include('Axioms/lift_machine_ieee+2.ax').
fof(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)))) )).
fof(mem_c_2Eiterate_2EFINREC,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => 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)))))) ) ) )).
fof(mem_c_2Eiterate_2EITSET,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => 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)))) ) ) )).
fof(mem_c_2Eiterate_2ESum,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Eiterate_2ESum(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal))) ) )).
fof(mem_c_2Eiterate_2Einf,axiom,( mem(c_2Eiterate_2Einf,arr(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal)) )).
fof(mem_c_2Eiterate_2Eiterate,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => 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)))) ) ) )).
fof(mem_c_2Eiterate_2Emonoidal,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Eiterate_2Emonoidal(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),bool)) ) )).
fof(mem_c_2Eiterate_2Eneutral,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Eiterate_2Eneutral(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),A_27a)) ) )).
fof(mem_c_2Eiterate_2Ensum,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Eiterate_2Ensum(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Enum_2Enum),ty_2Enum_2Enum))) ) )).
fof(mem_c_2Eiterate_2Epolynomial__function,axiom,( mem(c_2Eiterate_2Epolynomial__function,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool)) )).
fof(mem_c_2Eiterate_2Esupport,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => 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))))) ) ) )).
fof(conj_thm_2Eiterate_2EREAL__LT__BETWEEN,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V1b)) <=> ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V2x)) & p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1b)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ESIMP__REAL__ARCH,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ? [V1n] : ( mem(V1n,ty_2Enum_2Enum) & p(ap(ap(c_2Ereal_2Ereal__lte,V0x),ap(c_2Ereal_2Ereal__of__num,V1n))) ) ))).
fof(conj_thm_2Eiterate_2ELOWER__BOUND__FINITE__SET__REAL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2a),ap(V0f,V3x))) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2EUPPER__BOUND__FINITE__SET__REAL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3x)),V2a)) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2EREAL__WLOG__LT,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ( ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => p(ap(ap(V0P,V1x),V1x)) ) ) & ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(V0P,V2x),V3y)) <=> p(ap(ap(V0P,V3y),V2x)) ) ) ) ) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V5y)) => p(ap(ap(V0P,V4x),V5y)) ) ) ) ) ) => ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ! [V7y] : ( mem(V7y,ty_2Erealax_2Ereal) => p(ap(ap(V0P,V6x),V7y)) ) ) ) ))).
fof(ax_thm_2Eiterate_2EFINREC__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ( ( ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1b] : ( mem(V1b,A_27b) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3a] : ( mem(V3a,A_27b) => ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),c_2Enum_2E0)) <=> ( ( V2s = c_2Epred__set_2EEMPTY(A_27a) ) & ( V3a = V1b ) ) ) ) ) ) ) ) & ! [V4f] : ( mem(V4f,arr(A_27a,arr(A_27b,A_27b))) => ! [V5b] : ( mem(V5b,A_27b) => ! [V6s] : ( mem(V6s,arr(A_27a,bool)) => ! [V7a] : ( mem(V7a,A_27b) => ! [V8n] : ( mem(V8n,ty_2Enum_2Enum) => ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V4f),V5b),V6s),V7a),ap(c_2Enum_2ESUC,V8n))) <=> ? [V9x] : ( mem(V9x,A_27a) & ? [V10c] : ( 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),V8n)) & ( V7a = ap(ap(V4f,V9x),V10c) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Eiterate_2EFINREC__def__compute,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( ( ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1b] : ( mem(V1b,A_27b) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3a] : ( mem(V3a,A_27b) => ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),c_2Enum_2E0)) <=> ( ( V2s = c_2Epred__set_2EEMPTY(A_27a) ) & ( V3a = V1b ) ) ) ) ) ) ) ) & ( ( ! [V4f] : ( mem(V4f,arr(A_27a,arr(A_27b,A_27b))) => ! [V5b] : ( mem(V5b,A_27b) => ! [V6s] : ( mem(V6s,arr(A_27a,bool)) => ! [V7a] : ( mem(V7a,A_27b) => ! [V8n] : ( mem(V8n,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,V8n)))) <=> ? [V9x] : ( mem(V9x,A_27a) & ? [V10c] : ( 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,V8n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) & ( V7a = ap(ap(V4f,V9x),V10c) ) ) ) ) ) ) ) ) ) ) ) ) & ! [V11f] : ( mem(V11f,arr(A_27a,arr(A_27b,A_27b))) => ! [V12b] : ( mem(V12b,A_27b) => ! [V13s] : ( mem(V13s,arr(A_27a,bool)) => ! [V14a] : ( mem(V14a,A_27b) => ! [V15n] : ( mem(V15n,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,V15n)))) <=> ? [V16x] : ( mem(V16x,A_27a) & ? [V17c] : ( 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,V15n)))) & ( V14a = ap(ap(V11f,V16x),V17c) ) ) ) ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EFINREC__1__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1b] : ( mem(V1b,A_27b) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3a] : ( 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,c_2Enum_2E0))) <=> ? [V4x] : ( 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) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EFINREC__SUC__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1b] : ( mem(V1b,A_27b) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ! [V4s] : ( 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] : ( mem(V5n,ty_2Enum_2Enum) => ! [V6s] : ( mem(V6s,arr(A_27a,bool)) => ! [V7z] : ( 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,V5n))) => ! [V8x] : ( mem(V8x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),V6s)) => ? [V9w] : ( 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),V5n)) & ( V7z = ap(ap(V0f,V8x),V9w) ) ) ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EFINREC__UNIQUE__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1b] : ( mem(V1b,A_27b) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ! [V4s] : ( 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] : ( mem(V5n1,ty_2Enum_2Enum) => ! [V6n2] : ( mem(V6n2,ty_2Enum_2Enum) => ! [V7s] : ( mem(V7s,arr(A_27a,bool)) => ! [V8a1] : ( mem(V8a1,A_27b) => ! [V9a2] : ( mem(V9a2,A_27b) => ( ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V7s),V8a1),V5n1)) & p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V7s),V9a2),V6n2)) ) => ( ( V8a1 = V9a2 ) & ( V5n1 = V6n2 ) ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EFINREC__EXISTS__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1b] : ( mem(V1b,A_27b) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) => ? [V3a] : ( mem(V3a,A_27b) & ? [V4n] : ( mem(V4n,ty_2Enum_2Enum) & p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),V4n)) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EFINREC__FUN__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1R] : ( mem(V1R,arr(A_27a,arr(A_27b,arr(A_27c,bool)))) => ( ( ( ! [V2s] : ( mem(V2s,A_27a) => ( p(ap(V0P,V2s)) => ? [V3a] : ( mem(V3a,A_27b) & ? [V4n] : ( mem(V4n,A_27c) & p(ap(ap(ap(V1R,V2s),V3a),V4n)) ) ) ) ) ) & ! [V5n1] : ( mem(V5n1,A_27c) => ! [V6n2] : ( mem(V6n2,A_27c) => ! [V7s] : ( mem(V7s,A_27a) => ! [V8a1] : ( mem(V8a1,A_27b) => ! [V9a2] : ( 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] : ( mem(V10f,arr(A_27a,A_27b)) & ! [V11s] : ( mem(V11s,A_27a) => ! [V12a] : ( mem(V12a,A_27b) => ( p(ap(V0P,V11s)) => ( ( ? [V13n] : ( mem(V13n,A_27c) & p(ap(ap(ap(V1R,V11s),V12a),V13n)) ) ) <=> ( ap(V10f,V11s) = V12a ) ) ) ) ) ) ) ) )))))).
fof(conj_thm_2Eiterate_2EFINREC__FUN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1b] : ( mem(V1b,A_27b) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ! [V4s] : ( 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] : ( mem(V5g,arr(arr(A_27a,bool),A_27b)) & ( ( ap(V5g,c_2Epred__set_2EEMPTY(A_27a)) = V1b ) & ! [V6s] : ( mem(V6s,arr(A_27a,bool)) => ! [V7x] : ( 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))) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESET__RECURSION__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1b] : ( mem(V1b,A_27b) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ! [V4s] : ( 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] : ( mem(V5g,arr(arr(A_27a,bool),A_27b)) & ( ( ap(V5g,c_2Epred__set_2EEMPTY(A_27a)) = V1b ) & ! [V6x] : ( mem(V6x,A_27a) => ! [V7s] : ( 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))) ) ) ) ) ) ) ) ) ))))).
