_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V3b_27))@(inj__ty_2Erealax_2Ereal @ V1b))) => (? [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V3b_27))@(inj__ty_2Erealax_2Ereal @ V4x)))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@V0s)) = V1b)))))).
thf(conj_thm_2Eiterate_2EREAL__SUP__LE__S,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b)))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Esup@V0s))@(inj__ty_2Erealax_2Ereal @ V1b)))))))).
thf(conj_thm_2Eiterate_2EREAL__SUP__LE__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (? [V2b:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1t)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2b)))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Esup@V0s))@(ap@c_2Ereal_2Esup@V1t))))))))).
thf(conj_thm_2Eiterate_2EREAL__LE__SUP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0s)) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V3y))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V2b))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Ereal_2Esup@V0s)))))))))).
thf(conj_thm_2Eiterate_2EREAL__SUP__BOUNDS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V3x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2b))))))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Ereal_2Esup@V0s))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Esup@V0s))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))).
thf(conj_thm_2Eiterate_2EREAL__ABS__SUP__LE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1a)))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@c_2Ereal_2Esup@V0s)))@(inj__ty_2Erealax_2Ereal @ V1a)))))))).
thf(conj_thm_2Eiterate_2EREAL__SUP__ASCLOSE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V2e)))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Esup@V0s))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V2e))))))))).
thf(conj_thm_2Eiterate_2ESUP__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) = V0a))).
thf(conj_thm_2Eiterate_2ESUP__UNIQUE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:tp__ty_2Erealax_2Ereal] : (( ! [V2c:tp__ty_2Erealax_2Ereal] : (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2c))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@V0s)) = V1b)))))).
thf(conj_thm_2Eiterate_2ESUP__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((? [V2b:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2b)))))) & (? [V4c:tp__ty_2Erealax_2Ereal] : ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V1t)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V4c))))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Emax@(ap@c_2Ereal_2Esup@V0s))@(ap@c_2Ereal_2Esup@V1t)))))))))).
thf(ax_thm_2Eiterate_2Einf,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Eiterate_2Einf@V0s)) = (surj__ty_2Erealax_2Ereal @ (ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1a:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@c_2Ereal_2Ereal__lte@V1a)@V2x))))))@(ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3b:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V0s))@(ap@(ap@c_2Ereal_2Ereal__lte@V3b)@V4x))))))@(ap@(ap@c_2Ereal_2Ereal__lte@V3b)@V1a)))))))))))))).
thf(conj_thm_2Eiterate_2EINF__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2a:tp__ty_2Erealax_2Ereal] : (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3x))))) <=> ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1t)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V4x))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Eiterate_2Einf@V0s)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Eiterate_2Einf@V1t))))))))).
thf(conj_thm_2Eiterate_2EINF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (? [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2x))))))) => (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V3x))))) & ( ! [V4b:tp__ty_2Erealax_2Ereal] : (( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4b))@(inj__ty_2Erealax_2Ereal @ V5x))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4b))@(ap@c_2Eiterate_2Einf@V0s)))))))))).
thf(conj_thm_2Eiterate_2EINF__FINITE__LEMMA,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@V0s)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Eiterate_2EINF__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Eiterate_2Einf@V0s))@V0s)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V1x)))))))))).
thf(conj_thm_2Eiterate_2EREAL__LE__INF__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Eiterate_2Einf@V0s))) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Eiterate_2EREAL__INF__LE__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V1a))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1a))))))))))).
thf(conj_thm_2Eiterate_2EREAL__LT__INF__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Eiterate_2Einf@V0s))) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Eiterate_2EREAL__INF__LT__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V1a))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1a))))))))))).
thf(conj_thm_2Eiterate_2EREAL__INF__UNIQUE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2x))))) & ( ! [V3b_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V3b_27))) => (? [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V3b_27)))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Eiterate_2Einf@V0s)) = V1b)))))).
thf(conj_thm_2Eiterate_2EREAL__LE__INF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2x)))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@c_2Eiterate_2Einf@V0s)))))))).
thf(conj_thm_2Eiterate_2EREAL__LE__INF__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & (? [V2b:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2b))@(inj__ty_2Erealax_2Ereal @ V3x)))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Eiterate_2Einf@V0s))@(ap@c_2Eiterate_2Einf@V1t))))))))).
thf(conj_thm_2Eiterate_2EREAL__INF__BOUNDS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V3x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2b))))))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Eiterate_2Einf@V0s))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))).
thf(conj_thm_2Eiterate_2EREAL__ABS__INF__LE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1a)))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@c_2Eiterate_2Einf@V0s)))@(inj__ty_2Erealax_2Ereal @ V1a)))))))).
thf(conj_thm_2Eiterate_2EREAL__INF__ASCLOSE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V2e)))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V2e))))))))).
thf(conj_thm_2Eiterate_2ESUP__UNIQUE__FINITE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1s)) & (~~ (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@V1s)) = V0a) <=> ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V1s)) & ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V0a)))))))))))).
thf(conj_thm_2Eiterate_2EINF__UNIQUE__FINITE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1s)) & (~~ (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (((surj__ty_2Erealax_2Ereal @ (ap@c_2Eiterate_2Einf@V1s)) = V0a) <=> ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V1s)) & ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2y)))))))))))).
thf(conj_thm_2Eiterate_2ESUP__INSERT__FINITE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V1s))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@V1s)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@c_2Ereal_2Emax@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Esup@V1s)))))))))).
thf(conj_thm_2Eiterate_2EINF__INSERT__FINITE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Eiterate_2Einf@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V1s))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@V1s)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@c_2Ereal_2Emin@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Eiterate_2Einf@V1s)))))))))).
thf(conj_thm_2Eiterate_2EREAL__SUP__EQ__INF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (? [V1B:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1B))))))) => (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@V0s)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Eiterate_2Einf@V0s))) <=> (? [V3a:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Eiterate_2EINF__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Eiterate_2Einf@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) = V0a))).
thf(ax_thm_2Eiterate_2Enumseg,axiom,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)) = (ap@(c_2Epred__set_2EGSPEC@ty_2Enum_2Enum@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Enum_2Enum@bool)@V2x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@V2x))@(ap@(ap@c_2Earithmetic_2E_3C_3D@V2x)@(inj__ty_2Enum_2Enum @ V1n))))))))))).
thf(conj_thm_2Eiterate_2EFINITE__NUMSEG,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Eiterate_2ENUMSEG__COMBINE__R,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1p:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1p))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1p))@(inj__ty_2Enum_2Enum @ V2n)))) => ((ap@(ap@(c_2Epred__set_2EUNION@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1p)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1p))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V2n))) = (ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V2n)))))))).
thf(conj_thm_2Eiterate_2ENUMSEG__COMBINE__L,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1p:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1p))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1p))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) => ((ap@(ap@(c_2Epred__set_2EUNION@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V1p))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1p))@(inj__ty_2Enum_2Enum @ V2n))) = (ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V2n)))))))).
thf(conj_thm_2Eiterate_2ENUMSEG__LREC,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))) => ((ap@(ap@(c_2Epred__set_2EINSERT@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V1n))) = (ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Eiterate_2ENUMSEG__RREC,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))) => ((ap@(ap@(c_2Epred__set_2EINSERT@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V1n))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V1n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) = (ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Eiterate_2ENUMSEG__REC,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V1n)))) => ((ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V1n))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Enum_2Enum)@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V1n)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))))))).
thf(conj_thm_2Eiterate_2EIN__NUMSEG,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ( ! [V2p:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V2p))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))) <=> ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V2p))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2p))@(inj__ty_2Enum_2Enum @ V1n))))))))).
thf(conj_thm_2Eiterate_2EIN__NUMSEG__0,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V1n)))) <=> (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Eiterate_2ENUMSEG__SING,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ((ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0n))@(inj__ty_2Enum_2Enum @ V0n)) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V0n))@(c_2Epred__set_2EEMPTY@ty_2Enum_2Enum))))).
thf(conj_thm_2Eiterate_2ENUMSEG__EMPTY,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : (((ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)) = (c_2Epred__set_2EEMPTY@ty_2Enum_2Enum)) <=> (p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V0m))))))).
thf(conj_thm_2Eiterate_2ECARD__NUMSEG__LEMMA,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1d:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1d))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1d))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))).
thf(conj_thm_2Eiterate_2ECARD__NUMSEG,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2D@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V0m))))))).
thf(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))@(ap@(ap@c_2Earithmetic_2E_2D@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V0m))))))).
thf(conj_thm_2Eiterate_2ECARD__NUMSEG__1,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V0n)))) = V0n))).
thf(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG__1,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V0n)))@(inj__ty_2Enum_2Enum @ V0n))))).
thf(conj_thm_2Eiterate_2ENUMSEG__CLAUSES,lemma,(( ! [V0m:tp__ty_2Enum_2Enum] : ((ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Enum_2Enum@bool))@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(c_2Epred__set_2EEMPTY@ty_2Enum_2Enum)))@(c_2Epred__set_2EEMPTY@ty_2Enum_2Enum)))) & ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V2n))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Enum_2Enum@bool))@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V2n))))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Enum_2Enum)@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V2n)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))))))).
thf(conj_thm_2Eiterate_2EFINITE__INDEX__NUMSEG,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) <=> (? [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@A_27a)) & (( ! [V2i:tp__ty_2Enum_2Enum] : ( ! [V3j:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V2i))@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s)))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3j))@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s)))) & ((ap@V1f@(inj__ty_2Enum_2Enum @ V2i)) = (ap@V1f@(inj__ty_2Enum_2Enum @ V3j))))) => (V2i = V3j)))) & (V0s = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@A_27a)@V1f)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s)))))))))))).
