include('Axioms/bit+2.ax').
fof(mem_c_2Epred__set_2EBIGINTER,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EBIGINTER(A_27a),arr(arr(arr(A_27a,bool),bool),arr(A_27a,bool))) ) )).
fof(mem_c_2Epred__set_2EBIGUNION,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EBIGUNION(A_27a),arr(arr(arr(A_27a,bool),bool),arr(A_27a,bool))) ) )).
fof(mem_c_2Epred__set_2EBIJ,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2EBIJ(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool)))) ) ) )).
fof(mem_c_2Epred__set_2ECARD,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2ECARD(A_27a),arr(arr(A_27a,bool),ty_2Enum_2Enum)) ) )).
fof(mem_c_2Epred__set_2ECHOICE,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2ECHOICE(A_27a),arr(arr(A_27a,bool),A_27a)) ) )).
fof(mem_c_2Epred__set_2ECOMPL,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2ECOMPL(A_27a),arr(arr(A_27a,bool),arr(A_27a,bool))) ) )).
fof(mem_c_2Epred__set_2ECROSS,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2ECROSS(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),arr(ty_2Epair_2Eprod(A_27a,A_27b),bool)))) ) ) )).
fof(mem_c_2Epred__set_2EDELETE,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EDELETE(A_27a),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool)))) ) )).
fof(mem_c_2Epred__set_2EDFUNSET,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2EDFUNSET(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(A_27a,A_27b),bool)))) ) ) )).
fof(mem_c_2Epred__set_2EDIFF,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EDIFF(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) )).
fof(mem_c_2Epred__set_2EDISJOINT,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EDISJOINT(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Epred__set_2EEMPTY,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EEMPTY(A_27a),arr(A_27a,bool)) ) )).
fof(mem_c_2Epred__set_2EFINITE,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EFINITE(A_27a),arr(arr(A_27a,bool),bool)) ) )).
fof(mem_c_2Epred__set_2EFUNSET,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2EFUNSET(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),arr(arr(A_27a,A_27b),bool)))) ) ) )).
fof(mem_c_2Epred__set_2EGSPEC,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2EGSPEC(A_27a,A_27b),arr(arr(A_27b,ty_2Epair_2Eprod(A_27a,bool)),arr(A_27a,bool))) ) ) )).
fof(mem_c_2Epred__set_2EIMAGE,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2EIMAGE(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(A_27b,bool)))) ) ) )).
fof(mem_c_2Epred__set_2EINJ,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2EINJ(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool)))) ) ) )).
fof(mem_c_2Epred__set_2EINSERT,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EINSERT(A_27a),arr(A_27a,arr(arr(A_27a,bool),arr(A_27a,bool)))) ) )).
fof(mem_c_2Epred__set_2EINTER,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EINTER(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) )).
fof(mem_c_2Epred__set_2EITSET,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2EITSET(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,A_27b)),arr(arr(A_27a,bool),arr(A_27b,A_27b)))) ) ) )).
fof(mem_c_2Epred__set_2ELINV,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2ELINV(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(A_27b,A_27a)))) ) ) )).
fof(mem_c_2Epred__set_2ELINV__OPT,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2ELINV__OPT(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(A_27b,ty_2Eoption_2Eoption(A_27a))))) ) ) )).
fof(mem_c_2Epred__set_2EMAX__SET,axiom,( mem(c_2Epred__set_2EMAX__SET,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) )).
fof(mem_c_2Epred__set_2EMIN__SET,axiom,( mem(c_2Epred__set_2EMIN__SET,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) )).
fof(mem_c_2Epred__set_2EPOW,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EPOW(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Epred__set_2EPREIMAGE,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27b,bool),arr(A_27a,bool)))) ) ) )).
fof(mem_c_2Epred__set_2EPROD__IMAGE,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EPROD__IMAGE(A_27a),arr(arr(A_27a,ty_2Enum_2Enum),arr(arr(A_27a,bool),ty_2Enum_2Enum))) ) )).
fof(mem_c_2Epred__set_2EPROD__SET,axiom,( mem(c_2Epred__set_2EPROD__SET,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) )).
fof(mem_c_2Epred__set_2EPSUBSET,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EPSUBSET(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Epred__set_2EREL__RESTRICT,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EREL__RESTRICT(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))) ) )).
fof(mem_c_2Epred__set_2EREST,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EREST(A_27a),arr(arr(A_27a,bool),arr(A_27a,bool))) ) )).
fof(mem_c_2Epred__set_2ERINV,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2ERINV(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(A_27b,A_27a)))) ) ) )).
fof(mem_c_2Epred__set_2ESING,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2ESING(A_27a),arr(arr(A_27a,bool),bool)) ) )).
fof(mem_c_2Epred__set_2ESUBSET,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2ESUBSET(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Epred__set_2ESUM__IMAGE,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2ESUM__IMAGE(A_27a),arr(arr(A_27a,ty_2Enum_2Enum),arr(arr(A_27a,bool),ty_2Enum_2Enum))) ) )).
fof(mem_c_2Epred__set_2ESUM__SET,axiom,( mem(c_2Epred__set_2ESUM__SET,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) )).
fof(mem_c_2Epred__set_2ESURJ,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Epred__set_2ESURJ(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool)))) ) ) )).
fof(mem_c_2Epred__set_2EUNION,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EUNION(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) )).
fof(mem_c_2Epred__set_2EUNIV,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2EUNIV(A_27a),arr(A_27a,bool)) ) )).
fof(mem_c_2Epred__set_2Echooser,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2Echooser(A_27a),arr(arr(A_27a,bool),arr(ty_2Enum_2Enum,A_27a))) ) )).
fof(mem_c_2Epred__set_2Ecount,axiom,( mem(c_2Epred__set_2Ecount,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) )).
fof(mem_c_2Epred__set_2Ecountable,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2Ecountable(A_27a),arr(arr(A_27a,bool),bool)) ) )).
fof(mem_c_2Epred__set_2Eenumerate,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2Eenumerate(A_27a),arr(arr(A_27a,bool),arr(ty_2Enum_2Enum,A_27a))) ) )).
fof(mem_c_2Epred__set_2Eequiv__on,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2Eequiv__on(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Epred__set_2Eis__measure__maximal,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2Eis__measure__maximal(A_27a),arr(arr(A_27a,ty_2Enum_2Enum),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) )).
fof(mem_c_2Epred__set_2Enum__to__pair,axiom,( mem(c_2Epred__set_2Enum__to__pair,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum))) )).
fof(mem_c_2Epred__set_2Epair__to__num,axiom,( mem(c_2Epred__set_2Epair__to__num,arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum)) )).
fof(mem_c_2Epred__set_2Epairwise,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2Epairwise(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Epred__set_2Epartition,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2Epartition(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))) ) )).
fof(mem_c_2Epred__set_2Eschroeder__close,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Epred__set_2Eschroeder__close(A_27a),arr(arr(A_27a,A_27a),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) )).
fof(conj_thm_2Epred__set_2ESPECIFICATION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0P)) <=> p(ap(V0P,V1x)) ) ) )))).
fof(conj_thm_2Epred__set_2EIN__APP,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1P)) <=> p(ap(V1P,V0x)) ) ) )))).
fof(conj_thm_2Epred__set_2EIN__ABS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),f252(A_27a,V1P))) <=> p(ap(V1P,V0x)) ) ) )))).
fof(conj_thm_2Epred__set_2EEXTENSION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( V0s = V1t ) <=> ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) <=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ENOT__EQUAL__SETS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( ~~ ( V0s = V1t ) ) <=> ? [V2x] : ( mem(V2x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) <=> ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ENUM__SET__WOP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( ( ? [V1n] : ( mem(V1n,ty_2Enum_2Enum) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1n),V0s)) ) ) <=> ? [V2n] : ( mem(V2n,ty_2Enum_2Enum) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2n),V0s)) & ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3m),V0s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V2n),V3m)) ) ) ) ) ) ))).