fof(ax_thm_2Eiterate_2EITSET__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27b,arr(A_27a,A_27a))) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2b] : ( 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)),f3262(A_27a,A_27b,V2b,V0f)),V1s) ) ) ) ) ) ) )).
fof(conj_thm_2Eiterate_2EFINITE__RECURSION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1b] : ( mem(V1b,A_27b) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ! [V4s] : ( 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] : ( mem(V5x,A_27a) => ! [V6s] : ( 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))) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ECARD__UNION__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2u] : ( 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 ) ) ) => ( ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27a),V1t)) = ap(c_2Epred__set_2ECARD(A_27a),V2u) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUBSET__RESTRICT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f1884(A_27a,V1P,V0s))),V0s)) ) )))).
fof(conj_thm_2Eiterate_2EFINITE__RESTRICT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1P] : ( 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),f1884(A_27a,V1P,V0s)))) ) ) )))).
fof(conj_thm_2Eiterate_2EFINITE__REAL__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3263(V0a)))) ) ) ) & ( ( ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3264(V2a)))) ) ) ) & ( ( ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3265(V4b)))) ) ) ) & ( ( ! [V6b] : ( mem(V6b,ty_2Erealax_2Ereal) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3266(V6b)))) ) ) ) & ( ( ! [V8a] : ( mem(V8a,ty_2Erealax_2Ereal) => ! [V9b] : ( mem(V9b,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3267(V9b,V8a)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V9b),V8a)) ) ) ) ) & ( ( ! [V11a] : ( mem(V11a,ty_2Erealax_2Ereal) => ! [V12b] : ( mem(V12b,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3268(V12b,V11a)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V12b),V11a)) ) ) ) ) & ( ( ! [V14a] : ( mem(V14a,ty_2Erealax_2Ereal) => ! [V15b] : ( mem(V15b,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3269(V15b,V14a)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V15b),V14a)) ) ) ) ) & ! [V17a] : ( mem(V17a,ty_2Erealax_2Ereal) => ! [V18b] : ( mem(V18b,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3270(V18b,V17a)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V18b),V17a)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2Ereal__INFINITE,lemma,(( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ))).
fof(conj_thm_2Eiterate_2ESET__PROVE__CASES,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( ( p(ap(V0P,c_2Epred__set_2EEMPTY(A_27a))) & ! [V1a] : ( mem(V1a,A_27a) => ! [V2s] : ( 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] : ( mem(V3s,arr(A_27a,bool)) => p(ap(V0P,V3s)) ) ) )))).
fof(conj_thm_2Eiterate_2ECHOOSE__SUBSET__STRONG,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V0n),ap(c_2Epred__set_2ECARD(A_27a),V1s))) ) => ? [V2t] : ( 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),V0n)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ECHOOSE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),ap(c_2Epred__set_2ECARD(A_27a),V0s))) => ? [V2t] : ( 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),V1n)) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2EREAL__COMPLETE,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ? [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) & p(ap(V0P,V1x)) ) ) & ? [V2M] : ( mem(V2M,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(V0P,V3x)) => p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2M)) ) ) ) ) => ? [V4M] : ( mem(V4M,ty_2Erealax_2Ereal) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(V0P,V5x)) => p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V4M)) ) ) ) & ! [V6M_27] : ( mem(V6M_27,ty_2Erealax_2Ereal) => ( ( ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ( p(ap(V0P,V7x)) => p(ap(ap(c_2Ereal_2Ereal__lte,V7x),V6M_27)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V4M),V6M_27)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2Esup__alt,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal_2Esup,V0s) = ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3274(V0s)) ) ))).
fof(conj_thm_2Eiterate_2ESUP__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) <=> ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1t)) => p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V2b)) ) ) ) ) ) => ( ap(c_2Ereal_2Esup,V0s) = ap(c_2Ereal_2Esup,V1t) ) ) ) ))).
fof(conj_thm_2Eiterate_2ESUP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ? [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) ) ) ) ) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V3x),ap(c_2Ereal_2Esup,V0s))) ) ) ) & ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V4b)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V4b)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ESUP__FINITE__LEMMA,lemma,(! [V0s] : ( 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] : ( mem(V1b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1b),V0s)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ESUP__FINITE,lemma,(! [V0s] : ( 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] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1x),ap(c_2Ereal_2Esup,V0s))) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__LE__SUP__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,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,V1a),ap(c_2Ereal_2Esup,V0s))) <=> ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__SUP__LE__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,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)),V1a)) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1a)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__LT__SUP__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,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,V1a),ap(c_2Ereal_2Esup,V0s))) <=> ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__SUP__LT__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,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)),V1a)) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1a)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__SUP__LE__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ? [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) ) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V1y)) <=> ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V1y)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__SUP__UNIQUE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) ) ) ) & ! [V3b_27] : ( mem(V3b_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V3b_27),V1b)) => ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,V3b_27),V4x)) ) ) ) ) ) => ( ap(c_2Ereal_2Esup,V0s) = V1b ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__SUP__LE__S,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V1b)) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__SUP__LE__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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] : ( mem(V2b,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1t)) => p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),ap(c_2Ereal_2Esup,V1t))) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__LE__SUP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3y)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V2b)) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V1a),ap(c_2Ereal_2Esup,V0s))) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__SUP__BOUNDS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) ) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),ap(c_2Ereal_2Esup,V0s))) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V2b)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__ABS__SUP__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V2x)),V1a)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(c_2Ereal_2Esup,V0s))),V1a)) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__SUP__ASCLOSE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V3x),V1l))),V2e)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Esup,V0s)),V1l))),V2e)) ) ) ) ))).
fof(conj_thm_2Eiterate_2ESUP__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = V0a ) ))).
fof(conj_thm_2Eiterate_2ESUP__UNIQUE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2c)) ) ) ) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2c)) ) ) ) => ( ap(c_2Ereal_2Esup,V0s) = V1b ) ) ) ))).
fof(conj_thm_2Eiterate_2ESUP__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ? [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) ) & ? [V4c] : ( mem(V4c,ty_2Erealax_2Ereal) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V1t)) => p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V4c)) ) ) ) ) ) ) => ( ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)) = ap(ap(c_2Ereal_2Emax,ap(c_2Ereal_2Esup,V0s)),ap(c_2Ereal_2Esup,V1t)) ) ) ) ))).
fof(ax_thm_2Eiterate_2Einf,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Eiterate_2Einf,V0s) = ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3278(V0s)) ) )).
fof(conj_thm_2Eiterate_2EINF__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2a),V3x)) ) ) ) <=> ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1t)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2a),V4x)) ) ) ) ) ) => ( ap(c_2Eiterate_2Einf,V0s) = ap(c_2Eiterate_2Einf,V1t) ) ) ) ))).
fof(conj_thm_2Eiterate_2EINF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ? [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2x)) ) ) ) ) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V3x)) ) ) ) & ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V4b),V5x)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V4b),ap(c_2Eiterate_2Einf,V0s))) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EINF__FINITE__LEMMA,lemma,(! [V0s] : ( 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] : ( mem(V1b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1b),V0s)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EINF__FINITE,lemma,(! [V0s] : ( 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] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V1x)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__LE__INF__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,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,V1a),ap(c_2Eiterate_2Einf,V0s))) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__INF__LE__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,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)),V1a)) <=> ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1a)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__LT__INF__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,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,V1a),ap(c_2Eiterate_2Einf,V0s))) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__INF__LT__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,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)),V1a)) <=> ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1a)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__INF__UNIQUE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2x)) ) ) ) & ! [V3b_27] : ( mem(V3b_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V1b),V3b_27)) => ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V3b_27)) ) ) ) ) ) => ( ap(c_2Eiterate_2Einf,V0s) = V1b ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__LE__INF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2x)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V1b),ap(c_2Eiterate_2Einf,V0s))) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__LE__INF__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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] : ( mem(V2b,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2b),V3x)) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),ap(c_2Eiterate_2Einf,V1t))) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__INF__BOUNDS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) ) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),ap(c_2Eiterate_2Einf,V0s))) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V2b)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__ABS__INF__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V2x)),V1a)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(c_2Eiterate_2Einf,V0s))),V1a)) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__INF__ASCLOSE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V3x),V1l))),V2e)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Eiterate_2Einf,V0s)),V1l))),V2e)) ) ) ) ))).