thf(conj_thm_2Eiterate_2EFINITE__INDEX__NUMBERS,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) <=> (? [V1k:$i] : ((mem@V1k@(arr@ty_2Enum_2Enum@bool)) & (? [V2f:$i] : ((mem@V2f@(arr@ty_2Enum_2Enum@A_27a)) & (( ! [V3i:tp__ty_2Enum_2Enum] : ( ! [V4j:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3i))@V1k)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V4j))@V1k)) & ((ap@V2f@(inj__ty_2Enum_2Enum @ V3i)) = (ap@V2f@(inj__ty_2Enum_2Enum @ V4j))))) => (V3i = V4j)))) & ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@V1k)) & (V0s = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@A_27a)@V2f)@V1k))))))))))))).
thf(conj_thm_2Eiterate_2EDISJOINT__NUMSEG,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ( ! [V2p:tp__ty_2Enum_2Enum] : ( ! [V3q:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Epred__set_2EDISJOINT@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2p))@(inj__ty_2Enum_2Enum @ V3q)))) <=> ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V2p))) | ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V3q))@(inj__ty_2Enum_2Enum @ V0m))) | ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V0m))) | (p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V3q))@(inj__ty_2Enum_2Enum @ V2p)))))))))))).
thf(conj_thm_2Eiterate_2ENUMSEG__ADD__SPLIT,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ( ! [V2p:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) => ((ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V2p))) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V2p)))))))))).
thf(conj_thm_2Eiterate_2ENUMSEG__OFFSET__IMAGE,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ( ! [V2p:tp__ty_2Enum_2Enum] : ((ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V2p)))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V2p))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@(ap@c_2Earithmetic_2E_2B@V3i)@(inj__ty_2Enum_2Enum @ V2p)))))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))))))).
thf(conj_thm_2Eiterate_2ESUBSET__NUMSEG,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ( ! [V2p:tp__ty_2Enum_2Enum] : ( ! [V3q:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2p))@(inj__ty_2Enum_2Enum @ V3q)))) <=> ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V0m))) | ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2p))@(inj__ty_2Enum_2Enum @ V0m))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V3q))))))))))).
thf(conj_thm_2Eiterate_2ENUMSEG__LE,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ((ap@(c_2Epred__set_2EGSPEC@ty_2Enum_2Enum@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Enum_2Enum@bool)@V1x)@(ap@(ap@c_2Earithmetic_2E_3C_3D@V1x)@(inj__ty_2Enum_2Enum @ V0n)))))) = (ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V0n))))).
thf(conj_thm_2Eiterate_2ENUMSEG__LT,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ((ap@(c_2Epred__set_2EGSPEC@ty_2Enum_2Enum@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Enum_2Enum@bool)@V1x)@(ap@(ap@c_2Eprim__rec_2E_3C@V1x)@(inj__ty_2Enum_2Enum @ V0n)))))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Enum_2Enum@bool))@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V0n))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(c_2Epred__set_2EEMPTY@ty_2Enum_2Enum))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V0n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))).
thf(conj_thm_2Eiterate_2ETOPOLOGICAL__SORT,lemma,( ! [A_27a:del]: ( ! [V0_3C_3C:$i] : ((mem@V0_3C_3C@(arr@A_27a@(arr@A_27a@bool))) => ((( ! [V1x:$i] : ((mem@V1x@A_27a) => ( ! [V2y:$i] : ((mem@V2y@A_27a) => (((p@(ap@(ap@V0_3C_3C@V1x)@V2y)) & (p@(ap@(ap@V0_3C_3C@V2y)@V1x))) => (V1x = V2y)))))) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ( ! [V4y:$i] : ((mem@V4y@A_27a) => ( ! [V5z:$i] : ((mem@V5z@A_27a) => (((p@(ap@(ap@V0_3C_3C@V3x)@V4y)) & (p@(ap@(ap@V0_3C_3C@V4y)@V5z))) => (p@(ap@(ap@V0_3C_3C@V3x)@V5z)))))))))) => ( ! [V6n:tp__ty_2Enum_2Enum] : ( ! [V7s:$i] : ((mem@V7s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@A_27a)@V7s)@(inj__ty_2Enum_2Enum @ V6n))) => (? [V8f:$i] : ((mem@V8f@(arr@ty_2Enum_2Enum@A_27a)) & ((V7s = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@A_27a)@V8f)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V6n)))) & ( ! [V9j:tp__ty_2Enum_2Enum] : ( ! [V10k:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V9j))@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V6n)))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V10k))@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V6n)))) & (p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V9j))@(inj__ty_2Enum_2Enum @ V10k))))) => (~~ (p@(ap@(ap@V0_3C_3C@(ap@V8f@(inj__ty_2Enum_2Enum @ V10k)))@(ap@V8f@(inj__ty_2Enum_2Enum @ V9j)))))))))))))))))))).
thf(ax_thm_2Eiterate_2Eneutral,axiom,( ! [A_27a:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((ap@(c_2Eiterate_2Eneutral@A_27a)@V0op) = (ap@(c_2Emin_2E_40@A_27a)@(lam@A_27a@(^ [V1x:$i] : (ap@(c_2Ebool_2E_21@A_27a)@(lam@A_27a@(^ [V2y:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Emin_2E_3D@A_27a)@(ap@(ap@V0op@V1x)@V2y))@V2y))@(ap@(ap@(c_2Emin_2E_3D@A_27a)@(ap@(ap@V0op@V2y)@V1x))@V2y))))))))))))).
thf(ax_thm_2Eiterate_2Emonoidal,axiom,( ! [A_27a:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V0op)) <=> (( ! [V1x:$i] : ((mem@V1x@A_27a) => ( ! [V2y:$i] : ((mem@V2y@A_27a) => ((ap@(ap@V0op@V1x)@V2y) = (ap@(ap@V0op@V2y)@V1x)))))) & (( ! [V3x:$i] : ((mem@V3x@A_27a) => ( ! [V4y:$i] : ((mem@V4y@A_27a) => ( ! [V5z:$i] : ((mem@V5z@A_27a) => ((ap@(ap@V0op@V3x)@(ap@(ap@V0op@V4y)@V5z)) = (ap@(ap@V0op@(ap@(ap@V0op@V3x)@V4y))@V5z)))))))) & ( ! [V6x:$i] : ((mem@V6x@A_27a) => ((ap@(ap@V0op@(ap@(c_2Eiterate_2Eneutral@A_27a)@V0op))@V6x) = V6x)))))))))).
thf(conj_thm_2Eiterate_2EMONOIDAL__AC,lemma,( ! [A_27a:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V0op)) => (( ! [V1a:$i] : ((mem@V1a@A_27a) => ((ap@(ap@V0op@(ap@(c_2Eiterate_2Eneutral@A_27a)@V0op))@V1a) = V1a))) & (( ! [V2a:$i] : ((mem@V2a@A_27a) => ((ap@(ap@V0op@V2a)@(ap@(c_2Eiterate_2Eneutral@A_27a)@V0op)) = V2a))) & (( ! [V3a:$i] : ((mem@V3a@A_27a) => ( ! [V4b:$i] : ((mem@V4b@A_27a) => ((ap@(ap@V0op@V3a)@V4b) = (ap@(ap@V0op@V4b)@V3a)))))) & (( ! [V5a:$i] : ((mem@V5a@A_27a) => ( ! [V6b:$i] : ((mem@V6b@A_27a) => ( ! [V7c:$i] : ((mem@V7c@A_27a) => ((ap@(ap@V0op@(ap@(ap@V0op@V5a)@V6b))@V7c) = (ap@(ap@V0op@V5a)@(ap@(ap@V0op@V6b)@V7c))))))))) & ( ! [V8a:$i] : ((mem@V8a@A_27a) => ( ! [V9b:$i] : ((mem@V9b@A_27a) => ( ! [V10c:$i] : ((mem@V10c@A_27a) => ((ap@(ap@V0op@V8a)@(ap@(ap@V0op@V9b)@V10c)) = (ap@(ap@V0op@V9b)@(ap@(ap@V0op@V8a)@V10c)))))))))))))))))).
thf(ax_thm_2Eiterate_2Esupport,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@V2s) = (ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@A_27b)@(ap@V1f@V3x))@(ap@(c_2Eiterate_2Eneutral@A_27b)@V0op)))))))))))))))))).
thf(ax_thm_2Eiterate_2Eiterate,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@A_27b)) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V1s)@V2f) = (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27b)@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V2f)@V1s)))@(ap@(ap@(ap@(c_2Eiterate_2EITSET@A_27b@A_27a)@(lam@A_27a@(^ [V3x:$i] : (lam@A_27b@(^ [V4a:$i] : (ap@(ap@V0op@(ap@V2f@V3x))@V4a))))))@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V2f)@V1s))@(ap@(c_2Eiterate_2Eneutral@A_27b)@V0op)))@(ap@(c_2Eiterate_2Eneutral@A_27b)@V0op)))))))))))).
thf(conj_thm_2Eiterate_2EIN__SUPPORT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2x:$i] : ((mem@V2x@A_27b) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27b@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V2x)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@V3s))) <=> ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V2x)@V3s)) & (~~ ((ap@V1f@V2x) = (ap@(c_2Eiterate_2Eneutral@A_27a)@V0op)))))))))))))))).
thf(conj_thm_2Eiterate_2ESUPPORT__SUPPORT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@V2s)) = (ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@V2s))))))))))).
thf(conj_thm_2Eiterate_2ESUPPORT__EMPTY,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => (( ! [V3x:$i] : ((mem@V3x@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3x)@V2s)) => ((ap@V1f@V3x) = (ap@(c_2Eiterate_2Eneutral@A_27a)@V0op))))) <=> ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@V2s) = (c_2Epred__set_2EEMPTY@A_27b)))))))))))).
thf(conj_thm_2Eiterate_2ESUPPORT__SUBSET,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@V2s))@V2s))))))))))).
thf(conj_thm_2Eiterate_2EFINITE__SUPPORT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2s)) => (p@(ap@(c_2Epred__set_2EFINITE@A_27b)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@V2s))))))))))))).