fof(ax_thm_2Epred__set_2EGSPECIFICATION,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27b,ty_2Epair_2Eprod(A_27a,bool))) => ! [V1v] : ( mem(V1v,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1v),ap(c_2Epred__set_2EGSPEC(A_27a,A_27b),V0f))) <=> ? [V2x] : ( mem(V2x,A_27b) & ( ap(ap(c_2Epair_2E_2C(A_27a,bool),V1v),c_2Ebool_2ET) = ap(V0f,V2x) ) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EGSPECIFICATION__applied,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27b,ty_2Epair_2Eprod(A_27a,bool))) => ! [V1v] : ( mem(V1v,A_27a) => ( p(ap(ap(c_2Epred__set_2EGSPEC(A_27a,A_27b),V0f),V1v)) <=> ? [V2x] : ( mem(V2x,A_27b) & ( ap(ap(c_2Epair_2E_2C(A_27a,bool),V1v),c_2Ebool_2ET) = ap(V0f,V2x) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EGSPEC__ETA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f253(A_27a,V0P)) = V0P ) )))).
fof(conj_thm_2Epred__set_2EGSPEC__PAIR__ETA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,arr(A_27b,bool))) => ( 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)),f255(A_27a,A_27b,V0P))) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),V0P) ) ))))).
fof(conj_thm_2Epred__set_2EIN__GSPEC__IFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0y] : ( mem(V0y,A_27a) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0y),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f256(A_27a,V1P)))) <=> p(ap(V1P,V0y)) ) ) )))).
fof(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__IFF,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27b) => ! [V2P] : ( mem(V2P,arr(A_27a,arr(A_27b,bool))) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y)),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)),f258(A_27a,A_27b,V2P))))) <=> p(ap(ap(V2P,V0x),V1y)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIN__GSPEC,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1y] : ( mem(V1y,A_27a) => ! [V2x] : ( mem(V2x,A_27b) => ! [V3P] : ( mem(V3P,arr(A_27a,bool)) => ( ( p(ap(V3P,V1y)) & ( V2x = ap(V0f,V1y) ) ) => p(ap(ap(c_2Ebool_2EIN(A_27b),V2x),ap(c_2Epred__set_2EGSPEC(A_27b,A_27a),f259(A_27b,A_27a,V0f,V3P)))) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__1,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,A_27a) => ! [V1b] : ( mem(V1b,A_27b) => ! [V2x] : ( mem(V2x,A_27b) => ! [V3P] : ( mem(V3P,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0a),V1b)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),f260(A_27b,A_27a,V2x,V3P)))) <=> ( p(ap(V3P,V0a)) & ( V1b = V2x ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__2,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,A_27a) => ! [V1b] : ( mem(V1b,A_27b) => ! [V2x] : ( mem(V2x,A_27a) => ! [V3P] : ( mem(V3P,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0a),V1b)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27b),A_27b),f261(A_27a,A_27b,V2x,V3P)))) <=> ( p(ap(V3P,V1b)) & ( V0a = V2x ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__same,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,A_27a) => ! [V1b] : ( mem(V1b,A_27a) => ! [V2P] : ( mem(V2P,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0a),V1b)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27a),A_27a),f262(A_27a,V2P)))) <=> ( p(ap(V2P,V0a)) & ( V0a = V1b ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESET__MINIMUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1M] : ( mem(V1M,arr(A_27a,ty_2Enum_2Enum)) => ( ( ? [V2x] : ( mem(V2x,A_27a) & p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) ) ) <=> ? [V3x] : ( mem(V3x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) & ! [V4y] : ( mem(V4y,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V0s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V1M,V3x)),ap(V1M,V4y))) ) ) ) ) ) ) )))).
fof(ax_thm_2Epred__set_2EEMPTY__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ( c_2Epred__set_2EEMPTY(A_27a) = k(A_27a,c_2Ebool_2EF) ) ) )).
fof(conj_thm_2Epred__set_2ENOT__IN__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))) ) )))).
fof(conj_thm_2Epred__set_2EMEMBER__NOT__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ? [V1x] : ( mem(V1x,A_27a) & p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) ) ) <=> ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) )))).
fof(conj_thm_2Epred__set_2EEMPTY__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( p(ap(c_2Epred__set_2EEMPTY(A_27a),V0x)) <=> $false ) )))).
fof(ax_thm_2Epred__set_2EUNIV__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ( c_2Epred__set_2EUNIV(A_27a) = k(A_27a,c_2Ebool_2ET) ) ) )).
fof(conj_thm_2Epred__set_2EIN__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),c_2Epred__set_2EUNIV(A_27a))) )))).
fof(conj_thm_2Epred__set_2EUNIV__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => p(ap(c_2Epred__set_2EUNIV(A_27a),V0x)) )))).
fof(conj_thm_2Epred__set_2EUNIV__NOT__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ( ~~ ( c_2Epred__set_2EUNIV(A_27a) = c_2Epred__set_2EEMPTY(A_27a) ) )))).
fof(conj_thm_2Epred__set_2EEMPTY__NOT__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ( ~~ ( c_2Epred__set_2EEMPTY(A_27a) = c_2Epred__set_2EUNIV(A_27a) ) )))).
fof(conj_thm_2Epred__set_2EEQ__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ! [V1x] : ( mem(V1x,A_27a) => p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) ) ) <=> ( V0s = c_2Epred__set_2EUNIV(A_27a) ) ) )))).
fof(conj_thm_2Epred__set_2EIN__EQ__UNIV__IMP,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( V0s = c_2Epred__set_2EUNIV(A_27a) ) => ! [V1v] : ( mem(V1v,A_27a) => p(ap(ap(c_2Ebool_2EIN(A_27a),V1v),V0s)) ) ) )))).
fof(ax_thm_2Epred__set_2ESUBSET__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) <=> ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) => p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2ESUBSET__THM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0P),V1Q)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P)) => p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1Q)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) <=> ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(V0s,V2x)) => p(ap(V1t,V2x)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__TRANS,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(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V2u)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V2u)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V0s)) )))).
fof(conj_thm_2Epred__set_2ESUBSET__ANTISYM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) ) => ( V0s = V1t ) ) ) )))).
fof(conj_thm_2Epred__set_2EEMPTY__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0s)) )))).
fof(conj_thm_2Epred__set_2ESUBSET__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),c_2Epred__set_2EEMPTY(A_27a))) <=> ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),c_2Epred__set_2EUNIV(A_27a))) )))).
fof(conj_thm_2Epred__set_2EUNIV__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),c_2Epred__set_2EUNIV(A_27a)),V0s)) <=> ( V0s = c_2Epred__set_2EUNIV(A_27a) ) ) )))).
fof(conj_thm_2Epred__set_2EEQ__SUBSET__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( V0s = V1t ) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__SUBSET__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) ) <=> ( V0s = V1t ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__ADD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ! [V2d] : ( mem(V2d,ty_2Enum_2Enum) => ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V0f,V3n)),ap(V0f,ap(c_2Enum_2ESUC,V3n)))) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V0f,V1n)),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V1n),V2d)))) ) ) ) )))).
fof(conj_thm_2Epred__set_2EK__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,bool) => ! [V1y] : ( mem(V1y,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Ecombin_2EK(bool,A_27a),V0x)),V1y)) <=> ( ( ~~ p(V0x) ) | p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),c_2Epred__set_2EUNIV(A_27a)),V1y)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__K,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ! [V1y] : ( mem(V1y,bool) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0x),ap(c_2Ecombin_2EK(bool,A_27a),V1y))) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))) | p(V1y) ) ) ) )))).