fof(conj_thm_2Eiterate_2ESUP__UNIQUE__FINITE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( 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) ) ) ) => ( ( ap(c_2Ereal_2Esup,V1s) = V0a ) <=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),V1s)) & ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2y),V0a)) ) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EINF__UNIQUE__FINITE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( 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) ) ) ) => ( ( ap(c_2Eiterate_2Einf,V1s) = V0a ) <=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),V1s)) & ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V2y)) ) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ESUP__INSERT__FINITE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s)) => ( ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),V1s)) = 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))),V0x),ap(ap(c_2Ereal_2Emax,V0x),ap(c_2Ereal_2Esup,V1s))) ) ) ) ))).
fof(conj_thm_2Eiterate_2EINF__INSERT__FINITE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s)) => ( ap(c_2Eiterate_2Einf,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),V1s)) = 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))),V0x),ap(ap(c_2Ereal_2Emin,V0x),ap(c_2Eiterate_2Einf,V1s))) ) ) ) ))).
fof(conj_thm_2Eiterate_2EREAL__SUP__EQ__INF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ? [V1B] : ( mem(V1B,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V2x)),V1B)) ) ) ) ) => ( ( ap(c_2Ereal_2Esup,V0s) = ap(c_2Eiterate_2Einf,V0s) ) <=> ? [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) & ( V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EINF__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Eiterate_2Einf,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = V0a ) ))).
fof(ax_thm_2Eiterate_2Enumseg,axiom, ! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n) = ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3279(V1n,V0m)) ) ) )).
fof(conj_thm_2Eiterate_2EFINITE__NUMSEG,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n))) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__COMBINE__R,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1p] : ( mem(V1p,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),ap(ap(c_2Earithmetic_2E_2B,V1p),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V1p),V2n)) ) => ( ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1p)),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V1p),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2n)) = ap(ap(c_2Eiterate_2E_2E_2E,V0m),V2n) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__COMBINE__L,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1p] : ( mem(V1p,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1p)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V1p),ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) => ( ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(ap(c_2Earithmetic_2E_2D,V1p),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eiterate_2E_2E_2E,V1p),V2n)) = ap(ap(c_2Eiterate_2E_2E_2E,V0m),V2n) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__LREC,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) => ( ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0m),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V0m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V1n)) = ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__RREC,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) => ( ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V1n),ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(ap(c_2Earithmetic_2E_2D,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) = ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__REC,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),ap(c_2Enum_2ESUC,V1n))) => ( ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(c_2Enum_2ESUC,V1n)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),ap(c_2Enum_2ESUC,V1n)),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)) ) ) ) ))).
fof(conj_thm_2Eiterate_2EIN__NUMSEG,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ! [V2p] : ( mem(V2p,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2p),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n))) <=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V2p)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V2p),V1n)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2EIN__NUMSEG__0,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0m),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V1n))) <=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) ) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__SING,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2E_2E_2E,V0n),V0n) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0n),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__EMPTY,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ( ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n) = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) <=> p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V0m)) ) ) ))).
fof(conj_thm_2Eiterate_2ECARD__NUMSEG__LEMMA,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1d] : ( mem(V1d,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(ap(c_2Earithmetic_2E_2B,V0m),V1d))) = ap(ap(c_2Earithmetic_2E_2B,V1d),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ))).
fof(conj_thm_2Eiterate_2ECARD__NUMSEG,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)) = ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0m) ) ) ))).
fof(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => p(ap(ap(c_2Ecardinal_2EHAS__SIZE(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0m))) ) ))).
fof(conj_thm_2Eiterate_2ECARD__NUMSEG__1,lemma,(! [V0n] : ( mem(V0n,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,c_2Earithmetic_2EZERO))),V0n)) = V0n ) ))).
fof(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG__1,lemma,(! [V0n] : ( mem(V0n,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,c_2Earithmetic_2EZERO))),V0n)),V0n)) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__CLAUSES,lemma,(( ( ! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2E_2E_2E,V0m),c_2Enum_2E0) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V0m),c_2Enum_2E0)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),c_2Enum_2E0),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) ) ) ) & ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2E_2E_2E,V1m),ap(c_2Enum_2ESUC,V2n)) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Earithmetic_2E_3C_3D,V1m),ap(c_2Enum_2ESUC,V2n))),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),ap(c_2Enum_2ESUC,V2n)),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n))),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)) ) ) ) ))).
fof(conj_thm_2Eiterate_2EFINITE__INDEX__NUMSEG,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) <=> ? [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,A_27a)) & ( ( ! [V2i] : ( mem(V2i,ty_2Enum_2Enum) => ! [V3j] : ( mem(V3j,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(c_2Epred__set_2ECARD(A_27a),V0s)))) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3j),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(c_2Epred__set_2ECARD(A_27a),V0s)))) & ( ap(V1f,V2i) = ap(V1f,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,c_2Earithmetic_2EZERO))),ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2EFINITE__INDEX__NUMBERS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) <=> ? [V1k] : ( mem(V1k,arr(ty_2Enum_2Enum,bool)) & ? [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,A_27a)) & ( ( ! [V3i] : ( mem(V3i,ty_2Enum_2Enum) => ! [V4j] : ( mem(V4j,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3i),V1k)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4j),V1k)) & ( ap(V2f,V3i) = ap(V2f,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) ) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2EDISJOINT__NUMSEG,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ! [V2p] : ( mem(V2p,ty_2Enum_2Enum) => ! [V3q] : ( mem(V3q,ty_2Enum_2Enum) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)),ap(ap(c_2Eiterate_2E_2E_2E,V2p),V3q))) <=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V2p)) | ( p(ap(ap(c_2Eprim__rec_2E_3C,V3q),V0m)) | ( p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V0m)) | p(ap(ap(c_2Eprim__rec_2E_3C,V3q),V2p)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__ADD__SPLIT,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ! [V2p] : ( mem(V2p,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),ap(ap(c_2Earithmetic_2E_2B,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) => ( ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(ap(c_2Earithmetic_2E_2B,V1n),V2p)) = ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Earithmetic_2E_2B,V1n),V2p))) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__OFFSET__IMAGE,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ! [V2p] : ( mem(V2p,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V0m),V2p)),ap(ap(c_2Earithmetic_2E_2B,V1n),V2p)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),f3280(V2p)),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)) ) ) ) ))).
fof(conj_thm_2Eiterate_2ESUBSET__NUMSEG,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ! [V2p] : ( mem(V2p,ty_2Enum_2Enum) => ! [V3q] : ( mem(V3q,ty_2Enum_2Enum) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)),ap(ap(c_2Eiterate_2E_2E_2E,V2p),V3q))) <=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V0m)) | ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2p),V0m)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V3q)) ) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__LE,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3281(V0n)) = ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n) ) ))).
fof(conj_thm_2Eiterate_2ENUMSEG__LT,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3282(V0n)) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V0n),c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ))).
fof(conj_thm_2Eiterate_2ETOPOLOGICAL__SORT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0_3C_3C] : ( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool))) => ( ( ( ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( ( p(ap(ap(V0_3C_3C,V1x),V2y)) & p(ap(ap(V0_3C_3C,V2y),V1x)) ) => ( V1x = V2y ) ) ) ) ) & ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ! [V5z] : ( 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] : ( mem(V6n,ty_2Enum_2Enum) => ! [V7s] : ( mem(V7s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V7s),V6n)) => ? [V8f] : ( 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,c_2Earithmetic_2EZERO))),V6n)) ) & ! [V9j] : ( mem(V9j,ty_2Enum_2Enum) => ! [V10k] : ( mem(V10k,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V9j),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V6n))) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V10k),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V6n))) & p(ap(ap(c_2Eprim__rec_2E_3C,V9j),V10k)) ) ) => ( ~~ p(ap(ap(V0_3C_3C,ap(V8f,V10k)),ap(V8f,V9j))) ) ) ) ) ) ) ) ) ) ) )))).