thf(conj_thm_2Eiterate_2ESUPPORT__CLAUSES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [A_27d:del]: ( ! [A_27e:del]: ( ! [A_27f:del]: ( ! [A_27g:del]: ( ! [A_27h:del]: ( ! [A_27i:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => (( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@(c_2Epred__set_2EEMPTY@A_27a)) = (c_2Epred__set_2EEMPTY@A_27a)))) & (( ! [V2f:$i] : ((mem@V2f@(arr@A_27c@A_27b)) => ( ! [V3x:$i] : ((mem@V3x@A_27c) => ( ! [V4s:$i] : ((mem@V4s@(arr@A_27c@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27c@A_27b)@V0op)@V2f)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27c)@V3x)@V4s)) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@A_27c@bool))@(ap@(ap@(c_2Emin_2E_3D@A_27b)@(ap@V2f@V3x))@(ap@(c_2Eiterate_2Eneutral@A_27b)@V0op)))@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27c@A_27b)@V0op)@V2f)@V4s))@(ap@(ap@(c_2Epred__set_2EINSERT@A_27c)@V3x)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27c@A_27b)@V0op)@V2f)@V4s)))))))))) & (( ! [V5f:$i] : ((mem@V5f@(arr@A_27d@A_27b)) => ( ! [V6x:$i] : ((mem@V6x@A_27d) => ( ! [V7s:$i] : ((mem@V7s@(arr@A_27d@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27d@A_27b)@V0op)@V5f)@(ap@(ap@(c_2Epred__set_2EDELETE@A_27d)@V7s)@V6x)) = (ap@(ap@(c_2Epred__set_2EDELETE@A_27d)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27d@A_27b)@V0op)@V5f)@V7s))@V6x)))))))) & (( ! [V8f:$i] : ((mem@V8f@(arr@A_27e@A_27b)) => ( ! [V9s:$i] : ((mem@V9s@(arr@A_27e@bool)) => ( ! [V10t:$i] : ((mem@V10t@(arr@A_27e@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27e@A_27b)@V0op)@V8f)@(ap@(ap@(c_2Epred__set_2EUNION@A_27e)@V9s)@V10t)) = (ap@(ap@(c_2Epred__set_2EUNION@A_27e)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27e@A_27b)@V0op)@V8f)@V9s))@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27e@A_27b)@V0op)@V8f)@V10t))))))))) & (( ! [V11f:$i] : ((mem@V11f@(arr@A_27f@A_27b)) => ( ! [V12s:$i] : ((mem@V12s@(arr@A_27f@bool)) => ( ! [V13t:$i] : ((mem@V13t@(arr@A_27f@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27f@A_27b)@V0op)@V11f)@(ap@(ap@(c_2Epred__set_2EINTER@A_27f)@V12s)@V13t)) = (ap@(ap@(c_2Epred__set_2EINTER@A_27f)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27f@A_27b)@V0op)@V11f)@V12s))@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27f@A_27b)@V0op)@V11f)@V13t))))))))) & (( ! [V14f:$i] : ((mem@V14f@(arr@A_27g@A_27b)) => ( ! [V15s:$i] : ((mem@V15s@(arr@A_27g@bool)) => ( ! [V16t:$i] : ((mem@V16t@(arr@A_27g@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27g@A_27b)@V0op)@V14f)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27g)@V15s)@V16t)) = (ap@(ap@(c_2Epred__set_2EDIFF@A_27g)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27g@A_27b)@V0op)@V14f)@V15s))@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27g@A_27b)@V0op)@V14f)@V16t))))))))) & ( ! [V17f:$i] : ((mem@V17f@(arr@A_27h@A_27i)) => ( ! [V18g:$i] : ((mem@V18g@(arr@A_27i@A_27b)) => ( ! [V19s:$i] : ((mem@V19s@(arr@A_27h@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27i@A_27b)@V0op)@V18g)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27h@A_27i)@V17f)@V19s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@A_27h@A_27i)@V17f)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27h@A_27b)@V0op)@(ap@(ap@(c_2Ecombin_2Eo@A_27h@A_27b@A_27i)@V18g)@V17f))@V19s))))))))))))))))))))))))))).
thf(conj_thm_2Eiterate_2ESUPPORT__DELTA,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27b@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27b@A_27a)) => ( ! [V3a:$i] : ((mem@V3a@A_27b) => ((ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@(lam@A_27b@(^ [V4x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27a)@(ap@(ap@(c_2Emin_2E_3D@A_27b)@V4x)@V3a))@(ap@V2f@V4x))@(ap@(c_2Eiterate_2Eneutral@A_27a)@V0op)))))@V1s) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@A_27b@bool))@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3a)@V1s))@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V2f)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27b)@V3a)@(c_2Epred__set_2EEMPTY@A_27b))))@(c_2Epred__set_2EEMPTY@A_27b)))))))))))))).
thf(conj_thm_2Eiterate_2EFINITE__SUPPORT__DELTA,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27b@bool)) => ( ! [V1op:$i] : ((mem@V1op@(arr@A_27a@(arr@A_27a@A_27a))) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27b@A_27a)) => ( ! [V3a:$i] : ((mem@V3a@A_27b) => (p@(ap@(c_2Epred__set_2EFINITE@A_27b)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V1op)@(lam@A_27b@(^ [V4x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27a)@(ap@(ap@(c_2Emin_2E_3D@A_27b)@V4x)@V3a))@(ap@V2f@V4x))@(ap@(c_2Eiterate_2Eneutral@A_27a)@V1op)))))@V0s)))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__SUPPORT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@V2s))@V1f) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V2s)@V1f))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__EXPAND__CASES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V2s)@V1f) = (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27a)@(ap@(c_2Epred__set_2EFINITE@A_27b)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@V2s)))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27b@A_27a)@V0op)@V1f)@V2s))@V1f))@(ap@(c_2Eiterate_2Eneutral@A_27a)@V0op)))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__CLAUSES__GEN,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => (( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@(c_2Epred__set_2EEMPTY@A_27a))@V1f) = (ap@(c_2Eiterate_2Eneutral@A_27b)@V0op)))) & ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@A_27b)) => ( ! [V3x:$i] : ((mem@V3x@A_27a) => ( ! [V4s:$i] : ((mem@V4s@(arr@A_27a@bool)) => (((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) & (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V2f)@V4s)))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V3x)@V4s))@V2f) = (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27b)@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V4s))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V4s)@V2f))@(ap@(ap@V0op@(ap@V2f@V3x))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V4s)@V2f)))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__CLAUSES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V0op)) => (( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@(c_2Epred__set_2EEMPTY@A_27b))@V1f) = (ap@(c_2Eiterate_2Eneutral@A_27a)@V0op)))) & ( ! [V2f:$i] : ((mem@V2f@(arr@A_27b@A_27a)) => ( ! [V3x:$i] : ((mem@V3x@A_27b) => ( ! [V4s:$i] : ((mem@V4s@(arr@A_27b@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V4s)) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27b)@V3x)@V4s))@V2f) = (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27a)@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3x)@V4s))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V4s)@V2f))@(ap@(ap@V0op@(ap@V2f@V3x))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V4s)@V2f)))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__UNION,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2s)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V3t)) & (p@(ap@(ap@(c_2Epred__set_2EDISJOINT@A_27b)@V2s)@V3t)))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@(ap@(ap@(c_2Epred__set_2EUNION@A_27b)@V2s)@V3t))@V1f) = (ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V2s)@V1f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V3t)@V1f)))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__UNION__GEN,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@V2s))) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@V3t))) & (p@(ap@(ap@(c_2Epred__set_2EDISJOINT@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@V2s))@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@V3t))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V2s)@V3t))@V1f) = (ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V2s)@V1f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3t)@V1f)))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__DIFF,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@V3t)@V2s))) => ((ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27b)@V2s)@V3t))@V1f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V3t)@V1f)) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V2s)@V1f))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__DIFF__GEN,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@V2s))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@V3t))@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@V2s)))) => ((ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@V2s)@V3t))@V1f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3t)@V1f)) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V2s)@V1f))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__INCL__EXCL,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V0op)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27b@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27b@bool)) => ( ! [V3f:$i] : ((mem@V3f@(arr@A_27b@A_27a)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V1s)) & (p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2t))) => ((ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V1s)@V3f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V2t)@V3f)) = (ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@(ap@(ap@(c_2Epred__set_2EUNION@A_27b)@V1s)@V2t))@V3f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@(ap@(ap@(c_2Epred__set_2EINTER@A_27b)@V1s)@V2t))@V3f)))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__CLOSED,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1P:$i] : ((mem@V1P@(arr@A_27b@bool)) => (((p@(ap@V1P@(ap@(c_2Eiterate_2Eneutral@A_27b)@V0op))) & ( ! [V2x:$i] : ((mem@V2x@A_27b) => ( ! [V3y:$i] : ((mem@V3y@A_27b) => (((p@(ap@V1P@V2x)) & (p@(ap@V1P@V3y))) => (p@(ap@V1P@(ap@(ap@V0op@V2x)@V3y))))))))) => ( ! [V4f:$i] : ((mem@V4f@(arr@A_27a@A_27b)) => ( ! [V5s:$i] : ((mem@V5s@(arr@A_27a@bool)) => (( ! [V6x:$i] : ((mem@V6x@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6x)@V5s)) & (~~ ((ap@V4f@V6x) = (ap@(c_2Eiterate_2Eneutral@A_27b)@V0op)))) => (p@(ap@V1P@(ap@V4f@V6x)))))) => (p@(ap@V1P@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V5s)@V4f))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__RELATED,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1R:$i] : ((mem@V1R@(arr@A_27b@(arr@A_27b@bool))) => (((p@(ap@(ap@V1R@(ap@(c_2Eiterate_2Eneutral@A_27b)@V0op))@(ap@(c_2Eiterate_2Eneutral@A_27b)@V0op))) & ( ! [V2x1:$i] : ((mem@V2x1@A_27b) => ( ! [V3y1:$i] : ((mem@V3y1@A_27b) => ( ! [V4x2:$i] : ((mem@V4x2@A_27b) => ( ! [V5y2:$i] : ((mem@V5y2@A_27b) => (((p@(ap@(ap@V1R@V2x1)@V4x2)) & (p@(ap@(ap@V1R@V3y1)@V5y2))) => (p@(ap@(ap@V1R@(ap@(ap@V0op@V2x1)@V3y1))@(ap@(ap@V0op@V4x2)@V5y2))))))))))))) => ( ! [V6f:$i] : ((mem@V6f@(arr@A_27a@A_27b)) => ( ! [V7g:$i] : ((mem@V7g@(arr@A_27a@A_27b)) => ( ! [V8s:$i] : ((mem@V8s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V8s)) & ( ! [V9x:$i] : ((mem@V9x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V9x)@V8s)) => (p@(ap@(ap@V1R@(ap@V6f@V9x))@(ap@V7g@V9x))))))) => (p@(ap@(ap@V1R@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V8s)@V6f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V8s)@V7g))))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__EQ__NEUTRAL,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => ((ap@V1f@V3x) = (ap@(c_2Eiterate_2Eneutral@A_27b)@V0op))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V2s)@V1f) = (ap@(c_2Eiterate_2Eneutral@A_27b)@V0op))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__SING,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2x:$i] : ((mem@V2x@A_27a) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V2x)@(c_2Epred__set_2EEMPTY@A_27a)))@V1f) = (ap@V1f@V2x)))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__DELETE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3a:$i] : ((mem@V3a@A_27a) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3a)@V2s))) => ((ap@(ap@V0op@(ap@V1f@V3a))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@(ap@(ap@(c_2Epred__set_2EDELETE@A_27a)@V2s)@V3a))@V1f)) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V2s)@V1f))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__DELTA,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2a:$i] : ((mem@V2a@A_27b) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27b@bool)) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V3s)@(lam@A_27b@(^ [V4x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27a)@(ap@(ap@(c_2Emin_2E_3D@A_27b)@V4x)@V2a))@(ap@V1f@V4x))@(ap@(c_2Eiterate_2Eneutral@A_27a)@V0op))))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27a)@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V2a)@V3s))@(ap@V1f@V2a))@(ap@(c_2Eiterate_2Eneutral@A_27a)@V0op))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27c@(arr@A_27c@A_27c))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27c)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27b@A_27c)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27a@bool)) => (( ! [V4x:$i] : ((mem@V4x@A_27a) => ( ! [V5y:$i] : ((mem@V5y@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V3s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5y)@V3s)) & ((ap@V1f@V4x) = (ap@V1f@V5y)))) => (V4x = V5y)))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27c)@V0op)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V1f)@V3s))@V2g) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27c)@V0op)@V3s)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@A_27c@A_27b)@V2g)@V1f))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__BIJECTION,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2p:$i] : ((mem@V2p@(arr@A_27a@A_27a)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27a@bool)) => ((( ! [V4x:$i] : ((mem@V4x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V3s)) => (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@(ap@V2p@V4x))@V3s))))) & ( ! [V5y:$i] : ((mem@V5y@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5y)@V3s)) => (p@(ap@(c_2Ebool_2E_3F_21@A_27a)@(lam@A_27a@(^ [V6x:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6x)@V3s))@(ap@(ap@(c_2Emin_2E_3D@A_27a)@(ap@V2p@V6x))@V5y)))))))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3s)@V1f) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3s)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@A_27b@A_27a)@V1f)@V2p)))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__ITERATE__PRODUCT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27c@(arr@A_27c@A_27c))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27c)@V0op)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@(arr@A_27b@bool))) => ( ! [V3x:$i] : ((mem@V3x@(arr@A_27a@(arr@A_27b@A_27c))) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ( ! [V4i:$i] : ((mem@V4i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4i)@V1s)) => (p@(ap@(c_2Epred__set_2EFINITE@A_27b)@(ap@V2t@V4i))))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27c)@V0op)@V1s)@(lam@A_27a@(^ [V5i:$i] : (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27c)@V0op)@(ap@V2t@V5i))@(ap@V3x@V5i))))) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@(ty_2Epair_2Eprod@A_27a@A_27b)@A_27c)@V0op)@(ap@(c_2Epred__set_2EGSPEC@(ty_2Epair_2Eprod@A_27a@A_27b)@(ty_2Epair_2Eprod@A_27a@A_27b))@(ap@(c_2Epair_2EUNCURRY@A_27a@A_27b@(ty_2Epair_2Eprod@(ty_2Epair_2Eprod@A_27a@A_27b)@bool))@(lam@A_27a@(^ [V6i:$i] : (lam@A_27b@(^ [V7j:$i] : (ap@(ap@(c_2Epair_2E_2C@(ty_2Epair_2Eprod@A_27a@A_27b)@bool)@(ap@(ap@(c_2Epair_2E_2C@A_27a@A_27b)@V6i)@V7j))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6i)@V1s))@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V7j)@(ap@V2t@V6i)))))))))))@(ap@(c_2Epair_2EUNCURRY@A_27a@A_27b@A_27c)@(lam@A_27a@(^ [V8i:$i] : (lam@A_27b@(^ [V9j:$i] : (ap@(ap@V3x@V8i)@V9j)))))))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__EQ,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@A_27b)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27a@bool)) => (( ! [V4x:$i] : ((mem@V4x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V3s)) => ((ap@V1f@V4x) = (ap@V2g@V4x))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3s)@V1f) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3s)@V2g))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27c@(arr@A_27c@A_27c))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27c)@V0op)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27b@bool)) => ( ! [V3f:$i] : ((mem@V3f@(arr@A_27a@A_27c)) => ( ! [V4g:$i] : ((mem@V4g@(arr@A_27b@A_27c)) => ( ! [V5h:$i] : ((mem@V5h@(arr@A_27a@A_27b)) => ((( ! [V6y:$i] : ((mem@V6y@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V6y)@V2t)) => (p@(ap@(c_2Ebool_2E_3F_21@A_27a)@(lam@A_27a@(^ [V7x:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V7x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@A_27b)@(ap@V5h@V7x))@V6y))))))))) & ( ! [V8x:$i] : ((mem@V8x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V8x)@V1s)) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@(ap@V5h@V8x))@V2t)) & ((ap@V4g@(ap@V5h@V8x)) = (ap@V3f@V8x))))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27c)@V0op)@V1s)@V3f) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27c)@V0op)@V2t)@V4g)))))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL__INVERSES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27c@(arr@A_27c@A_27c))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27c)@V0op)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27b@bool)) => ( ! [V3f:$i] : ((mem@V3f@(arr@A_27a@A_27c)) => ( ! [V4g:$i] : ((mem@V4g@(arr@A_27b@A_27c)) => ( ! [V5h:$i] : ((mem@V5h@(arr@A_27a@A_27b)) => ( ! [V6k:$i] : ((mem@V6k@(arr@A_27b@A_27a)) => ((( ! [V7y:$i] : ((mem@V7y@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V7y)@V2t)) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@(ap@V6k@V7y))@V1s)) & ((ap@V5h@(ap@V6k@V7y)) = V7y))))) & ( ! [V8x:$i] : ((mem@V8x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V8x)@V1s)) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@(ap@V5h@V8x))@V2t)) & (((ap@V6k@(ap@V5h@V8x)) = V8x) & ((ap@V4g@(ap@V5h@V8x)) = (ap@V3f@V8x)))))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27c)@V0op)@V1s)@V3f) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27c)@V0op)@V2t)@V4g)))))))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__INJECTION,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2p:$i] : ((mem@V2p@(arr@A_27a@A_27a)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3s)) & (( ! [V4x:$i] : ((mem@V4x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V3s)) => (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@(ap@V2p@V4x))@V3s))))) & ( ! [V5x:$i] : ((mem@V5x@A_27a) => ( ! [V6y:$i] : ((mem@V6y@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V3s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6y)@V3s)) & ((ap@V2p@V5x) = (ap@V2p@V6y)))) => (V5x = V6y)))))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3s)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@A_27b@A_27a)@V1f)@V2p)) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3s)@V1f))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__UNION__NONZERO,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3t)) & ( ! [V4x:$i] : ((mem@V4x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@(ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V2s)@V3t))) => ((ap@V1f@V4x) = (ap@(c_2Eiterate_2Eneutral@A_27b)@V0op))))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V2s)@V3t))@V1f) = (ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V2s)@V1f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3t)@V1f)))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__OP,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27b@A_27a)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27b@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V3s)) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V3s)@(lam@A_27b@(^ [V4x:$i] : (ap@(ap@V0op@(ap@V1f@V4x))@(ap@V2g@V4x))))) = (ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V3s)@V1f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27a)@V0op)@V3s)@V2g)))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__SUPERSET,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2u:$i] : ((mem@V2u@(arr@A_27a@bool)) => ( ! [V3v:$i] : ((mem@V3v@(arr@A_27a@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V2u)@V3v)) & ( ! [V4x:$i] : ((mem@V4x@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V3v)) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2u)))) => ((ap@V1f@V4x) = (ap@(c_2Eiterate_2Eneutral@A_27b)@V0op)))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3v)@V1f) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V2u)@V1f))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__IMAGE__NONZERO,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27c@(arr@A_27c@A_27c))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27c)@V0op)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27b@A_27c)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@A_27b)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3s)) & ( ! [V4x:$i] : ((mem@V4x@A_27a) => ( ! [V5y:$i] : ((mem@V5y@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V3s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5y)@V3s)) & ((~~ (V4x = V5y)) & ((ap@V2f@V4x) = (ap@V2f@V5y))))) => ((ap@V1g@(ap@V2f@V4x)) = (ap@(c_2Eiterate_2Eneutral@A_27c)@V0op)))))))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27b@A_27c)@V0op)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V2f)@V3s))@V1g) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27c)@V0op)@V3s)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@A_27c@A_27b)@V1g)@V2f))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__CASES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2P:$i] : ((mem@V2P@(arr@A_27a@bool)) => ( ! [V3f:$i] : ((mem@V3f@(arr@A_27a@A_27b)) => ( ! [V4g:$i] : ((mem@V4g@(arr@A_27a@A_27b)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V1s)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27b)@(ap@V2P@V5x))@(ap@V3f@V5x))@(ap@V4g@V5x))))) = (ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6x)@V1s))@(ap@V2P@V6x)))))))@V3f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V7x)@V1s))@(ap@c_2Ebool_2E_7E@(ap@V2P@V7x))))))))@V4g)))))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__OP__GEN,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27b@(arr@A_27b@A_27b))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27b)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@A_27b)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V1f)@V3s))) & (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@A_27b)@V0op)@V2g)@V3s)))) => ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3s)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@V0op@(ap@V1f@V4x))@(ap@V2g@V4x))))) = (ap@(ap@V0op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3s)@V1f))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@A_27a@A_27b)@V0op)@V3s)@V2g)))))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__CLAUSES__NUMSEG,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@A_27a)) => ( ! [V1op:$i] : ((mem@V1op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V1op)) => (( ! [V2m:tp__ty_2Enum_2Enum] : ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@ty_2Enum_2Enum@A_27a)@V1op)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0f) = (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27a)@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V0f@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(c_2Eiterate_2Eneutral@A_27a)@V1op)))) & ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@ty_2Enum_2Enum@A_27a)@V1op)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V3m))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V4n))))@V0f) = (ap@(ap@(ap@(c_2Ebool_2ECOND@A_27a)@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V4n))))@(ap@(ap@V1op@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@ty_2Enum_2Enum@A_27a)@V1op)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n)))@V0f))@(ap@V0f@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V4n)))))@(ap@(ap@(ap@(c_2Eiterate_2Eiterate@ty_2Enum_2Enum@A_27a)@V1op)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n)))@V0f))))))))))))).