fof(ax_thm_2Epred__set_2EPSUBSET__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V1t)) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) & ( ~~ ( V0s = V1t ) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EPSUBSET__TRANS,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(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V1t)) & p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V1t),V2u)) ) => p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V2u)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EPSUBSET__IRREFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ~~ p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V0s)) ) )))).
fof(conj_thm_2Epred__set_2ENOT__PSUBSET__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ~~ p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),c_2Epred__set_2EEMPTY(A_27a))) ) )))).
fof(conj_thm_2Epred__set_2ENOT__UNIV__PSUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ~~ p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),c_2Epred__set_2EUNIV(A_27a)),V0s)) ) )))).
fof(conj_thm_2Epred__set_2EPSUBSET__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),c_2Epred__set_2EUNIV(A_27a))) <=> ? [V1x] : ( mem(V1x,A_27a) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) ) ) ) )))).
fof(ax_thm_2Epred__set_2EUNION__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f263(A_27a,V1t,V0s)) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t))) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) | p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EUNION__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t),V2x)) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) | p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EUNION__ASSOC,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)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),ap(ap(c_2Epred__set_2EUNION(A_27a),V1t),V2u)) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)),V2u) ) ) ) )))).
fof(conj_thm_2Epred__set_2EUNION__IDEMPOT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V0s) = V0s ) )))).
fof(conj_thm_2Epred__set_2EUNION__COMM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t) = ap(ap(c_2Epred__set_2EUNION(A_27a),V1t),V0s) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t))) ) ) ) & ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2s),ap(ap(c_2Epred__set_2EUNION(A_27a),V3t),V2s))) ) ) )))).
fof(conj_thm_2Epred__set_2EUNION__SUBSET,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(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)),V2u)) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V2u)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V2u)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__UNION__ABSORPTION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) <=> ( ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t) = V1t ) ) ) )))).
fof(conj_thm_2Epred__set_2EUNION__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0s) = V0s ) ) ) & ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),c_2Epred__set_2EEMPTY(A_27a)) = V1s ) ) )))).
fof(conj_thm_2Epred__set_2EUNION__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),c_2Epred__set_2EUNIV(A_27a)),V0s) = c_2Epred__set_2EUNIV(A_27a) ) ) ) & ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),c_2Epred__set_2EUNIV(A_27a)) = c_2Epred__set_2EUNIV(A_27a) ) ) )))).
fof(conj_thm_2Epred__set_2EEMPTY__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t) = c_2Epred__set_2EEMPTY(A_27a) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) & ( V1t = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFORALL__IN__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) => p(ap(V0P,V3x)) ) ) ) <=> ( ( ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1s)) => p(ap(V0P,V4x)) ) ) ) & ! [V5x] : ( mem(V5x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2t)) => p(ap(V0P,V5x)) ) ) ) ) ) ) )))).
fof(ax_thm_2Epred__set_2EINTER__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f264(A_27a,V1t,V0s)) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t))) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t),V2x)) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__ASSOC,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)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),ap(ap(c_2Epred__set_2EINTER(A_27a),V1t),V2u)) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t)),V2u) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__IDEMPOT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V0s) = V0s ) )))).
fof(conj_thm_2Epred__set_2EINTER__COMM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t) = ap(ap(c_2Epred__set_2EINTER(A_27a),V1t),V0s) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t)),V0s)) ) ) ) & ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V3t),V2s)),V2s)) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__INTER,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(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),ap(ap(c_2Epred__set_2EINTER(A_27a),V1t),V2u))) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V2u)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__INTER__ABSORPTION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) <=> ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t) = V0s ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__INTER1,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t) = V0s ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__INTER2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V1t),V0s) = V0s ) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0s) = c_2Epred__set_2EEMPTY(A_27a) ) ) ) & ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),c_2Epred__set_2EEMPTY(A_27a)) = c_2Epred__set_2EEMPTY(A_27a) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),c_2Epred__set_2EUNIV(A_27a)),V0s) = V0s ) ) ) & ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),c_2Epred__set_2EUNIV(A_27a)) = V1s ) ) )))).
fof(conj_thm_2Epred__set_2EUNION__OVER__INTER,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)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),ap(ap(c_2Epred__set_2EUNION(A_27a),V1t),V2u)) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t)),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V2u)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__OVER__UNION,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)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),ap(ap(c_2Epred__set_2EINTER(A_27a),V1t),V2u)) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V2u)) ) ) ) )))).
fof(ax_thm_2Epred__set_2EDISJOINT__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V1t)) <=> ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t) = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__DISJOINT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V1t)) <=> ( ~~ ? [V2x] : ( mem(V2x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__SYM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V1t)) <=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1t),V0s)) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__ALT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V1t)) <=> ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) => ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0s)) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),c_2Epred__set_2EEMPTY(A_27a))) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__EMPTY__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) <=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V0s)) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__EMPTY__REFL__RWT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V0s)) <=> ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__UNION,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(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)),V2u)) <=> ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V2u)) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1t),V2u)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__UNION__BOTH,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(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)),V2u)) <=> ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V2u)) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1t),V2u)) ) ) & ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V2u),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t))) <=> ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V2u)) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1t),V2u)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__SUBSET,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(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2u),V1t)) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V2u)) ) ) ) )))).
fof(ax_thm_2Epred__set_2EDIFF__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f265(A_27a,V1t,V0s)) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t))) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t),V2x)) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),c_2Epred__set_2EEMPTY(A_27a)) = V0s ) )))).
fof(conj_thm_2Epred__set_2EEMPTY__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0s) = c_2Epred__set_2EEMPTY(A_27a) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),c_2Epred__set_2EUNIV(A_27a)) = c_2Epred__set_2EEMPTY(A_27a) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t)),V1t) = ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__EQ__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V0s) = c_2Epred__set_2EEMPTY(A_27a) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t)),V0s)) ) )))).
fof(conj_thm_2Epred__set_2EUNION__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) => ( ( ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1t),V0s)) = V1t ) & ( ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1t),V0s)),V0s) = V1t ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__DIFF__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t)) = V1t ) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ! [V1y] : ( mem(V1y,arr(A_27a,bool)) => ! [V2z] : ( mem(V2z,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0x),ap(ap(c_2Epred__set_2EUNION(A_27a),V1y),V2z)) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0x),V1y)),V2z) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__COMM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ! [V1y] : ( mem(V1y,arr(A_27a,bool)) => ! [V2z] : ( mem(V2z,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0x),V1y)),V2z) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0x),V2z)),V1y) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__SAME__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ! [V1y] : ( mem(V1y,arr(A_27a,bool)) => ( ( ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0x),V1y)),V0x) = ap(ap(c_2Epred__set_2EDIFF(A_27a),V1y),V0x) ) & ( ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0x),V1y)),V1y) = ap(ap(c_2Epred__set_2EDIFF(A_27a),V0x),V1y) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2g] : ( mem(V2g,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t)),V2g) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V2g)),V1t) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__INTER2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),ap(ap(c_2Epred__set_2EINTER(A_27a),V1t),V0s)) = ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1t),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t))) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t)),V1t)) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__DIFFS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g] : ( mem(V0g,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ( ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,V4n)),ap(V1f,ap(c_2Enum_2ESUC,V4n)))) ) ) & ( ( ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( ap(V0g,V5n) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(V1f,ap(c_2Enum_2ESUC,V5n))),ap(V1f,V5n)) ) ) ) & ( ~~ ( V2m = V3n ) ) ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0g,V2m)),ap(V0g,V3n))) ) ) ) ) )))).
fof(ax_thm_2Epred__set_2EINSERT__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f266(A_27a,V1s,V0x)) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1y),V2s))) <=> ( ( V0x = V1y ) | p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2s)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINSERT(A_27a),V1y),V2s),V0x)) <=> ( ( V0x = V1y ) | p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2s)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECOMPONENT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s))) ) )))).