fof(ax_thm_2Eiterate_2Eneutral,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( ap(c_2Eiterate_2Eneutral(A_27a),V0op) = ap(c_2Emin_2E_40(A_27a),f3284(A_27a,V0op)) ) ) ) )).
fof(ax_thm_2Eiterate_2Emonoidal,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op)) <=> ( ( ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( ap(ap(V0op,V1x),V2y) = ap(ap(V0op,V2y),V1x) ) ) ) ) & ( ( ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ! [V5z] : ( mem(V5z,A_27a) => ( ap(ap(V0op,V3x),ap(ap(V0op,V4y),V5z)) = ap(ap(V0op,ap(ap(V0op,V3x),V4y)),V5z) ) ) ) ) ) & ! [V6x] : ( mem(V6x,A_27a) => ( ap(ap(V0op,ap(c_2Eiterate_2Eneutral(A_27a),V0op)),V6x) = V6x ) ) ) ) ) ) ) )).
fof(conj_thm_2Eiterate_2EMONOIDAL__AC,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op)) => ( ( ! [V1a] : ( mem(V1a,A_27a) => ( ap(ap(V0op,ap(c_2Eiterate_2Eneutral(A_27a),V0op)),V1a) = V1a ) ) ) & ( ( ! [V2a] : ( mem(V2a,A_27a) => ( ap(ap(V0op,V2a),ap(c_2Eiterate_2Eneutral(A_27a),V0op)) = V2a ) ) ) & ( ( ! [V3a] : ( mem(V3a,A_27a) => ! [V4b] : ( mem(V4b,A_27a) => ( ap(ap(V0op,V3a),V4b) = ap(ap(V0op,V4b),V3a) ) ) ) ) & ( ( ! [V5a] : ( mem(V5a,A_27a) => ! [V6b] : ( mem(V6b,A_27a) => ! [V7c] : ( mem(V7c,A_27a) => ( ap(ap(V0op,ap(ap(V0op,V5a),V6b)),V7c) = ap(ap(V0op,V5a),ap(ap(V0op,V6b),V7c)) ) ) ) ) ) & ! [V8a] : ( mem(V8a,A_27a) => ! [V9b] : ( mem(V9b,A_27a) => ! [V10c] : ( mem(V10c,A_27a) => ( ap(ap(V0op,V8a),ap(ap(V0op,V9b),V10c)) = ap(ap(V0op,V9b),ap(ap(V0op,V8a),V10c)) ) ) ) ) ) ) ) ) ) )))).
fof(ax_thm_2Eiterate_2Esupport,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2s] : ( 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),f3285(A_27b,A_27a,V2s,V0op,V1f)) ) ) ) ) ) ) )).
fof(ax_thm_2Eiterate_2Eiterate,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( 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),f3287(A_27a,A_27b,V0op,V2f)),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)) ) ) ) ) ) ) )).
fof(conj_thm_2Eiterate_2EIN__SUPPORT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2x] : ( mem(V2x,A_27b) => ! [V3s] : ( 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) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUPPORT__SUPPORT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( 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) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUPPORT__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( ! [V3x] : ( 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) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUPPORT__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( 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)) ) ) ))))).
fof(conj_thm_2Eiterate_2EFINITE__SUPPORT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( 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))) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUPPORT__CLAUSES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) =>  ! [A_27e]: ( ne(A_27e) =>  ! [A_27f]: ( ne(A_27f) =>  ! [A_27g]: ( ne(A_27g) =>  ! [A_27h]: ( ne(A_27h) =>  ! [A_27i]: ( ne(A_27i) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( ( ! [V1f] : ( 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] : ( mem(V2f,arr(A_27c,A_27b)) => ! [V3x] : ( mem(V3x,A_27c) => ! [V4s] : ( 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] : ( mem(V5f,arr(A_27d,A_27b)) => ! [V6x] : ( mem(V6x,A_27d) => ! [V7s] : ( 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] : ( mem(V8f,arr(A_27e,A_27b)) => ! [V9s] : ( mem(V9s,arr(A_27e,bool)) => ! [V10t] : ( 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] : ( mem(V11f,arr(A_27f,A_27b)) => ! [V12s] : ( mem(V12s,arr(A_27f,bool)) => ! [V13t] : ( 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] : ( mem(V14f,arr(A_27g,A_27b)) => ! [V15s] : ( mem(V15s,arr(A_27g,bool)) => ! [V16t] : ( 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] : ( mem(V17f,arr(A_27h,A_27i)) => ! [V18g] : ( mem(V18g,arr(A_27i,A_27b)) => ! [V19s] : ( 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)) ) ) ) ) ) ) ) ) ) ) )))))))))))).
fof(conj_thm_2Eiterate_2ESUPPORT__DELTA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2f] : ( mem(V2f,arr(A_27b,A_27a)) => ! [V3a] : ( mem(V3a,A_27b) => ( ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),f3288(A_27b,A_27a,V2f,V3a,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)) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EFINITE__SUPPORT__DELTA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27b,bool)) => ! [V1op] : ( mem(V1op,arr(A_27a,arr(A_27a,A_27a))) => ! [V2f] : ( mem(V2f,arr(A_27b,A_27a)) => ! [V3a] : ( mem(V3a,A_27b) => p(ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V1op),f3289(A_27b,A_27a,V2f,V3a,V1op)),V0s))) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__SUPPORT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( 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) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__EXPAND__CASES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( 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)) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__CLAUSES__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ( ( ! [V1f] : ( 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] : ( mem(V2f,arr(A_27a,A_27b)) => ! [V3x] : ( mem(V3x,A_27a) => ! [V4s] : ( 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))) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__CLAUSES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op)) => ( ( ! [V1f] : ( 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] : ( mem(V2f,arr(A_27b,A_27a)) => ! [V3x] : ( mem(V3x,A_27b) => ! [V4s] : ( 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))) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__UNION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ! [V3t] : ( 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)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__UNION__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( 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)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ! [V3t] : ( 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) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__DIFF__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( 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) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__INCL__EXCL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ! [V3f] : ( 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)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__CLOSED,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1P] : ( mem(V1P,arr(A_27b,bool)) => ( ( p(ap(V1P,ap(c_2Eiterate_2Eneutral(A_27b),V0op))) & ! [V2x] : ( mem(V2x,A_27b) => ! [V3y] : ( mem(V3y,A_27b) => ( ( p(ap(V1P,V2x)) & p(ap(V1P,V3y)) ) => p(ap(V1P,ap(ap(V0op,V2x),V3y))) ) ) ) ) => ! [V4f] : ( mem(V4f,arr(A_27a,A_27b)) => ! [V5s] : ( mem(V5s,arr(A_27a,bool)) => ( ( ! [V6x] : ( 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))) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__RELATED,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1R] : ( 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] : ( mem(V2x1,A_27b) => ! [V3y1] : ( mem(V3y1,A_27b) => ! [V4x2] : ( mem(V4x2,A_27b) => ! [V5y2] : ( 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] : ( mem(V6f,arr(A_27a,A_27b)) => ! [V7g] : ( mem(V7g,arr(A_27a,A_27b)) => ! [V8s] : ( mem(V8s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V8s)) & ! [V9x] : ( 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))) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__EQ__NEUTRAL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ! [V3x] : ( 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) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__SING,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2x] : ( 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) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3a] : ( 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) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__DELTA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2a] : ( mem(V2a,A_27b) => ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ( ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V3s),f3290(A_27b,A_27a,V1f,V2a,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)) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0op] : ( mem(V0op,arr(A_27c,arr(A_27c,A_27c))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2g] : ( mem(V2g,arr(A_27b,A_27c)) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( ( ! [V4x] : ( mem(V4x,A_27a) => ! [V5y] : ( 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)) ) ) ) ) ) ) )))))).