thf(conj_thm_2Eiterate_2EITERATE__PAIR,lemma,( ! [A_27a:del]: ( ! [V0op:$i] : ((mem@V0op@(arr@A_27a@(arr@A_27a@A_27a))) => ((p@(ap@(c_2Eiterate_2Emonoidal@A_27a)@V0op)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@A_27a)) => ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((ap@(ap@(ap@(c_2Eiterate_2Eiterate@ty_2Enum_2Enum@A_27a)@V0op)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2A@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V2m)))@(ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@c_2Earithmetic_2E_2A@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V3n)))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@V1f) = (ap@(ap@(ap@(c_2Eiterate_2Eiterate@ty_2Enum_2Enum@A_27a)@V0op)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@(lam@ty_2Enum_2Enum@(^ [V4i:$i] : (ap@(ap@V0op@(ap@V1f@(ap@(ap@c_2Earithmetic_2E_2A@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@V4i)))@(ap@V1f@(ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@c_2Earithmetic_2E_2A@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@V4i))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))))))))))).
thf(ax_thm_2Eiterate_2Ensum__def,axiom,( ! [A_27a:del]: ((c_2Eiterate_2Ensum@A_27a) = (ap@(c_2Eiterate_2Eiterate@A_27a@ty_2Enum_2Enum)@c_2Earithmetic_2E_2B)))).
thf(conj_thm_2Eiterate_2ENEUTRAL__ADD,lemma,((surj__ty_2Enum_2Enum @ (ap@(c_2Eiterate_2Eneutral@ty_2Enum_2Enum)@c_2Earithmetic_2E_2B)) = fo__c_2Enum_2E0)).
thf(conj_thm_2Eiterate_2ENEUTRAL__MUL,lemma,((surj__ty_2Enum_2Enum @ (ap@(c_2Eiterate_2Eneutral@ty_2Enum_2Enum)@c_2Earithmetic_2E_2A)) = (surj__ty_2Enum_2Enum @ (ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))).
thf(conj_thm_2Eiterate_2EMONOIDAL__ADD,lemma,(p@(ap@(c_2Eiterate_2Emonoidal@ty_2Enum_2Enum)@c_2Earithmetic_2E_2B))).
thf(conj_thm_2Eiterate_2EMONOIDAL__MUL,lemma,(p@(ap@(c_2Eiterate_2Emonoidal@ty_2Enum_2Enum)@c_2Earithmetic_2E_2A))).
thf(conj_thm_2Eiterate_2ENSUM__DEGENERATE,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((~~ (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V2x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(ap@V0f@V2x))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f)) = fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eiterate_2ENSUM__CLAUSES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: (( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(c_2Epred__set_2EEMPTY@A_27a))@V0f)) = fo__c_2Enum_2E0))) & ( ! [V1x:$i] : ((mem@V1x@A_27b) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27b@ty_2Enum_2Enum)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27b@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V3s)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27b)@V1x)@V3s))@V2f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Enum_2Enum)@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V1x)@V3s))@(ap@(ap@(c_2Eiterate_2Ensum@A_27b)@V3s)@V2f))@(ap@(ap@c_2Earithmetic_2E_2B@(ap@V2f@V1x))@(ap@(ap@(c_2Eiterate_2Ensum@A_27b)@V3s)@V2f)))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__UNION,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2t)) & (p@(ap@(ap@(c_2Epred__set_2EDISJOINT@A_27a)@V1s)@V2t)))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V1s)@V2t))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2t)@V0f))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__DIFF,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V2t)@V1s))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@V1s)@V2t))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2D@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2t)@V0f))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__INCL__EXCL,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Enum_2Enum)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1t))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@V2f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1t)@V2f))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V0s)@V1t))@V2f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V0s)@V1t))@V2f))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__SUPPORT,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@ty_2Enum_2Enum)@c_2Earithmetic_2E_2B)@V0f)@V1s))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f))))))))).
thf(conj_thm_2Eiterate_2ENSUM__ADD,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Earithmetic_2E_2B@(ap@V0f@V3x))@(ap@V1g@V3x)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1g))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__ADD__GEN,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(ap@V0f@V3x))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))) & (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(ap@V1g@V4x))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Earithmetic_2E_2B@(ap@V0f@V5x))@(ap@V1g@V5x)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1g))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__EQ__0,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => ((surj__ty_2Enum_2Enum @ (ap@V0f@V2x)) = fo__c_2Enum_2E0)))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f)) = fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eiterate_2ENSUM__0,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@(lam@A_27a@(^ [V1n:$i] : (inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) = fo__c_2Enum_2E0))))).
thf(conj_thm_2Eiterate_2ENSUM__LMUL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1c:tp__ty_2Enum_2Enum] : ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Earithmetic_2E_2A@(inj__ty_2Enum_2Enum @ V1c))@(ap@V0f@V3x)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2A@(inj__ty_2Enum_2Enum @ V1c))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__RMUL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1c:tp__ty_2Enum_2Enum] : ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Earithmetic_2E_2A@(ap@V0f@V3x))@(inj__ty_2Enum_2Enum @ V1c)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2A@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))@(inj__ty_2Enum_2Enum @ V1c))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__LE,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@V0f@V3x))@(ap@V1g@V3x))))))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1g)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__LT,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & (( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@V0f@V3x))@(ap@V1g@V3x)))))) & (? [V4x:$i] : ((mem@V4x@A_27a) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2s)) & (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V0f@V4x))@(ap@V1g@V4x)))))))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1g)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__LT__ALL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & ((~~ (V2s = (c_2Epred__set_2EEMPTY@A_27a))) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V0f@V3x))@(ap@V1g@V3x)))))))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1g)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__EQ,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => ((surj__ty_2Enum_2Enum @ (ap@V0f@V3x)) = (surj__ty_2Enum_2Enum @ (ap@V1g@V3x)))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1g)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__CONST,lemma,( ! [A_27a:del]: ( ! [V0c:tp__ty_2Enum_2Enum] : ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@(lam@A_27a@(^ [V2n:$i] : (inj__ty_2Enum_2Enum @ V0c))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2A@(ap@(c_2Epred__set_2ECARD@A_27a)@V1s))@(inj__ty_2Enum_2Enum @ V0c)))))))))).
thf(conj_thm_2Eiterate_2ENSUM__POS__BOUND,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1b:tp__ty_2Enum_2Enum] : ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))@(inj__ty_2Enum_2Enum @ V1b)))) => ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@V0f@V3x))@(inj__ty_2Enum_2Enum @ V1b)))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) => (((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f)) = fo__c_2Enum_2E0) <=> ( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => ((surj__ty_2Enum_2Enum @ (ap@V0f@V2x)) = fo__c_2Enum_2E0)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__POS__LT,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & (? [V2x:$i] : ((mem@V2x@A_27a) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) & (p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(ap@V0f@V2x))))))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f)))))))))).
thf(conj_thm_2Eiterate_2ENSUM__POS__LT__ALL,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Enum_2Enum)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@A_27a))) & ( ! [V2i:$i] : ((mem@V2i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2i)@V0s)) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(ap@V1f@V2i)))))))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@V1f)))))))))).
thf(conj_thm_2Eiterate_2ENSUM__DELETE,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2a:$i] : ((mem@V2a@A_27a) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2a)@V1s))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@V0f@V2a))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(c_2Epred__set_2EDELETE@A_27a)@V1s)@V2a))@V0f))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__SING,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1x:$i] : ((mem@V1x@A_27a) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1x)@(c_2Epred__set_2EEMPTY@A_27a)))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@V0f@V1x))))))))).