fof(conj_thm_2Epred__set_2ESET__CASES,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) | ? [V1x] : ( mem(V1x,A_27a) & ? [V2t] : ( mem(V2t,arr(A_27a,bool)) & ( ( V0s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V2t) ) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDECOMPOSITION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) <=> ? [V2t] : ( mem(V2t,arr(A_27a,bool)) & ( ( V0s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V2t) ) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EABSORPTION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) <=> ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s) = V1s ) ) ) )))).
fof(conj_thm_2Epred__set_2EABSORPTION__RWT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) => ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s) = V1s ) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s)) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__COMM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1y),V2s)) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V1y),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V2s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EUNIV(A_27a)) = c_2Epred__set_2EUNIV(A_27a) ) )))).
fof(conj_thm_2Epred__set_2ENOT__INSERT__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ~~ ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s) = c_2Epred__set_2EEMPTY(A_27a) ) ) ) )))).
fof(conj_thm_2Epred__set_2ENOT__EMPTY__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ~~ ( c_2Epred__set_2EEMPTY(A_27a) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s)),V2t) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,bool)),ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2t)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__UNION__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s)),V2t) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s)),V2t) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,bool)),ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2t)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s)),V2t)) <=> ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__INSERT_27,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V2t),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s))) <=> ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V2t),V1s)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s)),V2t)) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) ) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V2t))) <=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V0s)),V1t) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,bool)),ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t))) ) ) ) )))).
fof(conj_thm_2Epred__set_2EUNIV__BOOL,lemma,(( c_2Epred__set_2EUNIV(bool) = ap(ap(c_2Epred__set_2EINSERT(bool),c_2Ebool_2ET),ap(ap(c_2Epred__set_2EINSERT(bool),c_2Ebool_2EF),c_2Epred__set_2EEMPTY(bool))) ))).
fof(conj_thm_2Epred__set_2EFORALL__IN__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1a),V2s))) => p(ap(V0P,V3x)) ) ) ) <=> ( p(ap(V0P,V1a)) & ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s)) => p(ap(V0P,V4x)) ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EEXISTS__IN__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ? [V3x] : ( mem(V3x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1a),V2s))) & p(ap(V0P,V3x)) ) ) ) <=> ( p(ap(V0P,V1a)) | ? [V4x] : ( mem(V4x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s)) & p(ap(V0P,V4x)) ) ) ) ) ) ) )))).
fof(ax_thm_2Epred__set_2EDELETE__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V1x) = ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V2y))) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) & ( ~~ ( V1x = V2y ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( p(ap(ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V2y),V1x)) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) & ( ~~ ( V1x = V2y ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__NON__ELEMENT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) ) <=> ( ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V0x) = V1s ) ) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__NON__ELEMENT__RWT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) ) => ( ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V1x) = V0s ) ) ) )))).
fof(conj_thm_2Epred__set_2EIN__DELETE__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2x_27] : ( mem(V2x_27,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) <=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x_27),V0s)) ) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V2x_27))) <=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x_27),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V1x))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EEMPTY__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( ap(ap(c_2Epred__set_2EDELETE(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0x) = c_2Epred__set_2EEMPTY(A_27a) ) )))).
fof(conj_thm_2Epred__set_2EELT__IN__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V0x))) ) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V0x)),V0x) = ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V0x) ) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__COMM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V0x)),V1y) = ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V1y)),V0x) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V0x)),V1s)) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2t),V0x))) <=> ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) ) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__INSERT__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V2t))) <=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V0x)),V2t)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__OF__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s))) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V1t)) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V2x)),V1t) ) ) ) )))).
fof(conj_thm_2Epred__set_2EPSUBSET__INSERT__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V1t)) <=> ? [V2x] : ( mem(V2x,A_27a) & ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) ) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V0s)),V1t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EPSUBSET__MEMBER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V1t)) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) & ? [V2y] : ( mem(V2y,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),V1t)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),V0s)) ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V2s)),V1y) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,bool)),ap(ap(c_2Emin_2E_3D(A_27a),V0x),V1y)),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V1y)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V1y))) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) => ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V0x)) = V1s ) ) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V2x)),V1t) = ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t)),V2x) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__DELETE__SYM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V2x)),V1t)) <=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1t),V2x)),V0s)) ) ) ) )))).
fof(ax_thm_2Epred__set_2ECHOICE__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) => p(ap(ap(c_2Ebool_2EIN(A_27a),ap(c_2Epred__set_2ECHOICE(A_27a),V0s)),V0s)) ) ) ) )).
fof(ax_thm_2Epred__set_2EREST__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EREST(A_27a),V0s) = ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),ap(c_2Epred__set_2ECHOICE(A_27a),V0s)) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__REST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Epred__set_2EREST(A_27a),V1s))) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) & ( ~~ ( V0x = ap(c_2Epred__set_2ECHOICE(A_27a),V1s) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EREST__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EREST(A_27a),V1s),V0x)) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) & ( ~~ ( V0x = ap(c_2Epred__set_2ECHOICE(A_27a),V1s) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECHOICE__NOT__IN__REST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),ap(c_2Epred__set_2ECHOICE(A_27a),V0s)),ap(c_2Epred__set_2EREST(A_27a),V0s))) ) )))).
fof(conj_thm_2Epred__set_2ECHOICE__INSERT__REST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) => ( ap(ap(c_2Epred__set_2EINSERT(A_27a),ap(c_2Epred__set_2ECHOICE(A_27a),V0s)),ap(c_2Epred__set_2EREST(A_27a),V0s)) = V0s ) ) )))).
fof(conj_thm_2Epred__set_2EREST__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Epred__set_2EREST(A_27a),V0s)),V0s)) )))).
fof(conj_thm_2Epred__set_2EREST__PSUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) => p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),ap(c_2Epred__set_2EREST(A_27a),V0s)),V0s)) ) )))).
fof(ax_thm_2Epred__set_2ESING__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2ESING(A_27a),V0s)) <=> ? [V1x] : ( mem(V1x,A_27a) & ( V0s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a)) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2ESING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => p(ap(c_2Epred__set_2ESING(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a)))) )))).
fof(conj_thm_2Epred__set_2ESING__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ( p(ap(c_2Epred__set_2ESING(A_27a),c_2Epred__set_2EEMPTY(A_27a))) <=> $false )))).
fof(conj_thm_2Epred__set_2ESING__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2ESING(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s))) <=> ( ( V1s = c_2Epred__set_2EEMPTY(A_27a) ) | ( V1s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESING__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2ESING(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t))) <=> ( ( p(ap(c_2Epred__set_2ESING(A_27a),V0s)) & ( V1t = c_2Epred__set_2EEMPTY(A_27a) ) ) | ( ( p(ap(c_2Epred__set_2ESING(A_27a),V1t)) & ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) | ( p(ap(c_2Epred__set_2ESING(A_27a),V0s)) & ( p(ap(c_2Epred__set_2ESING(A_27a),V1t)) & ( V0s = V1t ) ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EIN__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1y),c_2Epred__set_2EEMPTY(A_27a)))) <=> ( V0x = V1y ) ) ) )))).
fof(conj_thm_2Epred__set_2ESING__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27a) => ( p(ap(ap(ap(c_2Epred__set_2EINSERT(A_27a),V1y),c_2Epred__set_2EEMPTY(A_27a)),V0x)) <=> ( V0x = V1y ) ) ) )))).
fof(conj_thm_2Epred__set_2ENOT__SING__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( ~~ ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a)) = c_2Epred__set_2EEMPTY(A_27a) ) ) )))).
fof(conj_thm_2Epred__set_2ENOT__EMPTY__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( ~~ ( c_2Epred__set_2EEMPTY(A_27a) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a)) ) ) )))).