fof(conj_thm_2Eiterate_2EITERATE__BIJECTION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2p] : ( mem(V2p,arr(A_27a,A_27a)) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( ( ( ! [V4x] : ( 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] : ( mem(V5y,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V3s)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f3291(A_27a,V5y,V2p,V3s))) ) ) ) => ( 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)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__ITERATE__PRODUCT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0op] : ( mem(V0op,arr(A_27c,arr(A_27c,A_27c))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,arr(A_27b,bool))) => ! [V3x] : ( mem(V3x,arr(A_27a,arr(A_27b,A_27c))) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & ! [V4i] : ( 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),f3292(A_27b,A_27c,A_27a,V0op,V2t,V3x)) = 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)),f3294(A_27b,A_27a,V1s,V2t)))),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),f3296(A_27a,A_27c,A_27b,V3x))) ) ) ) ) ) ) )))))).
fof(conj_thm_2Eiterate_2EITERATE__EQ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2g] : ( mem(V2g,arr(A_27a,A_27b)) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( ( ! [V4x] : ( 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) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0op] : ( mem(V0op,arr(A_27c,arr(A_27c,A_27c))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ! [V3f] : ( mem(V3f,arr(A_27a,A_27c)) => ! [V4g] : ( mem(V4g,arr(A_27b,A_27c)) => ! [V5h] : ( mem(V5h,arr(A_27a,A_27b)) => ( ( ( ! [V6y] : ( mem(V6y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V6y),V2t)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f3297(A_27b,A_27a,V6y,V5h,V1s))) ) ) ) & ! [V8x] : ( 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) ) ) ) ) ) ) ) ) )))))).
fof(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL__INVERSES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0op] : ( mem(V0op,arr(A_27c,arr(A_27c,A_27c))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ! [V3f] : ( mem(V3f,arr(A_27a,A_27c)) => ! [V4g] : ( mem(V4g,arr(A_27b,A_27c)) => ! [V5h] : ( mem(V5h,arr(A_27a,A_27b)) => ! [V6k] : ( mem(V6k,arr(A_27b,A_27a)) => ( ( ( ! [V7y] : ( 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] : ( 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) ) ) ) ) ) ) ) ) ) )))))).
fof(conj_thm_2Eiterate_2EITERATE__INJECTION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2p] : ( mem(V2p,arr(A_27a,A_27a)) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3s)) & ( ( ! [V4x] : ( 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] : ( mem(V5x,A_27a) => ! [V6y] : ( 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) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__UNION__NONZERO,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( 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] : ( 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)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__OP,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2g] : ( mem(V2g,arr(A_27b,A_27a)) => ! [V3s] : ( 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),f3298(A_27a,A_27b,V0op,V1f,V2g)) = 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)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__SUPERSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2u] : ( mem(V2u,arr(A_27a,bool)) => ! [V3v] : ( mem(V3v,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2u),V3v)) & ! [V4x] : ( 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) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__IMAGE__NONZERO,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0op] : ( mem(V0op,arr(A_27c,arr(A_27c,A_27c))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op)) => ! [V1g] : ( mem(V1g,arr(A_27b,A_27c)) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3s)) & ! [V4x] : ( mem(V4x,A_27a) => ! [V5y] : ( 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)) ) ) ) ) ) ) )))))).
fof(conj_thm_2Eiterate_2EITERATE__CASES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2P] : ( mem(V2P,arr(A_27a,bool)) => ! [V3f] : ( mem(V3f,arr(A_27a,A_27b)) => ! [V4g] : ( 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),f3299(A_27b,A_27a,V3f,V2P,V4g)) = ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3300(A_27a,V2P,V1s))),V3f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3301(A_27a,V2P,V1s))),V4g)) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__OP__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0op] : ( mem(V0op,arr(A_27b,arr(A_27b,A_27b))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2g] : ( mem(V2g,arr(A_27a,A_27b)) => ! [V3s] : ( 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),f3302(A_27b,A_27a,V0op,V1f,V2g)) = 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)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EITERATE__CLAUSES__NUMSEG,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,A_27a)) => ! [V1op] : ( mem(V1op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V1op)) => ( ( ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ( ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,V2m),c_2Enum_2E0)),V0f) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V2m),c_2Enum_2E0)),ap(V0f,c_2Enum_2E0)),ap(c_2Eiterate_2Eneutral(A_27a),V1op)) ) ) ) & ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,V3m),ap(c_2Enum_2ESUC,V4n))),V0f) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Earithmetic_2E_3C_3D,V3m),ap(c_2Enum_2ESUC,V4n))),ap(ap(V1op,ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,V3m),V4n)),V0f)),ap(V0f,ap(c_2Enum_2ESUC,V4n)))),ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,V3m),V4n)),V0f)) ) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2EITERATE__PAIR,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0op] : ( mem(V0op,arr(A_27a,arr(A_27a,A_27a))) => ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op)) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,A_27a)) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,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,c_2Earithmetic_2EZERO))),V2m)),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V3n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V1f) = ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V0op),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3303(A_27a,V0op,V1f)) ) ) ) ) ) )))).
fof(ax_thm_2Eiterate_2Ensum__def,axiom,( ! [A_27a] : ( ne(A_27a) => ( c_2Eiterate_2Ensum(A_27a) = ap(c_2Eiterate_2Eiterate(A_27a,ty_2Enum_2Enum),c_2Earithmetic_2E_2B) ) ) )).
fof(conj_thm_2Eiterate_2ENEUTRAL__ADD,lemma,(( ap(c_2Eiterate_2Eneutral(ty_2Enum_2Enum),c_2Earithmetic_2E_2B) = c_2Enum_2E0 ))).
fof(conj_thm_2Eiterate_2ENEUTRAL__MUL,lemma,(( ap(c_2Eiterate_2Eneutral(ty_2Enum_2Enum),c_2Earithmetic_2E_2A) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ))).
fof(conj_thm_2Eiterate_2EMONOIDAL__ADD,lemma,(p(ap(c_2Eiterate_2Emonoidal(ty_2Enum_2Enum),c_2Earithmetic_2E_2B)))).
fof(conj_thm_2Eiterate_2EMONOIDAL__MUL,lemma,(p(ap(c_2Eiterate_2Emonoidal(ty_2Enum_2Enum),c_2Earithmetic_2E_2A)))).
fof(conj_thm_2Eiterate_2ENSUM__DEGENERATE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3304(A_27a,V0f,V1s)))) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) = c_2Enum_2E0 ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__CLAUSES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( ( ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0f) = c_2Enum_2E0 ) ) ) & ! [V1x] : ( mem(V1x,A_27b) => ! [V2f] : ( mem(V2f,arr(A_27b,ty_2Enum_2Enum)) => ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),V3s)) => ( ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27b),V1x),V3s)),V2f) = 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))) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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)) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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)) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__INCL__EXCL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2f] : ( 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)) ) => ( ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V2f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V1t),V2f)) = 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)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__SUPPORT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( 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) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__ADD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f453(A_27a,V0f,V1g)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__ADD__GEN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3305(A_27a,V0f,V2s)))) & p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3306(A_27a,V1g,V2s)))) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3307(A_27a,V0f,V1g)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__EQ__0,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) => ( ap(V0f,V2x) = c_2Enum_2E0 ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) = c_2Enum_2E0 ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__0,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),k(A_27a,c_2Enum_2E0)) = c_2Enum_2E0 ) )))).