thf(conj_thm_2Eiterate_2ENSUM__DELTA,lemma,( ! [A_27a:del]: ( ! [V0b:tp__ty_2Enum_2Enum] : ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2a:$i] : ((mem@V2a@A_27a) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Enum_2Enum)@(ap@(ap@(c_2Emin_2E_3D@A_27a)@V3x)@V2a))@(inj__ty_2Enum_2Enum @ V0b))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Enum_2Enum)@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2a)@V1s))@(inj__ty_2Enum_2Enum @ V0b))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__SWAP,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@ty_2Enum_2Enum))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & (p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2t))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@(lam@A_27a@(^ [V3i:$i] : (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@V2t)@(ap@V0f@V3i)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@V2t)@(lam@A_27b@(^ [V4j:$i] : (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@(lam@A_27a@(^ [V5i:$i] : (ap@(ap@V0f@V5i)@V4j))))))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@A_27b)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27b@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (( ! [V3x:$i] : ((mem@V3x@A_27a) => ( ! [V4y:$i] : ((mem@V4y@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4y)@V2s)) & ((ap@V0f@V3x) = (ap@V0f@V4y)))) => (V3x = V4y)))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@V2s))@V1g)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@ty_2Enum_2Enum@A_27b)@V1g)@V0f)))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__SUPERSET,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => ( ! [V2v:$i] : ((mem@V2v@(arr@A_27a@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V1u)@V2v)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2v)) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1u)))) => ((surj__ty_2Enum_2Enum @ (ap@V0f@V3x)) = fo__c_2Enum_2E0))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2v)@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1u)@V0f)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__UNION__RZERO,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => ( ! [V2v:$i] : ((mem@V2v@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1u)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2v)) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1u)))) => ((surj__ty_2Enum_2Enum @ (ap@V0f@V3x)) = fo__c_2Enum_2E0))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V1u)@V2v))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1u)@V0f)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__UNION__LZERO,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => ( ! [V2v:$i] : ((mem@V2v@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2v)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1u)) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2v)))) => ((surj__ty_2Enum_2Enum @ (ap@V0f@V3x)) = fo__c_2Enum_2E0))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V1u)@V2v))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2v)@V0f)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__RESTRICT,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@(lam@A_27a@(^ [V2x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Enum_2Enum)@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s))@(ap@V0f@V2x))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f)))))))))).
thf(conj_thm_2Eiterate_2ENSUM__BOUND,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2b:tp__ty_2Enum_2Enum] : (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V0s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@V1f@V3x))@(inj__ty_2Enum_2Enum @ V2b))))))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@V1f))@(ap@(ap@c_2Earithmetic_2E_2A@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s))@(inj__ty_2Enum_2Enum @ V2b)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__BOUND__GEN,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2b:tp__ty_2Enum_2Enum] : (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@A_27a))) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V0s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@V1f@V3x))@(ap@(ap@c_2Earithmetic_2EDIV@(inj__ty_2Enum_2Enum @ V2b))@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s))))))))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@V1f))@(inj__ty_2Enum_2Enum @ V2b))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__BOUND__LT,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2b:tp__ty_2Enum_2Enum] : (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & (( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V0s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@V1f@V3x))@(inj__ty_2Enum_2Enum @ V2b)))))) & (? [V4x:$i] : ((mem@V4x@A_27a) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V0s)) & (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V1f@V4x))@(inj__ty_2Enum_2Enum @ V2b)))))))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@V1f))@(ap@(ap@c_2Earithmetic_2E_2A@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s))@(inj__ty_2Enum_2Enum @ V2b)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__BOUND__LT__ALL,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2b:tp__ty_2Enum_2Enum] : (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@A_27a))) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V0s)) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V1f@V3x))@(inj__ty_2Enum_2Enum @ V2b)))))))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@V1f))@(ap@(ap@c_2Earithmetic_2E_2A@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s))@(inj__ty_2Enum_2Enum @ V2b)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__BOUND__LT__GEN,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2b:tp__ty_2Enum_2Enum] : (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@A_27a))) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V0s)) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V1f@V3x))@(ap@(ap@c_2Earithmetic_2EDIV@(inj__ty_2Enum_2Enum @ V2b))@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s))))))))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@V1f))@(inj__ty_2Enum_2Enum @ V2b))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__UNION__EQ,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3u)) & (((ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V1s)@V2t) = (c_2Epred__set_2EEMPTY@A_27a)) & ((ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V1s)@V2t) = V3u))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2t)@V0f))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V3u)@V0f)))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__EQ__SUPERSET,lemma,( ! [A_27a:del]: ( ! [V0g:$i] : ((mem@V0g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V3t)@V2s)) & (( ! [V4x:$i] : ((mem@V4x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V3t)) => ((surj__ty_2Enum_2Enum @ (ap@V1f@V4x)) = (surj__ty_2Enum_2Enum @ (ap@V0g@V4x)))))) & ( ! [V5x:$i] : ((mem@V5x@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V2s)) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V3t)))) => ((surj__ty_2Enum_2Enum @ (ap@V1f@V5x)) = fo__c_2Enum_2E0))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V3t)@V0g)))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__RESTRICT__SET,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Enum_2Enum)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1s))@(ap@V0P@V3x)))))))@V2f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Enum_2Enum)@(ap@V0P@V4x))@(ap@V2f@V4x))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__NSUM__RESTRICT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0R:$i] : ((mem@V0R@(arr@A_27a@(arr@A_27b@bool))) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@A_27b@ty_2Enum_2Enum))) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & (p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V3t))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@(ap@(c_2Epred__set_2EGSPEC@A_27b@A_27b)@(lam@A_27b@(^ [V5y:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@V5y)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V5y)@V3t))@(ap@(ap@V0R@V4x)@V5y)))))))@(lam@A_27b@(^ [V6y:$i] : (ap@(ap@V1f@V4x)@V6y)))))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@V3t)@(lam@A_27b@(^ [V7y:$i] : (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V8x)@V2s))@(ap@(ap@V0R@V8x)@V7y)))))))@(lam@A_27a@(^ [V9x:$i] : (ap@(ap@V1f@V9x)@V7y))))))))))))))))))))).
thf(conj_thm_2Eiterate_2ECARD__EQ__NSUM,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) => ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@A_27a)@V0s)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@(lam@A_27a@(^ [V1x:$i] : (ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__MULTICOUNT__GEN,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0R:$i] : ((mem@V0R@(arr@A_27a@(arr@A_27b@bool))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27b@bool)) => ( ! [V3k:$i] : ((mem@V3k@(arr@A_27b@ty_2Enum_2Enum)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2t)) & ( ! [V4j:$i] : ((mem@V4j@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V4j)@V2t)) => ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V5i:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V5i)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5i)@V1s))@(ap@(ap@V0R@V5i)@V4j)))))))) = (surj__ty_2Enum_2Enum @ (ap@V3k@V4j)))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@(lam@A_27a@(^ [V6i:$i] : (ap@(c_2Epred__set_2ECARD@A_27b)@(ap@(c_2Epred__set_2EGSPEC@A_27b@A_27b)@(lam@A_27b@(^ [V7j:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@V7j)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V7j)@V2t))@(ap@(ap@V0R@V6i)@V7j))))))))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@V2t)@(lam@A_27b@(^ [V8i:$i] : (ap@V3k@V8i)))))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__MULTICOUNT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0R:$i] : ((mem@V0R@(arr@A_27a@(arr@A_27b@bool))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27b@bool)) => ( ! [V3k:tp__ty_2Enum_2Enum] : (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2t)) & ( ! [V4j:$i] : ((mem@V4j@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V4j)@V2t)) => ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V5i:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V5i)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5i)@V1s))@(ap@(ap@V0R@V5i)@V4j)))))))) = V3k)))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@(lam@A_27a@(^ [V6i:$i] : (ap@(c_2Epred__set_2ECARD@A_27b)@(ap@(c_2Epred__set_2EGSPEC@A_27b@A_27b)@(lam@A_27b@(^ [V7j:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@V7j)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V7j)@V2t))@(ap@(ap@V0R@V6i)@V7j))))))))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2A@(inj__ty_2Enum_2Enum @ V3k))@(ap@(c_2Epred__set_2ECARD@A_27b)@V2t))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__IMAGE__GEN,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@A_27b)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1g)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@V2s))@(lam@A_27b@(^ [V3y:$i] : (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2s))@(ap@(ap@(c_2Emin_2E_3D@A_27b)@(ap@V0f@V4x))@V3y)))))))@V1g)))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__GROUP,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@A_27b)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@V2s))@V3t))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@V3t)@(lam@A_27b@(^ [V4y:$i] : (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V2s))@(ap@(ap@(c_2Emin_2E_3D@A_27b)@(ap@V0f@V5x))@V4y)))))))@V1g))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1g))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__SUBSET,lemma,( ! [A_27a:del]: ( ! [V0u:$i] : ((mem@V0u@(arr@A_27a@bool)) => ( ! [V1v:$i] : ((mem@V1v@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Enum_2Enum)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0u)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1v)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@V0u)@V1v))) => ((surj__ty_2Enum_2Enum @ (ap@V2f@V3x)) = fo__c_2Enum_2E0)))))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0u)@V2f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1v)@V2f)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__SUBSET__SIMPLE,lemma,( ! [A_27a:del]: ( ! [V0u:$i] : ((mem@V0u@(arr@A_27a@bool)) => ( ! [V1v:$i] : ((mem@V1v@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Enum_2Enum)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1v)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V0u)@V1v))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0u)@V2f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1v)@V2f)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__LE__GEN,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@V0f@V3x))@(ap@V1g@V3x)))))) & (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(ap@V1g@V4x))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1g)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__IMAGE__NONZERO,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0d:$i] : ((mem@V0d@(arr@A_27b@ty_2Enum_2Enum)) => ( ! [V1i:$i] : ((mem@V1i@(arr@A_27a@A_27b)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ( ! [V4y:$i] : ((mem@V4y@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4y)@V2s)) & ((~~ (V3x = V4y)) & ((ap@V1i@V3x) = (ap@V1i@V4y))))) => ((surj__ty_2Enum_2Enum @ (ap@V0d@(ap@V1i@V3x))) = fo__c_2Enum_2E0))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V1i)@V2s))@V0d)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@ty_2Enum_2Enum@A_27b)@V0d)@V1i)))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__BIJECTION,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1p:$i] : ((mem@V1p@(arr@A_27a@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@(ap@V1p@V3x))@V2s))))) & ( ! [V4y:$i] : ((mem@V4y@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4y)@V2s)) => (p@(ap@(c_2Ebool_2E_3F_21@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V2s))@(ap@(ap@(c_2Emin_2E_3D@A_27a)@(ap@V1p@V5x))@V4y)))))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@ty_2Enum_2Enum@A_27a)@V0f)@V1p))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__NSUM__PRODUCT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@(arr@A_27b@bool))) => ( ! [V2x:$i] : ((mem@V2x@(arr@A_27a@(arr@A_27b@ty_2Enum_2Enum))) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & ( ! [V3i:$i] : ((mem@V3i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3i)@V0s)) => (p@(ap@(c_2Epred__set_2EFINITE@A_27b)@(ap@V1t@V3i))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@(lam@A_27a@(^ [V4i:$i] : (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@(ap@V1t@V4i))@(ap@V2x@V4i)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@(ty_2Epair_2Eprod@A_27a@A_27b))@(ap@(c_2Epred__set_2EGSPEC@(ty_2Epair_2Eprod@A_27a@A_27b)@(ty_2Epair_2Eprod@A_27a@A_27b))@(ap@(c_2Epair_2EUNCURRY@A_27a@A_27b@(ty_2Epair_2Eprod@(ty_2Epair_2Eprod@A_27a@A_27b)@bool))@(lam@A_27a@(^ [V5i:$i] : (lam@A_27b@(^ [V6j:$i] : (ap@(ap@(c_2Epair_2E_2C@(ty_2Epair_2Eprod@A_27a@A_27b)@bool)@(ap@(ap@(c_2Epair_2E_2C@A_27a@A_27b)@V5i)@V6j))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5i)@V0s))@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V6j)@(ap@V1t@V5i)))))))))))@(ap@(c_2Epair_2EUNCURRY@A_27a@A_27b@ty_2Enum_2Enum)@(lam@A_27a@(^ [V7i:$i] : (lam@A_27b@(^ [V8j:$i] : (ap@(ap@V2x@V7i)@V8j))))))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27b@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V3g:$i] : ((mem@V3g@(arr@A_27b@ty_2Enum_2Enum)) => ( ! [V4h:$i] : ((mem@V4h@(arr@A_27a@A_27b)) => ((( ! [V5y:$i] : ((mem@V5y@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V5y)@V1t)) => (p@(ap@(c_2Ebool_2E_3F_21@A_27a)@(lam@A_27a@(^ [V6x:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6x)@V0s))@(ap@(ap@(c_2Emin_2E_3D@A_27b)@(ap@V4h@V6x))@V5y))))))))) & ( ! [V7x:$i] : ((mem@V7x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V7x)@V0s)) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@(ap@V4h@V7x))@V1t)) & ((surj__ty_2Enum_2Enum @ (ap@V3g@(ap@V4h@V7x))) = (surj__ty_2Enum_2Enum @ (ap@V2f@V7x)))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@V2f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@V1t)@V3g))))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL__INVERSES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27b@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V3g:$i] : ((mem@V3g@(arr@A_27b@ty_2Enum_2Enum)) => ( ! [V4h:$i] : ((mem@V4h@(arr@A_27a@A_27b)) => ( ! [V5k:$i] : ((mem@V5k@(arr@A_27b@A_27a)) => ((( ! [V6y:$i] : ((mem@V6y@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V6y)@V1t)) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@(ap@V5k@V6y))@V0s)) & ((ap@V4h@(ap@V5k@V6y)) = V6y))))) & ( ! [V7x:$i] : ((mem@V7x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V7x)@V0s)) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@(ap@V4h@V7x))@V1t)) & (((ap@V5k@(ap@V4h@V7x)) = V7x) & ((surj__ty_2Enum_2Enum @ (ap@V3g@(ap@V4h@V7x))) = (surj__ty_2Enum_2Enum @ (ap@V2f@V7x))))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@V2f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27b)@V1t)@V3g))))))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__INJECTION,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1p:$i] : ((mem@V1p@(arr@A_27a@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & (( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@(ap@V1p@V3x))@V2s))))) & ( ! [V4x:$i] : ((mem@V4x@A_27a) => ( ! [V5y:$i] : ((mem@V5y@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5y)@V2s)) & ((ap@V1p@V4x) = (ap@V1p@V5y)))) => (V4x = V5y)))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@ty_2Enum_2Enum@A_27a)@V0f)@V1p))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__UNION__NONZERO,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2t)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@(ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V1s)@V2t))) => ((surj__ty_2Enum_2Enum @ (ap@V0f@V3x)) = fo__c_2Enum_2E0)))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V1s)@V2t))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V1s)@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2t)@V0f))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__BIGUNION__NONZERO,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@(arr@A_27a@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@A_27a@bool))@V1s)) & (( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V2t)@V1s)) => (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2t))))) & ( ! [V3t1:$i] : ((mem@V3t1@(arr@A_27a@bool)) => ( ! [V4t2:$i] : ((mem@V4t2@(arr@A_27a@bool)) => ( ! [V5x:$i] : ((mem@V5x@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V3t1)@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V4t2)@V1s)) & ((~~ (V3t1 = V4t2)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V3t1)) & (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V4t2)))))) => ((surj__ty_2Enum_2Enum @ (ap@V0f@V5x)) = fo__c_2Enum_2E0)))))))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(c_2Epred__set_2EBIGUNION@A_27a)@V1s))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@(arr@A_27a@bool))@V1s)@(lam@(arr@A_27a@bool)@(^ [V6t:$i] : (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V6t)@V0f))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__CASES,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1P:$i] : ((mem@V1P@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V3g:$i] : ((mem@V3g@(arr@A_27a@ty_2Enum_2Enum)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V0s)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Enum_2Enum)@(ap@V1P@V4x))@(ap@V2f@V4x))@(ap@V3g@V4x)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V0s))@(ap@V1P@V5x)))))))@V2f))@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6x)@V0s))@(ap@c_2Ebool_2E_7E@(ap@V1P@V6x))))))))@V3g))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__CLOSED,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@V0P@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))) & (( ! [V3x:tp__ty_2Enum_2Enum] : ( ! [V4y:tp__ty_2Enum_2Enum] : (((p@(ap@V0P@(inj__ty_2Enum_2Enum @ V3x))) & (p@(ap@V0P@(inj__ty_2Enum_2Enum @ V4y)))) => (p@(ap@V0P@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V3x))@(inj__ty_2Enum_2Enum @ V4y))))))) & ( ! [V5a:$i] : ((mem@V5a@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5a)@V2s)) => (p@(ap@V0P@(ap@V1f@V5a)))))))) => (p@(ap@V0P@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V1f)))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__ADD__NUMSEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@(lam@ty_2Enum_2Enum@(^ [V4i:$i] : (ap@(ap@c_2Earithmetic_2E_2B@(ap@V0f@V4i))@(ap@V1g@V4i)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@V1g))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__LE__NUMSEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : (( ! [V4i:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V4i))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V4i))@(inj__ty_2Enum_2Enum @ V3n)))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@V0f@(inj__ty_2Enum_2Enum @ V4i)))@(ap@V1g@(inj__ty_2Enum_2Enum @ V4i)))))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@V1g))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__EQ__NUMSEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : (( ! [V4i:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V4i))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V4i))@(inj__ty_2Enum_2Enum @ V3n)))) => ((surj__ty_2Enum_2Enum @ (ap@V0f@(inj__ty_2Enum_2Enum @ V4i))) = (surj__ty_2Enum_2Enum @ (ap@V1g@(inj__ty_2Enum_2Enum @ V4i)))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@V1g))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__CONST__NUMSEG,lemma,( ! [V0c:tp__ty_2Enum_2Enum] : ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (inj__ty_2Enum_2Enum @ V0c))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2A@(ap@(ap@c_2Earithmetic_2E_2D@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V1m)))@(inj__ty_2Enum_2Enum @ V0c)))))))).
thf(conj_thm_2Eiterate_2ENSUM__EQ__0__NUMSEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : (( ! [V3i:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V3i))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3i))@(inj__ty_2Enum_2Enum @ V2n)))) => ((surj__ty_2Enum_2Enum @ (ap@V0f@(inj__ty_2Enum_2Enum @ V3i))) = fo__c_2Enum_2E0))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@V0f)) = fo__c_2Enum_2E0))))))).
thf(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF__NUMSEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : (((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@V0f)) = fo__c_2Enum_2E0) <=> ( ! [V3i:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V3i))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3i))@(inj__ty_2Enum_2Enum @ V2n)))) => ((surj__ty_2Enum_2Enum @ (ap@V0f@(inj__ty_2Enum_2Enum @ V3i))) = fo__c_2Enum_2E0))))))))).
thf(conj_thm_2Eiterate_2ENSUM__TRIV__NUMSEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V2n))@(inj__ty_2Enum_2Enum @ V1m))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@V0f)) = fo__c_2Enum_2E0))))))).
thf(conj_thm_2Eiterate_2ENSUM__SING__NUMSEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V1n)))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@V0f@(inj__ty_2Enum_2Enum @ V1n)))))))).
thf(conj_thm_2Eiterate_2ENSUM__CLAUSES__NUMSEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => (( ! [V1m:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Enum_2Enum)@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V0f@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) & ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V3n))))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Enum_2Enum)@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2m))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V3n))))@(ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@V0f))@(ap@V0f@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V3n)))))@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@V0f)))))))))).