fof(conj_thm_2Epred__set_2EEQUAL__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27a) => ( ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a)) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V1y),c_2Epred__set_2EEMPTY(A_27a)) ) <=> ( V0x = V1y ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__SING__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))),c_2Epred__set_2EEMPTY(A_27a))) )))).
fof(conj_thm_2Epred__set_2EINSERT__SING__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V0s) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),V0s) ) ) )))).
fof(conj_thm_2Epred__set_2ESING__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))),V0x) = c_2Epred__set_2EEMPTY(A_27a) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__EQ__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) => ( ( ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V1x) = c_2Epred__set_2EEMPTY(A_27a) ) <=> ( V0s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECHOICE__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( ap(c_2Epred__set_2ECHOICE(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))) = V0x ) )))).
fof(conj_thm_2Epred__set_2EREST__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( ap(c_2Epred__set_2EREST(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))) = c_2Epred__set_2EEMPTY(A_27a) ) )))).
fof(conj_thm_2Epred__set_2ESING__IFF__EMPTY__REST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2ESING(A_27a),V0s)) <=> ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ( ap(c_2Epred__set_2EREST(A_27a),V0s) = c_2Epred__set_2EEMPTY(A_27a) ) ) ) )))).
fof(ax_thm_2Epred__set_2EIMAGE__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s) = ap(c_2Epred__set_2EGSPEC(A_27b,A_27a),f267(A_27b,A_27a,V0f,V1s)) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0y] : ( mem(V0y,A_27b) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V0y),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f),V1s))) <=> ? [V3x] : ( mem(V3x,A_27a) & ( ( V0y = ap(V2f,V3x) ) & p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__applied,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0y] : ( mem(V0y,A_27b) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( p(ap(ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f),V1s),V0y)) <=> ? [V3x] : ( mem(V3x,A_27a) & ( ( V0y = ap(V2f,V3x) ) & p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__IN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V2f,V0x)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f),V1s))) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),c_2Epred__set_2EEMPTY(A_27a)) = c_2Epred__set_2EEMPTY(A_27b) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__ID,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),i(A_27a)),V0s) = V0s ) )))).
fof(conj_thm_2Epred__set_2EIMAGE__I,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),c_2Ecombin_2EI(A_27a)),V0s) = V0s ) )))).
fof(conj_thm_2Epred__set_2EIMAGE__II,lemma,( ! [A_27a]: ( ne(A_27a) => ( ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),c_2Ecombin_2EI(A_27a)) = c_2Ecombin_2EI(arr(A_27a,bool)) )))).
fof(conj_thm_2Epred__set_2EIMAGE__COMPOSE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27b,A_27c)) => ! [V1g] : ( mem(V1g,arr(A_27a,A_27b)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27c),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V0f),V1g)),V2s) = ap(ap(c_2Epred__set_2EIMAGE(A_27b,A_27c),V0f),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1g),V2s)) ) ) ) )))))).
fof(conj_thm_2Epred__set_2EIMAGE__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V2s)) = ap(ap(c_2Epred__set_2EINSERT(A_27b),ap(V0f,V1x)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__EQ__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ( ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),V0s) = c_2Epred__set_2EEMPTY(A_27b) ) <=> ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V2s)) ) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V1x)) = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__UNION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = ap(ap(c_2Epred__set_2EUNION(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2t)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f),V0s)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f),V1t))) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__INTER,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2t)))) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__11,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s1] : ( mem(V1s1,arr(A_27a,bool)) => ! [V2s2] : ( mem(V2s2,arr(A_27a,bool)) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( ap(V0f,V3x) = ap(V0f,V4y) ) <=> ( V3x = V4y ) ) ) ) ) => ( ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s1) = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s2) ) <=> ( V1s1 = V2s2 ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EDISJOINT__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s1] : ( mem(V1s1,arr(A_27a,bool)) => ! [V2s2] : ( mem(V2s2,arr(A_27a,bool)) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( ap(V0f,V3x) = ap(V0f,V4y) ) <=> ( V3x = V4y ) ) ) ) ) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s1)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s2))) <=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s1),V2s2)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__CONG,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f_27] : ( mem(V2f_27,arr(A_27a,A_27b)) => ! [V3s_27] : ( mem(V3s_27,arr(A_27a,bool)) => ( ( ( V1s = V3s_27 ) & ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3s_27)) => ( ap(V0f,V4x) = ap(V2f_27,V4x) ) ) ) ) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s) = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f_27),V3s_27) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EGSPEC__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27b,ty_2Epair_2Eprod(A_27a,bool))) => ( ap(c_2Epred__set_2EGSPEC(A_27a,A_27b),V0f) = ap(ap(c_2Epred__set_2EIMAGE(A_27b,A_27a),ap(ap(c_2Ecombin_2Eo(A_27b,A_27a,ty_2Epair_2Eprod(A_27a,bool)),c_2Epair_2EFST(A_27a,bool)),V0f)),ap(ap(c_2Ecombin_2Eo(A_27b,bool,ty_2Epair_2Eprod(A_27a,bool)),c_2Epair_2ESND(A_27a,bool)),V0f)) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27c,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27c,bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EIMAGE(A_27c,A_27a),V1g),V2s)) = ap(ap(c_2Epred__set_2EIMAGE(A_27c,A_27b),ap(ap(c_2Ecombin_2Eo(A_27c,A_27b,A_27a),V0f),V1g)),V2s) ) ) ) )))))).
fof(conj_thm_2Epred__set_2EFORALL__IN__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( ! [V3y] : ( mem(V3y,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),ap(ap(c_2Epred__set_2EIMAGE(A_27b,A_27a),V1f),V2s))) => p(ap(V0P,V3y)) ) ) ) <=> ! [V4x] : ( mem(V4x,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4x),V2s)) => p(ap(V0P,ap(V1f,V4x))) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EEXISTS__IN__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( ? [V3y] : ( mem(V3y,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),ap(ap(c_2Epred__set_2EIMAGE(A_27b,A_27a),V1f),V2s))) & p(ap(V0P,V3y)) ) ) ) <=> ? [V4x] : ( mem(V4x,A_27b) & ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4x),V2s)) & p(ap(V0P,ap(V1f,V4x))) ) ) ) ) ) ))))).
fof(ax_thm_2Epred__set_2EINJ__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) <=> ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V3x)),V2t)) ) ) ) & ! [V4x] : ( mem(V4x,A_27a) => ! [V5y] : ( mem(V5y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V1s)) ) => ( ( ap(V0f,V4x) = ap(V0f,V5y) ) => ( V4x = V5y ) ) ) ) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EINJ__IFF,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) <=> ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V3x)),V2t)) ) ) ) & ! [V4x] : ( mem(V4x,A_27a) => ! [V5y] : ( mem(V5y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V1s)) ) => ( ( ap(V0f,V4x) = ap(V0f,V5y) ) <=> ( V4x = V5y ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__ID,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27a),i(A_27a)),V0s),V0s)) )))).
fof(conj_thm_2Epred__set_2EINJ__COMPOSE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27b,A_27c)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ! [V4u] : ( mem(V4u,arr(A_27c,bool)) => ( ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V2s),V3t)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27b,A_27c),V1g),V3t),V4u)) ) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27c),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),V2s),V4u)) ) ) ) ) ) )))))).