fof(conj_thm_2Eiterate_2ENSUM__LMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1c] : ( mem(V1c,ty_2Enum_2Enum) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3308(A_27a,V0f,V1c)) = ap(ap(c_2Earithmetic_2E_2A,V1c),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__RMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1c] : ( mem(V1c,ty_2Enum_2Enum) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3309(A_27a,V0f,V1c)) = ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),V1c) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ! [V3x] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__LT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ( ! [V3x] : ( 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] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__LT__ALL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ( ~~ ( V2s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ! [V3x] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s)) => ( ap(V0f,V3x) = ap(V1g,V3x) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__CONST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0c] : ( mem(V0c,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),k(A_27a,V0c)) = ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V1s)),V0c) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__POS__BOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1b] : ( mem(V1b,ty_2Enum_2Enum) => ! [V2s] : ( 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)),V1b)) ) => ! [V3x] : ( 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)),V1b)) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) = c_2Enum_2E0 ) <=> ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) => ( ap(V0f,V2x) = c_2Enum_2E0 ) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__POS__LT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & ? [V2x] : ( mem(V2x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) & p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(V0f,V2x))) ) ) ) => p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f))) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__POS__LT__ALL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( 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] : ( mem(V2i,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2i),V0s)) => p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(V1f,V2i))) ) ) ) ) => p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V1f))) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2a] : ( mem(V2a,A_27a) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)) ) => ( 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)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1x] : ( mem(V1x,A_27a) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),V0f) = ap(V0f,V1x) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__DELTA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0b] : ( mem(V0b,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2a] : ( mem(V2a,A_27a) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3310(A_27a,V0b,V2a)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)),V0b),c_2Enum_2E0) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__SWAP,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3311(A_27a,A_27b,V0f,V2t)) = ap(ap(c_2Eiterate_2Ensum(A_27b),V2t),f3313(A_27b,A_27a,V0f,V1s)) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27b,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( 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 ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V1g) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27b),V1g),V0f)) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__SUPERSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ! [V2v] : ( mem(V2v,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1u),V2v)) & ! [V3x] : ( mem(V3x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) ) ) => ( ap(V0f,V3x) = c_2Enum_2E0 ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2v),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1u),V0f) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__UNION__RZERO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ! [V2v] : ( mem(V2v,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1u)) & ! [V3x] : ( mem(V3x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) ) ) => ( ap(V0f,V3x) = c_2Enum_2E0 ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1u),V0f) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__UNION__LZERO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ! [V2v] : ( mem(V2v,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2v)) & ! [V3x] : ( mem(V3x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) ) ) => ( ap(V0f,V3x) = c_2Enum_2E0 ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2v),V0f) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__RESTRICT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3314(A_27a,V0f,V1s)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__BOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Enum_2Enum)) => ! [V2b] : ( mem(V2b,ty_2Enum_2Enum) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ! [V3x] : ( 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)),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)),V2b))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__BOUND__GEN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Enum_2Enum)) => ! [V2b] : ( mem(V2b,ty_2Enum_2Enum) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ! [V3x] : ( 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,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)),V2b)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__BOUND__LT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Enum_2Enum)) => ! [V2b] : ( mem(V2b,ty_2Enum_2Enum) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ( ! [V3x] : ( 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)),V2b)) ) ) ) & ? [V4x] : ( 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)),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)),V2b))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__BOUND__LT__ALL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Enum_2Enum)) => ! [V2b] : ( mem(V2b,ty_2Enum_2Enum) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ! [V3x] : ( 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)),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)),V2b))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__BOUND__LT__GEN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Enum_2Enum)) => ! [V2b] : ( mem(V2b,ty_2Enum_2Enum) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ! [V3x] : ( 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,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)),V2b)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__UNION__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( 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 ) ) ) => ( ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V3u),V0f) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__EQ__SUPERSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g] : ( mem(V0g,arr(A_27a,ty_2Enum_2Enum)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( 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] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3t)) => ( ap(V1f,V4x) = ap(V0g,V4x) ) ) ) ) & ! [V5x] : ( mem(V5x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2s)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V3t)) ) ) => ( ap(V1f,V5x) = c_2Enum_2E0 ) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V3t),V0g) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__RESTRICT__SET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Enum_2Enum)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3315(A_27a,V0P,V1s))),V2f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3316(A_27a,V2f,V0P)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__NSUM__RESTRICT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27b,bool))) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum))) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V3t)) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3319(A_27a,A_27b,V0R,V3t,V1f)) = ap(ap(c_2Eiterate_2Ensum(A_27b),V3t),f3322(A_27a,A_27b,V0R,V2s,V1f)) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ECARD__EQ__NSUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ( ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),k(A_27a,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__MULTICOUNT__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27b,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ! [V3k] : ( 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] : ( mem(V4j,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t)) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3323(A_27b,A_27a,V1s,V0R,V4j))) = ap(V3k,V4j) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3325(A_27a,A_27b,V2t,V0R)) = ap(ap(c_2Eiterate_2Ensum(A_27b),V2t),f3326(A_27b,V3k)) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__MULTICOUNT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27b,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ! [V3k] : ( mem(V3k,ty_2Enum_2Enum) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) & ! [V4j] : ( mem(V4j,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t)) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3323(A_27b,A_27a,V1s,V0R,V4j))) = V3k ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3325(A_27a,A_27b,V2t,V0R)) = ap(ap(c_2Earithmetic_2E_2A,V3k),ap(c_2Epred__set_2ECARD(A_27b),V2t)) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__IMAGE__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g) = ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),f3328(A_27b,A_27a,V0f,V2s,V1g)) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__GROUP,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( 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)) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27b),V3t),f3330(A_27b,A_27a,V0f,V2s,V1g)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0u] : ( mem(V0u,arr(A_27a,bool)) => ! [V1v] : ( mem(V1v,arr(A_27a,bool)) => ! [V2f] : ( 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] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0u),V1v))) => ( ap(V2f,V3x) = 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__SUBSET__SIMPLE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0u] : ( mem(V0u,arr(A_27a,bool)) => ! [V1v] : ( mem(V1v,arr(A_27a,bool)) => ! [V2f] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__LE__GEN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ( ! [V3x] : ( 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),f3306(A_27a,V1g,V2s)))) ) => 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__IMAGE__NONZERO,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0d] : ( mem(V0d,arr(A_27b,ty_2Enum_2Enum)) => ! [V1i] : ( mem(V1i,arr(A_27a,A_27b)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( 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) ) ) ) ) => ( ap(V0d,ap(V1i,V3x)) = c_2Enum_2E0 ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1i),V2s)),V0d) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27b),V0d),V1i)) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__BIJECTION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1p] : ( mem(V1p,arr(A_27a,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ( ! [V3x] : ( 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] : ( mem(V4y,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f3331(A_27a,V4y,V1p,V2s))) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27a),V0f),V1p)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__NSUM__PRODUCT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,arr(A_27b,bool))) => ! [V2x] : ( mem(V2x,arr(A_27a,arr(A_27b,ty_2Enum_2Enum))) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ! [V3i] : ( 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))) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),f3332(A_27b,A_27a,V1t,V2x)) = 