thf(conj_thm_2Eiterate_2ENSUM__SWAP__NUMSEG,lemma,( ! [V0a:tp__ty_2Enum_2Enum] : ( ! [V1b:tp__ty_2Enum_2Enum] : ( ! [V2c:tp__ty_2Enum_2Enum] : ( ! [V3d:tp__ty_2Enum_2Enum] : ( ! [V4f:$i] : ((mem@V4f@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0a))@(inj__ty_2Enum_2Enum @ V1b)))@(lam@ty_2Enum_2Enum@(^ [V5i:$i] : (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2c))@(inj__ty_2Enum_2Enum @ V3d)))@(ap@V4f@V5i)))))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2c))@(inj__ty_2Enum_2Enum @ V3d)))@(lam@ty_2Enum_2Enum@(^ [V6j:$i] : (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0a))@(inj__ty_2Enum_2Enum @ V1b)))@(lam@ty_2Enum_2Enum@(^ [V7i:$i] : (ap@(ap@V4f@V7i)@V6j)))))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__ADD__SPLIT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ( ! [V3p:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2n))@(inj__ty_2Enum_2Enum @ V3p))))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@V0f))@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2n))@(inj__ty_2Enum_2Enum @ V3p))))@V0f))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__OFFSET,lemma,( ! [V0p:tp__ty_2Enum_2Enum] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V0p)))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V3n))@(inj__ty_2Enum_2Enum @ V0p))))@V1f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n)))@(lam@ty_2Enum_2Enum@(^ [V4i:$i] : (ap@V1f@(ap@(ap@c_2Earithmetic_2E_2B@V4i)@(inj__ty_2Enum_2Enum @ V0p)))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__OFFSET__0,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V2n))@(inj__ty_2Enum_2Enum @ V1m))))@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@V3i)@(inj__ty_2Enum_2Enum @ V1m)))))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__CLAUSES__LEFT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@V0f@(inj__ty_2Enum_2Enum @ V1m)))@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1m))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V2n)))@V0f)))))))))).
thf(conj_thm_2Eiterate_2ENSUM__CLAUSES__RIGHT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V2n))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V2n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@V0f))@(ap@V0f@(inj__ty_2Enum_2Enum @ V2n))))))))))).
thf(conj_thm_2Eiterate_2ENSUM__PAIR,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2A@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V1m)))@(ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@c_2Earithmetic_2E_2A@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V2n)))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@V0f)) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@(ap@c_2Earithmetic_2E_2B@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2A@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@V3i)))@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@c_2Earithmetic_2E_2A@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@V3i))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))))))))).
thf(conj_thm_2Eiterate_2EMOD__NSUM__MOD,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1n:tp__ty_2Enum_2Enum] : ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & (~~ (V1n = fo__c_2Enum_2E0))) => ((surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2EMOD@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@V0f))@(inj__ty_2Enum_2Enum @ V1n))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2EMOD@(ap@(ap@(c_2Eiterate_2Ensum@A_27a)@V2s)@(lam@A_27a@(^ [V3i:$i] : (ap@(ap@c_2Earithmetic_2EMOD@(ap@V0f@V3i))@(inj__ty_2Enum_2Enum @ V1n))))))@(inj__ty_2Enum_2Enum @ V1n)))))))))))).
thf(conj_thm_2Eiterate_2EMOD__NSUM__MOD__NUMSEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V1a:tp__ty_2Enum_2Enum] : ( ! [V2b:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((~~ (V3n = fo__c_2Enum_2E0)) => ((surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2EMOD@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1a))@(inj__ty_2Enum_2Enum @ V2b)))@V0f))@(inj__ty_2Enum_2Enum @ V3n))) = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2EMOD@(ap@(ap@(c_2Eiterate_2Ensum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1a))@(inj__ty_2Enum_2Enum @ V2b)))@(lam@ty_2Enum_2Enum@(^ [V4i:$i] : (ap@(ap@c_2Earithmetic_2EMOD@(ap@V0f@V4i))@(inj__ty_2Enum_2Enum @ V3n))))))@(inj__ty_2Enum_2Enum @ V3n))))))))))).
thf(conj_thm_2Eiterate_2ECARD__BIGUNION,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@(arr@A_27a@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@A_27a@bool))@V0s)) & (( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V1t)@V0s)) => (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1t))))) & ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@A_27a@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V2t)@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V3u)@V0s)) & (~~ (V2t = V3u)))) => ((ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V2t)@V3u) = (c_2Epred__set_2EEMPTY@A_27a))))))))) => ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@A_27a)@(ap@(c_2Epred__set_2EBIGUNION@A_27a)@V0s))) = (surj__ty_2Enum_2Enum @ (ap@(ap@(c_2Eiterate_2Ensum@(arr@A_27a@bool))@V0s)@(c_2Epred__set_2ECARD@A_27a))))))))).
thf(ax_thm_2Eiterate_2Esum__def,axiom,( ! [A_27a:del]: ((c_2Eiterate_2ESum@A_27a) = (ap@(c_2Eiterate_2Eiterate@A_27a@ty_2Erealax_2Ereal)@c_2Erealax_2Ereal__add)))).
thf(conj_thm_2Eiterate_2ENEUTRAL__REAL__ADD,lemma,((surj__ty_2Erealax_2Ereal @ (ap@(c_2Eiterate_2Eneutral@ty_2Erealax_2Ereal)@c_2Erealax_2Ereal__add)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))).
thf(conj_thm_2Eiterate_2ENEUTRAL__REAL__MUL,lemma,((surj__ty_2Erealax_2Ereal @ (ap@(c_2Eiterate_2Eneutral@ty_2Erealax_2Ereal)@c_2Erealax_2Ereal__mul)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))).
thf(conj_thm_2Eiterate_2EMONOIDAL__REAL__ADD,lemma,(p@(ap@(c_2Eiterate_2Emonoidal@ty_2Erealax_2Ereal)@c_2Erealax_2Ereal__add))).
thf(conj_thm_2Eiterate_2EMONOIDAL__REAL__MUL,lemma,(p@(ap@(c_2Eiterate_2Emonoidal@ty_2Erealax_2Ereal)@c_2Erealax_2Ereal__mul))).
thf(conj_thm_2Eiterate_2ESUM__DEGENERATE,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((~~ (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V2x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V2x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@V0f)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))).
thf(conj_thm_2Eiterate_2ESUM__CLAUSES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: (( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@(c_2Epred__set_2EEMPTY@A_27a))@V0f)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) & ( ! [V1x:$i] : ((mem@V1x@A_27b) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27b@ty_2Erealax_2Ereal)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27b@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V3s)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27b)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27b)@V1x)@V3s))@V2f)) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V1x)@V3s))@(ap@(ap@(c_2Eiterate_2ESum@A_27b)@V3s)@V2f))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@V2f@V1x))@(ap@(ap@(c_2Eiterate_2ESum@A_27b)@V3s)@V2f)))))))))))))))).
thf(conj_thm_2Eiterate_2ESUM__UNION,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2t)) & (p@(ap@(ap@(c_2Epred__set_2EDISJOINT@A_27a)@V1s)@V2t)))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V1s)@V2t))@V0f)) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@V0f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2t)@V0f))))))))))))).
thf(conj_thm_2Eiterate_2ESUM__DIFF,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V2t)@V1s))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@V1s)@V2t))@V0f)) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@V0f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2t)@V0f))))))))))))).
thf(conj_thm_2Eiterate_2ESUM__INCL__EXCL,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1t))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V0s)@V2f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1t)@V2f))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V0s)@V1t))@V2f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@(ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V0s)@V1t))@V2f))))))))))))).
thf(conj_thm_2Eiterate_2ESUM__SUPPORT,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@(ap@(ap@(ap@(c_2Eiterate_2Esupport@A_27a@ty_2Erealax_2Ereal)@c_2Erealax_2Ereal__add)@V0f)@V1s))@V0f)) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@V0f))))))))).
thf(conj_thm_2Eiterate_2ESUM__ADD,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0f@V3x))@(ap@V1g@V3x)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V1g))))))))))))).
thf(conj_thm_2Eiterate_2ESUM__ADD__GEN,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))) & (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V1g@V4x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0f@V5x))@(ap@V1g@V5x)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V1g))))))))))))).
thf(conj_thm_2Eiterate_2ESUM__EQ__0,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@V2x)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@V0f)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))).
thf(conj_thm_2Eiterate_2ESUM__0,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V0s)@(lam@A_27a@(^ [V1n:$i] : (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eiterate_2ESUM__LMUL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(ap@V0f@V3x)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V0f))))))))))).
thf(conj_thm_2Eiterate_2ESUM__RMUL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V0f@V3x))@(inj__ty_2Erealax_2Ereal @ V1c)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V0f))@(inj__ty_2Erealax_2Ereal @ V1c))))))))))).
thf(conj_thm_2Eiterate_2ESUM__NEG,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@(lam@A_27a@(^ [V2x:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V0f@V2x)))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Erealax_2Ereal__neg@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@V0f)))))))))).
thf(conj_thm_2Eiterate_2ESUM__SUB,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@V3x))@(ap@V1g@V3x)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V1g))))))))))))).
thf(conj_thm_2Eiterate_2ESUM__LE,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@V3x))@(ap@V1g@V3x))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V1g)))))))))))).
thf(conj_thm_2Eiterate_2ESUM__LT,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & (( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@V3x))@(ap@V1g@V3x)))))) & (? [V4x:$i] : ((mem@V4x@A_27a) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@V0f@V4x))@(ap@V1g@V4x)))))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V1g)))))))))))).
thf(conj_thm_2Eiterate_2ESUM__LT__ALL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & ((~~ (V2s = (c_2Epred__set_2EEMPTY@A_27a))) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@V0f@V3x))@(ap@V1g@V3x)))))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V0f))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V1g)))))))))))).
thf(conj_thm_2Eiterate_2ESUM__POS__LT,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & (( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V0f@V2x)))))) & (? [V3x:$i] : ((mem@V3x@A_27a) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V0f@V3x)))))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@V0f)))))))))).
thf(conj_thm_2Eiterate_2ESUM__POS__LT__ALL,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@A_27a))) & ( ! [V2i:$i] : ((mem@V2i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2i)@V0s)) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V1f@V2i)))))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V0s)@V1f)))))))))).
thf(conj_thm_2Eiterate_2ESUM__EQ,conjecture,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@V3x)) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@V3x)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V0f)) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V1g)))))))))))).