fof(conj_thm_2Epred__set_2EINJ__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ( ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),c_2Epred__set_2EEMPTY(A_27a)),V1s)) ) ) & ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V2s),c_2Epred__set_2EEMPTY(A_27b))) <=> ( V2s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__DELETE,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,A_27b)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),V0s),V1t)) => ! [V3e] : ( mem(V3e,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3e),V0s)) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V3e)),ap(ap(c_2Epred__set_2EDELETE(A_27b),V1t),ap(V2f,V3e)))) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V2s)),V3t)) <=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V2s),V3t)) & ( p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V1x)),V3t)) & ! [V4y] : ( mem(V4y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s)) & ( ap(V0f,V1x) = ap(V0f,V4y) ) ) => ( V1x = V4y ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__EXTEND,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0b] : ( mem(V0b,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27b) => ( ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0b),V1s),V2t)) & ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) ) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V2t)) ) ) ) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V3x),V4y),V0b)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V3x),V1s)),ap(ap(c_2Epred__set_2EINSERT(A_27b),V4y),V2t))) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ! [V3s0] : ( mem(V3s0,arr(A_27a,bool)) => ! [V4t0] : ( mem(V4t0,arr(A_27b,bool)) => ( ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3s0),V1s)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V2t),V4t0)) ) ) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V3s0),V4t0)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s))) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__IMAGE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s)),V2t)) ) ) ) ))))).
fof(ax_thm_2Epred__set_2ESURJ__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),V2t)) <=> ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V3x)),V2t)) ) ) ) & ! [V4x] : ( mem(V4x,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4x),V2t)) => ? [V5y] : ( mem(V5y,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V1s)) & ( ap(V0f,V5y) = V4x ) ) ) ) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2ESURJ__ID,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27a),i(A_27a)),V0s),V0s)) )))).
fof(conj_thm_2Epred__set_2ESURJ__COMPOSE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27b,A_27c)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ! [V4u] : ( mem(V4u,arr(A_27c,bool)) => ( ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V2s),V3t)) & p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27b,A_27c),V1g),V3t),V4u)) ) => p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27c),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),V2s),V4u)) ) ) ) ) ) )))))).
fof(conj_thm_2Epred__set_2ESURJ__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ( ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),c_2Epred__set_2EEMPTY(A_27a)),V1s)) <=> ( V1s = c_2Epred__set_2EEMPTY(A_27b) ) ) ) ) & ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V2s),c_2Epred__set_2EEMPTY(A_27b))) <=> ( V2s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIMAGE__SURJ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),V2t)) <=> ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s) = V2t ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ESURJ__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s))) ) ))))).
fof(conj_thm_2Epred__set_2ESURJ__IMP__INJ,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,A_27b)) & p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V2f),V0s),V1t)) ) ) => ? [V3g] : ( mem(V3g,arr(A_27b,A_27a)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27b,A_27a),V3g),V1t),V0s)) ) ) ) ))))).
fof(ax_thm_2Epred__set_2EBIJ__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) <=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) & p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),V2t)) ) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EBIJ__ID,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27a),i(A_27a)),V0s),V0s)) )))).
fof(conj_thm_2Epred__set_2EBIJ__IMP__11,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),c_2Epred__set_2EUNIV(A_27a)),c_2Epred__set_2EUNIV(A_27b))) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( ( ap(V0f,V1x) = ap(V0f,V2y) ) <=> ( V1x = V2y ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ( ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),c_2Epred__set_2EEMPTY(A_27a)),V1s)) <=> ( V1s = c_2Epred__set_2EEMPTY(A_27b) ) ) ) ) & ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V2s),c_2Epred__set_2EEMPTY(A_27b))) <=> ( V2s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__COMPOSE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27b,A_27c)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ! [V4u] : ( mem(V4u,arr(A_27c,bool)) => ( ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V2s),V3t)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27b,A_27c),V1g),V3t),V4u)) ) => p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27c),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),V2s),V4u)) ) ) ) ) ) )))))).
fof(conj_thm_2Epred__set_2EBIJ__DELETE,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,A_27b)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s),V1t)) => ! [V3e] : ( mem(V3e,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3e),V0s)) => p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V3e)),ap(ap(c_2Epred__set_2EDELETE(A_27b),V1t),ap(V2f,V3e)))) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__IMAGE__BIJ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ( ( ? [V2t] : ( mem(V2t,arr(A_27b,bool)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V1f),V0s),V2t)) ) ) => p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V1f),V0s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),V0s))) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__BIJ__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s0] : ( mem(V0s0,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s0),V1s)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),V1s),V3t)) ) => p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s0),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f),V0s0))) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__SYM__IMP,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,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s),V1t)) ) ) => ? [V3g] : ( mem(V3g,arr(A_27b,A_27a)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27b,A_27a),V3g),V1t),V0s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__SYM,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,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s),V1t)) ) ) <=> ? [V3g] : ( mem(V3g,arr(A_27b,A_27a)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27b,A_27a),V3g),V1t),V0s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__TRANS,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27c,bool)) => ! [V2u] : ( mem(V2u,arr(A_27b,bool)) => ( ( ( ? [V3f] : ( mem(V3f,arr(A_27a,A_27c)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27c),V3f),V0s),V1t)) ) ) & ? [V4g] : ( mem(V4g,arr(A_27c,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27c,A_27b),V4g),V1t),V2u)) ) ) => ? [V5h] : ( mem(V5h,arr(A_27a,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V5h),V0s),V2u)) ) ) ) ) )))))).
fof(conj_thm_2Epred__set_2EBIJ__INV,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) => ? [V3g] : ( mem(V3g,arr(A_27b,A_27a)) & ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27b,A_27a),V3g),V2t),V1s)) & ( ( ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1s)) => ( ap(ap(ap(c_2Ecombin_2Eo(A_27a,A_27a,A_27b),V3g),V0f),V4x) = V4x ) ) ) ) & ! [V5x] : ( mem(V5x,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5x),V2t)) => ( ap(ap(ap(c_2Ecombin_2Eo(A_27b,A_27b,A_27a),V0f),V3g),V5x) = V5x ) ) ) ) ) ) ) ) ) ))))).
fof(ax_thm_2Epred__set_2EFUNSET,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V0P),V1Q) = f269(A_27b,A_27a,V0P,V1Q) ) ) ) ) ) )).
fof(ax_thm_2Epred__set_2EDFUNSET,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,arr(A_27b,bool))) => ( ap(ap(c_2Epred__set_2EDFUNSET(A_27a,A_27b),V0P),V1Q) = f271(A_27b,A_27a,V0P,V1Q) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__FUNSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ! [V2Q] : ( mem(V2Q,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V1P),V2Q))) <=> ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1P)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V3x)),V2Q)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFUNSET__applied,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ! [V2Q] : ( mem(V2Q,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V1P),V2Q),V0f)) <=> ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1P)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V3x)),V2Q)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EIN__DFUNSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ! [V2Q] : ( mem(V2Q,arr(A_27a,arr(A_27b,bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Epred__set_2EDFUNSET(A_27a,A_27b),V1P),V2Q))) <=> ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1P)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V3x)),ap(V2Q,V3x))) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EDFUNSET__applied,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ! [V2Q] : ( mem(V2Q,arr(A_27a,arr(A_27b,bool))) => ( p(ap(ap(ap(c_2Epred__set_2EDFUNSET(A_27a,A_27b),V1P),V2Q),V0f)) <=> ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1P)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V3x)),ap(V2Q,V3x))) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFUNSET__THM,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,A_27b)) => ! [V3x] : ( mem(V3x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V0s),V1t))) & p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) ) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V2f,V3x)),V1t)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EUNIV__FUNSET__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),c_2Epred__set_2EUNIV(A_27a)),c_2Epred__set_2EUNIV(A_27b)) = c_2Epred__set_2EUNIV(arr(A_27a,A_27b)) ))))).
fof(conj_thm_2Epred__set_2EFUNSET__DFUNSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ! [V1y] : ( mem(V1y,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V0x),V1y) = ap(ap(c_2Epred__set_2EDFUNSET(A_27a,A_27b),V0x),ap(c_2Ecombin_2EK(arr(A_27b,bool),A_27a),V1y)) ) ) ))))).