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)),f3334(A_27b,A_27a,V0s,V1t)))),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Enum_2Enum),f3336(A_27a,A_27b,V2x))) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27b,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Enum_2Enum)) => ! [V3g] : ( mem(V3g,arr(A_27b,ty_2Enum_2Enum)) => ! [V4h] : ( mem(V4h,arr(A_27a,A_27b)) => ( ( ( ! [V5y] : ( mem(V5y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5y),V1t)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f3337(A_27b,A_27a,V5y,V4h,V0s))) ) ) ) & ! [V7x] : ( 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(V3g,ap(V4h,V7x)) = ap(V2f,V7x) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V2f) = ap(ap(c_2Eiterate_2Ensum(A_27b),V1t),V3g) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL__INVERSES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27b,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Enum_2Enum)) => ! [V3g] : ( mem(V3g,arr(A_27b,ty_2Enum_2Enum)) => ! [V4h] : ( mem(V4h,arr(A_27a,A_27b)) => ! [V5k] : ( mem(V5k,arr(A_27b,A_27a)) => ( ( ( ! [V6y] : ( 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] : ( 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 ) & ( ap(V3g,ap(V4h,V7x)) = ap(V2f,V7x) ) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V2f) = ap(ap(c_2Eiterate_2Ensum(A_27b),V1t),V3g) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ENSUM__INJECTION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1p] : ( mem(V1p,arr(A_27a,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ( ! [V3x] : ( 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] : ( mem(V4x,A_27a) => ! [V5y] : ( 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 ) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27a),V0f),V1p)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__UNION__NONZERO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))) => ( ap(V0f,V3x) = c_2Enum_2E0 ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__BIGUNION__NONZERO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s)) & ( ( ! [V2t] : ( 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] : ( mem(V3t1,arr(A_27a,bool)) => ! [V4t2] : ( mem(V4t2,arr(A_27a,bool)) => ! [V5x] : ( 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)) ) ) ) ) => ( ap(V0f,V5x) = c_2Enum_2E0 ) ) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s)),V0f) = ap(ap(c_2Eiterate_2Ensum(arr(A_27a,bool)),V1s),f3338(A_27a,V0f)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__CASES,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Enum_2Enum)) => ! [V3g] : ( mem(V3g,arr(A_27a,ty_2Enum_2Enum)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ( ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),f3339(A_27a,V2f,V1P,V3g)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3340(A_27a,V1P,V0s))),V2f)),ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3341(A_27a,V1P,V0s))),V3g)) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__CLOSED,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Enum_2Enum)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(V0P,c_2Enum_2E0)) & ( ( ! [V3x] : ( mem(V3x,ty_2Enum_2Enum) => ! [V4y] : ( mem(V4y,ty_2Enum_2Enum) => ( ( p(ap(V0P,V3x)) & p(ap(V0P,V4y)) ) => p(ap(V0P,ap(ap(c_2Earithmetic_2E_2B,V3x),V4y))) ) ) ) ) & ! [V5a] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ENSUM__ADD__NUMSEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3342(V0f,V1g)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__LE__NUMSEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ! [V4i] : ( mem(V4i,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V4i)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V4i),V3n)) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V0f,V4i)),ap(V1g,V4i))) ) ) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g))) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__EQ__NUMSEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ! [V4i] : ( mem(V4i,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V4i)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V4i),V3n)) ) => ( ap(V0f,V4i) = ap(V1g,V4i) ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__CONST__NUMSEG,lemma,(! [V0c] : ( mem(V0c,ty_2Enum_2Enum) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),k(ty_2Enum_2Enum,V0c)) = ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V1m)),V0c) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__EQ__0__NUMSEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( ! [V3i] : ( mem(V3i,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V3i)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V3i),V2n)) ) => ( ap(V0f,V3i) = c_2Enum_2E0 ) ) ) ) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = c_2Enum_2E0 ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF__NUMSEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = c_2Enum_2E0 ) <=> ! [V3i] : ( mem(V3i,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V3i)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V3i),V2n)) ) => ( ap(V0f,V3i) = c_2Enum_2E0 ) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__TRIV__NUMSEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V2n),V1m)) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = c_2Enum_2E0 ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__SING__NUMSEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1n),V1n)),V0f) = ap(V0f,V1n) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__CLAUSES__NUMSEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ( ( ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),c_2Enum_2E0)),V0f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1m),c_2Enum_2E0)),ap(V0f,c_2Enum_2E0)),c_2Enum_2E0) ) ) ) & ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),ap(c_2Enum_2ESUC,V3n))),V0f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_3C_3D,V2m),ap(c_2Enum_2ESUC,V3n))),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(V0f,ap(c_2Enum_2ESUC,V3n)))),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__SWAP__NUMSEG,lemma,(! [V0a] : ( mem(V0a,ty_2Enum_2Enum) => ! [V1b] : ( mem(V1b,ty_2Enum_2Enum) => ! [V2c] : ( mem(V2c,ty_2Enum_2Enum) => ! [V3d] : ( mem(V3d,ty_2Enum_2Enum) => ! [V4f] : ( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0a),V1b)),f3343(V2c,V3d,V4f)) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2c),V3d)),f3345(V0a,V1b,V4f)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__ADD__SPLIT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ! [V3p] : ( mem(V3p,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),ap(ap(c_2Earithmetic_2E_2B,V2n),V3p))),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Earithmetic_2E_2B,V2n),V3p))),V0f)) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__OFFSET,lemma,(! [V0p] : ( mem(V0p,ty_2Enum_2Enum) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V2m),V0p)),ap(ap(c_2Earithmetic_2E_2B,V3n),V0p))),V1f) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3346(V0p,V1f)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__OFFSET__0,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n)) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V2n),V1m))),f3347(V1m,V0f)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__CLAUSES__LEFT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n)) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,V1m)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V1m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2n)),V0f)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__CLAUSES__RIGHT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V2n)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n)) ) => ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0f)),ap(V0f,V2n)) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ENSUM__PAIR,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,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,c_2Earithmetic_2EZERO))),V1m)),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0f) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),f3348(V0f)) ) ) ) ))).
fof(conj_thm_2Eiterate_2EMOD__NSUM__MOD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ~~ ( V1n = c_2Enum_2E0 ) ) ) => ( ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),V1n) = ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3349(A_27a,V0f,V1n))),V1n) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2EMOD__NSUM__MOD__NUMSEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V1a] : ( mem(V1a,ty_2Enum_2Enum) => ! [V2b] : ( mem(V2b,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ~~ ( V3n = c_2Enum_2E0 ) ) => ( ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1a),V2b)),V0f)),V3n) = ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1a),V2b)),f3350(V0f,V3n))),V3n) ) ) ) ) ) ))).
fof(conj_thm_2Eiterate_2ECARD__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V0s)) & ( ( ! [V1t] : ( 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] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( 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) ) ) ) ) ) ) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s)) = ap(ap(c_2Eiterate_2Ensum(arr(A_27a,bool)),V0s),c_2Epred__set_2ECARD(A_27a)) ) ) )))).
fof(ax_thm_2Eiterate_2Esum__def,axiom,( ! [A_27a] : ( ne(A_27a) => ( c_2Eiterate_2ESum(A_27a) = ap(c_2Eiterate_2Eiterate(A_27a,ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add) ) ) )).
fof(conj_thm_2Eiterate_2ENEUTRAL__REAL__ADD,lemma,(( ap(c_2Eiterate_2Eneutral(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ))).
fof(conj_thm_2Eiterate_2ENEUTRAL__REAL__MUL,lemma,(( ap(c_2Eiterate_2Eneutral(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__mul) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ))).
fof(conj_thm_2Eiterate_2EMONOIDAL__REAL__ADD,lemma,(p(ap(c_2Eiterate_2Emonoidal(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add)))).
fof(conj_thm_2Eiterate_2EMONOIDAL__REAL__MUL,lemma,(p(ap(c_2Eiterate_2Emonoidal(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__mul)))).
fof(conj_thm_2Eiterate_2ESUM__DEGENERATE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3351(A_27a,V0f,V1s)))) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__CLAUSES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( ( ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) & ! [V1x] : ( mem(V1x,A_27b) => ! [V2f] : ( mem(V2f,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),V3s)) => ( ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27b),V1x),V3s)),V2f) = 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))) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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)) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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)) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__INCL__EXCL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2f] : ( 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)) ) => ( ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),V1t),V2f)) = 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)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__SUPPORT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( 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) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__ADD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3352(A_27a,V0f,V1g)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__ADD__GEN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3353(A_27a,V0f,V2s)))) & p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3354(A_27a,V1g,V2s)))) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3355(A_27a,V0f,V1g)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__EQ__0,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) => ( ap(V0f,V2x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__0,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V0s),k(A_27a,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) )))).