fof(conj_thm_2Epred__set_2EEMPTY__FUNSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),c_2Epred__set_2EEMPTY(A_27a)),V0s) = c_2Epred__set_2EUNIV(arr(A_27a,A_27b)) ) ))))).
fof(conj_thm_2Epred__set_2EFUNSET__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V0s),c_2Epred__set_2EEMPTY(A_27b)))) <=> ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFUNSET__INTER,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,arr(A_27a,bool)) => ! [V1b] : ( mem(V1b,arr(A_27b,bool)) => ! [V2c] : ( mem(V2c,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V0a),ap(ap(c_2Epred__set_2EINTER(A_27b),V1b),V2c)) = ap(ap(c_2Epred__set_2EINTER(arr(A_27a,A_27b)),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V0a),V1b)),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V0a),V2c)) ) ) ) ))))).
fof(ax_thm_2Epred__set_2Eschroeder__close__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(ap(c_2Epred__set_2Eschroeder__close(A_27a),V0f),V1s),V2x)) <=> ? [V3n] : ( mem(V3n,ty_2Enum_2Enum) & p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(ap(ap(c_2Earithmetic_2EFUNPOW(arr(A_27a,bool)),ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V0f)),V3n),V1s))) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2ESCHROEDER__CLOSE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(ap(c_2Epred__set_2Eschroeder__close(A_27a),V1f),V2s))) <=> ? [V3n] : ( mem(V3n,ty_2Enum_2Enum) & p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(ap(ap(c_2Earithmetic_2EFUNPOW(arr(A_27a,bool)),ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f)),V3n),V2s))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESCHROEDER__CLOSED,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V0f),ap(ap(c_2Epred__set_2Eschroeder__close(A_27a),V0f),V1s))),ap(ap(c_2Epred__set_2Eschroeder__close(A_27a),V0f),V1s))) ) )))).
fof(conj_thm_2Epred__set_2ESCHROEDER__CLOSE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(ap(c_2Epred__set_2Eschroeder__close(A_27a),V0f),V1s))) ) )))).
fof(conj_thm_2Epred__set_2ESCHROEDER__CLOSE__SET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27a)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27a),V1s),V1s))) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V1s)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2Eschroeder__close(A_27a),V0f),V2t)),V1s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESCHROEDER__BERNSTEIN__AUTO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) & ? [V2f] : ( mem(V2f,arr(A_27a,A_27a)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27a),V2f),V0s),V1t)) ) ) => ? [V3g] : ( mem(V3g,arr(A_27a,A_27a)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27a),V3g),V0s),V1t)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESCHROEDER__BERNSTEIN,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,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),V0s),V1t)) ) ) & ? [V3g] : ( mem(V3g,arr(A_27b,A_27a)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27b,A_27a),V3g),V1t),V0s)) ) ) => ? [V4h] : ( mem(V4h,arr(A_27a,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V4h),V0s),V1t)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__INJ__SURJ,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,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),V0s),V1t)) ) ) & ? [V3g] : ( mem(V3g,arr(A_27a,A_27b)) & p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V3g),V0s),V1t)) ) ) => ? [V4h] : ( mem(V4h,arr(A_27a,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V4h),V0s),V1t)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__ALT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) <=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V1s),V2t))) & ! [V3y] : ( mem(V3y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V2t)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f272(A_27b,A_27a,V3y,V0f,V1s))) ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__INSERT__IMP,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1e] : ( mem(V1e,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ( ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1e),V2s)) ) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s)),V3t)) ) => ? [V4u] : ( mem(V4u,arr(A_27b,bool)) & ( ( ap(ap(c_2Epred__set_2EINSERT(A_27b),ap(V0f,V1e)),V4u) = V3t ) & ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V1e)),V4u)) ) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V2s),V4u)) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) => ( V2t = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s) ) ) ) ) ))))).
fof(ax_thm_2Epred__set_2ELINV__OPT__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2y] : ( mem(V2y,A_27b) => ( ap(ap(ap(c_2Epred__set_2ELINV__OPT(A_27a,A_27b),V0f),V1s),V2y) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Ebool_2EIN(A_27b),V2y),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s))),ap(c_2Eoption_2ESOME(A_27a),ap(c_2Emin_2E_40(A_27a),f273(A_27b,A_27a,V2y,V0f,V1s)))),c_2Eoption_2ENONE(A_27a)) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2ELINV__OPT__THM,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2y] : ( mem(V2y,A_27b) => ! [V3x] : ( mem(V3x,A_27a) => ( ( ap(ap(ap(c_2Epred__set_2ELINV__OPT(A_27a,A_27b),V0f),V1s),V2y) = ap(c_2Eoption_2ESOME(A_27a),V3x) ) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) & ( ap(V0f,V3x) = V2y ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__LINV__OPT__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27b,A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Epred__set_2ELINV__OPT(A_27b,A_27a),V0f),V1s)),ap(ap(c_2Epred__set_2EIMAGE(A_27b,A_27a),V0f),V1s)),ap(ap(c_2Epred__set_2EIMAGE(A_27b,ty_2Eoption_2Eoption(A_27b)),c_2Eoption_2ESOME(A_27b)),V1s))) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__LINV__OPT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) => ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27b) => ( ( ap(ap(ap(c_2Epred__set_2ELINV__OPT(A_27a,A_27b),V0f),V1s),V4y) = ap(c_2Eoption_2ESOME(A_27a),V3x) ) <=> ( ( V4y = ap(V0f,V3x) ) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V2t)) ) ) ) ) ) ) ) ) ))))).
fof(ax_thm_2Epred__set_2ELINV__LO,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2y] : ( mem(V2y,A_27b) => ( ap(ap(ap(c_2Epred__set_2ELINV(A_27a,A_27b),V0f),V1s),V2y) = ap(c_2Eoption_2ETHE(A_27a),ap(ap(ap(c_2Epred__set_2ELINV__OPT(A_27a,A_27b),V0f),V1s),V2y)) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2ELINV__DEF,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) => ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => ( ap(ap(ap(c_2Epred__set_2ELINV(A_27a,A_27b),V0f),V1s),ap(V0f,V3x)) = V3x ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__LINV__INV,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) => ! [V3x] : ( mem(V3x,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3x),V2t)) => ( ap(V0f,ap(ap(ap(c_2Epred__set_2ELINV(A_27a,A_27b),V0f),V1s),V3x)) = V3x ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__LINV__BIJ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) => p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27b,A_27a),ap(ap(c_2Epred__set_2ELINV(A_27a,A_27b),V0f),V1s)),V2t),V1s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__IFF__INV,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) <=> ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V3x)),V2t)) ) ) ) & ? [V4g] : ( mem(V4g,arr(A_27b,A_27a)) & ( ( ! [V5x] : ( mem(V5x,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5x),V2t)) => p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V4g,V5x)),V1s)) ) ) ) & ( ( ! [V6x] : ( mem(V6x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V1s)) => ( ap(V4g,ap(V0f,V6x)) = V6x ) ) ) ) & ! [V7x] : ( mem(V7x,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V7x),V2t)) => ( ap(V0f,ap(V4g,V7x)) = V7x ) ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1e] : ( mem(V1e,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s)),V3t)) <=> ( ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1e),V2s)) ) & ( p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V1e)),V3t)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V2s),ap(ap(c_2Epred__set_2EDELETE(A_27b),V3t),ap(V0f,V1e)))) ) ) | ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1e),V2s)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V2s),V3t)) ) ) ) ) ) ) ))))).