fof(conj_thm_2Eiterate_2ESUM__LMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3356(A_27a,V0f,V1c)) = ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__RMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3357(A_27a,V0f,V1c)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),V1c) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__NEG,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3358(A_27a,V0f)) = ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__SUB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3359(A_27a,V0f,V1g)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ! [V3x] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__LT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ( ! [V3x] : ( 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] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__LT__ALL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ( ~~ ( V2s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ! [V3x] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__POS__LT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & ( ( ! [V2x] : ( 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,c_2Enum_2E0)),ap(V0f,V2x))) ) ) ) & ? [V3x] : ( 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,c_2Enum_2E0)),ap(V0f,V3x))) ) ) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f))) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__POS__LT__ALL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( 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] : ( 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,c_2Enum_2E0)),ap(V1f,V2i))) ) ) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f))) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s)) => ( ap(V0f,V3x) = ap(V1g,V3x) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__ABS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f))),ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3360(A_27a,V0f)))) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__ABS__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V3x))),ap(V1g,V3x))) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f))),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__CONST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),k(A_27a,V0c)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V1s))),V0c) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__POS__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ! [V2x] : ( 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,c_2Enum_2E0)),ap(V0f,V2x))) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f))) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__POS__BOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V3x))) ) ) ) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),V1b)) ) ) => ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4x)),V1b)) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__POS__EQ__0,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & ( ( ! [V2x] : ( 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,c_2Enum_2E0)),ap(V0f,V2x))) ) ) ) & ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => ( ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__ZERO__EXISTS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0u] : ( mem(V0u,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0u) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ( ! [V2i] : ( mem(V2i,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2i),V1s)) => ( ap(V0u,V2i) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) | ? [V3j] : ( mem(V3j,A_27a) & ? [V4k] : ( mem(V4k,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3j),V1s)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(V0u,V3j)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4k),V1s)) & p(ap(ap(c_2Ereal_2Ereal__gt,ap(V0u,V4k)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2a] : ( mem(V2a,A_27a) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2a)),V0f) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(V0f,V2a)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__DELETE__CASES,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2a] : ( mem(V2a,A_27a) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2a)),V0f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(V0f,V2a))),ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,A_27a) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),V0f) = ap(V0f,V1x) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__DELTA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0b] : ( mem(V0b,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2a] : ( mem(V2a,A_27a) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3361(A_27a,V0b,V2a)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)),V0b),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__SWAP,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3362(A_27a,A_27b,V0f,V2t)) = ap(ap(c_2Eiterate_2ESum(A_27b),V2t),f3363(A_27b,A_27a,V0f,V1s)) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( 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 ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V1g) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27b),V1g),V0f)) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__SUPERSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ! [V2v] : ( mem(V2v,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1u),V2v)) & ! [V3x] : ( mem(V3x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) ) ) => ( ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2v),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V1u),V0f) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__UNION__RZERO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ! [V2v] : ( mem(V2v,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1u)) & ! [V3x] : ( mem(V3x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) ) ) => ( ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V1u),V0f) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__UNION__LZERO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ! [V2v] : ( mem(V2v,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2v)) & ! [V3x] : ( mem(V3x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) ) ) => ( ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V2v),V0f) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__RESTRICT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3364(A_27a,V0f,V1s)) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__BOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),V2b)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))),V2b))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__BOUND__GEN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),ap(ap(c_2Ereal_2E_2F,V2b),ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))))) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),V2b)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__ABS__BOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V1f,V3x))),V2b)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))),V2b))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__BOUND__LT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),V2b)) ) ) ) & ? [V4x] : ( mem(V4x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(V1f,V4x)),V2b)) ) ) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))),V2b))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__BOUND__LT__ALL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(V1f,V3x)),V2b)) ) ) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))),V2b))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__BOUND__LT__GEN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(V1f,V3x)),ap(ap(c_2Ereal_2E_2F,V2b),ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))))) ) ) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),V2b)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__UNION__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( 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 ) ) ) => ( ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f)) = ap(ap(c_2Eiterate_2ESum(A_27a),V3u),V0f) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__EQ__SUPERSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g] : ( mem(V0g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( 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] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3t)) => ( ap(V1f,V4x) = ap(V0g,V4x) ) ) ) ) & ! [V5x] : ( mem(V5x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2s)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V3t)) ) ) => ( ap(V1f,V5x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1f) = ap(ap(c_2Eiterate_2ESum(A_27a),V3t),V0g) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__RESTRICT__SET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3315(A_27a,V0P,V1s))),V2f) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3365(A_27a,V2f,V0P)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__SUM__RESTRICT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27b,bool))) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal))) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V3t)) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3366(A_27a,A_27b,V0R,V3t,V1f)) = ap(ap(c_2Eiterate_2ESum(A_27b),V3t),f3367(A_27a,A_27b,V0R,V2s,V1f)) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ECARD__EQ__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ( ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s)) = ap(ap(c_2Eiterate_2ESum(A_27a),V0s),k(A_27a,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__MULTICOUNT__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27b,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ! [V3k] : ( 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] : ( mem(V4j,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t)) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3323(A_27b,A_27a,V1s,V0R,V4j))) = ap(V3k,V4j) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3368(A_27b,A_27a,V0R,V2t)) = ap(ap(c_2Eiterate_2ESum(A_27b),V2t),f3369(A_27b,V3k)) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__MULTICOUNT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27b,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ! [V3k] : ( mem(V3k,ty_2Enum_2Enum) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) & ! [V4j] : ( mem(V4j,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t)) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3323(A_27b,A_27a,V1s,V0R,V4j))) = V3k ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3368(A_27b,A_27a,V0R,V2t)) = ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2A,V3k),ap(c_2Epred__set_2ECARD(A_27b),V2t))) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__IMAGE__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g) = ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),f3370(A_27b,A_27a,V0f,V2s,V1g)) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__GROUP,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( 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)) ) => ( ap(ap(c_2Eiterate_2ESum(A_27b),V3t),f3371(A_27b,A_27a,V0f,V2s,V1g)) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2EREAL__OF__NUM__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3372(A_27a,V0f)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0u] : ( mem(V0u,arr(A_27a,bool)) => ! [V1v] : ( mem(V1v,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0u)) & ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1v)) & ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0u),V1v))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V2f,V3x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) & ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1v),V0u))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V2f,V4x))) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V0u),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),V1v),V2f))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__SUBSET__SIMPLE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0u] : ( mem(V0u,arr(A_27a,bool)) => ! [V1v] : ( mem(V1v,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1v)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0u),V1v)) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1v),V0u))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V2f,V3x))) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V0u),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),V1v),V2f))) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__IMAGE__NONZERO,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0d] : ( mem(V0d,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V1i] : ( mem(V1i,arr(A_27a,A_27b)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( 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) ) ) ) ) => ( ap(V0d,ap(V1i,V3x)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1i),V2s)),V0d) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27b),V0d),V1i)) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__BIJECTION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1p] : ( mem(V1p,arr(A_27a,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ( ! [V3x] : ( 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] : ( mem(V4y,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f3331(A_27a,V4y,V1p,V2s))) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27a),V0f),V1p)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__SUM__PRODUCT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,arr(A_27b,bool))) => ! [V2x] : ( mem(V2x,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal))) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ! [V3i] : ( 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))) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V0s),f3373(A_27b,A_27a,V1t,V2x)) = ap(ap(c_2Eiterate_2ESum(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)),f3334(A_27b,A_27a,V0s,V1t)))),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Erealax_2Ereal),f3336(A_27a,A_27b,V2x))) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__EQ__GENERAL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27b,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3g] : ( mem(V3g,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V4h] : ( mem(V4h,arr(A_27a,A_27b)) => ( ( ( ! [V5y] : ( mem(V5y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5y),V1t)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f3337(A_27b,A_27a,V5y,V4h,V0s))) ) ) ) & ! [V7x] : ( 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(V3g,ap(V4h,V7x)) = ap(V2f,V7x) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f) = ap(ap(c_2Eiterate_2ESum(A_27b),V1t),V3g) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__EQ__GENERAL__INVERSES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27b,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3g] : ( mem(V3g,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V4h] : ( mem(V4h,arr(A_27a,A_27b)) => ! [V5k] : ( mem(V5k,arr(A_27b,A_27a)) => ( ( ( ! [V6y] : ( 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] : ( 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 ) & ( ap(V3g,ap(V4h,V7x)) = ap(V2f,V7x) ) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f) = ap(ap(c_2Eiterate_2ESum(A_27b),V1t),V3g) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eiterate_2ESUM__INJECTION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1p] : ( mem(V1p,arr(A_27a,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ( ! [V3x] : ( 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] : ( mem(V4x,A_27a) => ! [V5y] : ( 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 ) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27a),V0f),V1p)) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__UNION__NONZERO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))) => ( ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f)) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__BIGUNION__NONZERO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s)) & ( ( ! [V2t] : ( 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] : ( mem(V3t1,arr(A_27a,bool)) => ! [V4t2] : ( mem(V4t2,arr(A_27a,bool)) => ! [V5x] : ( 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)) ) ) ) ) => ( ap(V0f,V5x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ) => ( ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s)),V0f) = ap(ap(c_2Eiterate_2ESum(arr(A_27a,bool)),V1s),f3374(A_27a,V0f)) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__CASES,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3g] : ( mem(V3g,arr(A_27a,ty_2Erealax_2Ereal)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ( ap(ap(c_2Eiterate_2ESum(A_27a),V0s),f3375(A_27a,V2f,V1P,V3g)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3340(A_27a,V1P,V0s))),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3341(A_27a,V1P,V0s))),V3g)) ) ) ) ) ) )))).
fof(conj_thm_2Eiterate_2ESUM__CASES__1,conjecture,( ! [A_27a]: ( ne(A_27a) => ! [V0y] : ( mem(V0y,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3a] : ( 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(c_2Eiterate_2ESum(A_27a),V2s),f3376(A_27a,V0y,V3a,V1f)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1f)),ap(ap(c_2Ereal_2Ereal__sub,V0y),ap(V1f,V3a))) ) ) ) ) ) )))).