fof(ax_thm_2Epred__set_2ERINV__LO,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2y] : ( mem(V2y,A_27b) => ( ap(ap(ap(c_2Epred__set_2ERINV(A_27a,A_27b),V0f),V1s),V2y) = ap(c_2Eoption_2ETHE(A_27a),ap(ap(ap(c_2Epred__set_2ELINV__OPT(A_27a,A_27b),V0f),V1s),V2y)) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2ERINV__DEF,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),V2t)) => ! [V3x] : ( mem(V3x,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3x),V2t)) => ( ap(V0f,ap(ap(ap(c_2Epred__set_2ERINV(A_27a,A_27b),V0f),V1s),V3x)) = V3x ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ESURJ__INJ__INV,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),V2t)) => ? [V3g] : ( mem(V3g,arr(A_27b,A_27a)) & ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27b,A_27a),V3g),V2t),V1s)) & ! [V4y] : ( mem(V4y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V2t)) => ( ap(V0f,ap(V3g,V4y)) = V4y ) ) ) ) ) ) ) ) ))))).
fof(ax_thm_2Epred__set_2EFINITE__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) <=> ! [V1P] : ( mem(V1P,arr(arr(A_27a,bool),bool)) => ( ( p(ap(V1P,c_2Epred__set_2EEMPTY(A_27a))) & ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(V1P,V2s)) => ! [V3e] : ( mem(V3e,A_27a) => p(ap(V1P,ap(ap(c_2Epred__set_2EINSERT(A_27a),V3e),V2s))) ) ) ) ) => p(ap(V1P,V0s)) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EFINITE__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EEMPTY(A_27a)))))).
fof(conj_thm_2Epred__set_2EFINITE__INDUCT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( ( p(ap(V0P,c_2Epred__set_2EEMPTY(A_27a))) & ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & p(ap(V0P,V1s)) ) => ! [V2e] : ( mem(V2e,A_27a) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2e),V1s)) ) => p(ap(V0P,ap(ap(c_2Epred__set_2EINSERT(A_27a),V2e),V1s))) ) ) ) ) ) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3s)) => p(ap(V0P,V3s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s))) <=> p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V0x))) <=> p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__REST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,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_2EREST(A_27a),V0s))) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__REST__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EREST(A_27a),V0s))) <=> p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t))) <=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t))) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) => p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__FINITE__I,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) ) )))).
fof(conj_thm_2Epred__set_2EPSUBSET__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V1t),V0s)) => p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t))) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__DIFF__down,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0P),V1Q))) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1Q)) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),V0P)) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a)))) )))).
fof(conj_thm_2Epred__set_2ESING__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2ESING(A_27a),V0s)) => p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) )))).
fof(conj_thm_2Epred__set_2EIMAGE__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => p(ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),V0s))) ) ) ))))).
fof(conj_thm_2Epred__set_2EFINITELY__INJECTIVE__IMAGE__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ( ! [V1x] : ( mem(V1x,A_27b) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f274(A_27b,A_27a,V1x,V0f)))) ) ) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V3s))) <=> p(ap(c_2Epred__set_2EFINITE(A_27a),V3s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJECTIVE__IMAGE__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ( ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( ( ap(V0f,V1x) = ap(V0f,V2y) ) <=> ( V1x = V2y ) ) ) ) ) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V3s))) <=> p(ap(c_2Epred__set_2EFINITE(A_27a),V3s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFINITE__INJ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) ) ))))).
fof(ax_thm_2Epred__set_2EREL__RESTRICT__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ( p(ap(ap(ap(ap(c_2Epred__set_2EREL__RESTRICT(A_27a),V0R),V1s),V2x),V3y)) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) & p(ap(ap(V0R,V2x),V3y)) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EREL__RESTRICT__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ( ap(ap(c_2Epred__set_2EREL__RESTRICT(A_27a),V0R),c_2Epred__set_2EEMPTY(A_27a)) = c_2Erelation_2EEMPTY__REL(A_27a) ) )))).
fof(conj_thm_2Epred__set_2EREL__RESTRICT__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27a,bool)) => ! [V2R] : ( mem(V2R,arr(A_27a,arr(A_27a,bool))) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s1),V1s2)) => p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27a),ap(ap(c_2Epred__set_2EREL__RESTRICT(A_27a),V2R),V0s1)),ap(ap(c_2Epred__set_2EREL__RESTRICT(A_27a),V2R),V1s2))) ) ) ) )))).
fof(ax_thm_2Epred__set_2ECARD__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ( ( ap(c_2Epred__set_2ECARD(A_27a),c_2Epred__set_2EEMPTY(A_27a)) = c_2Enum_2E0 ) & ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1x] : ( mem(V1x,A_27a) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V0s)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)),ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Enum_2ESUC,ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2ECARD__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ( ap(c_2Epred__set_2ECARD(A_27a),c_2Epred__set_2EEMPTY(A_27a)) = c_2Enum_2E0 )))).
fof(conj_thm_2Epred__set_2ECARD__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1x] : ( mem(V1x,A_27a) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V0s)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)),ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Enum_2ESUC,ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__EQ__0,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) = c_2Enum_2E0 ) <=> ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1x] : ( mem(V1x,A_27a) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V1x)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epred__set_2ECARD(A_27a),V0s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__INTER__LESS__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t))),ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) => ( ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t))),ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t))) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27a),V1t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__UNION__EQN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)) = ap(ap(c_2Earithmetic_2E_2D,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),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27a),V1t)),ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__PSUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V1t),V0s)) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(c_2Epred__set_2ECARD(A_27a),V1t)),ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__EQ__CARD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) & ( ( ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(c_2Epred__set_2ECARD(A_27a),V1t) ) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) ) ) => ( V0s = V1t ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ) )))).
fof(conj_thm_2Epred__set_2ESING__IFF__CARD1,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2ESING(A_27a),V0s)) <=> ( ( ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ) & p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0t] : ( mem(V0t,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0t)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V0t)) = ap(ap(c_2Earithmetic_2E_2D,ap(c_2Epred__set_2ECARD(A_27a),V1s)),ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V0t))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__DIFF__EQN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0t] : ( mem(V0t,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V0t)) = ap(ap(c_2Earithmetic_2E_2D,ap(c_2Epred__set_2ECARD(A_27a),V1s)),ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V0t))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ELESS__CARD__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0t] : ( mem(V0t,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0t)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( p(ap(ap(c_2Eprim__rec_2E_3C,ap(c_2Epred__set_2ECARD(A_27a),V0t)),ap(c_2Epred__set_2ECARD(A_27a),V1s))) => p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V0t)))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EBIJ__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) => p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIJ__FINITE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V1s)) & ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V1s)) ) ) => ? [V3N] : ( mem(V3N,ty_2Enum_2Enum) & ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3N),V4n)) => ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V0f,V4n)),V2t)) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__BIJ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [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(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) ) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) & ( ap(c_2Epred__set_2ECARD(A_27a),V1s) = ap(c_2Epred__set_2ECARD(A_27b),V2t) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFINITE__BIJ__CARD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [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(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t)) ) => ( ap(c_2Epred__set_2ECARD(A_27a),V1s) = ap(c_2Epred__set_2ECARD(A_27b),V2t) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFINITE__BIJ__CARD__EQ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0S] : ( mem(V0S,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0S)) => ! [V1t] : ( mem(V1t,arr(A_27b,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0S),V1t)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) => ( ap(c_2Epred__set_2ECARD(A_27a),V0S) = ap(c_2Epred__set_2ECARD(A_27b),V1t) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECARD__INJ__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) <=> ( V2x = V3y ) ) ) ) ) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) => ( ap(c_2Epred__set_2ECARD(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s)) = ap(c_2Epred__set_2ECARD(A_27a),V1s) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECARD__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s))),ap(c_2Epred__set_2ECARD(A_27a),V1s))) ) ) ))))).
fof(conj_thm_2Epred__set_2ESURJ__CARD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),V2t)) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27b),V2t)),ap(c_2Epred__set_2ECARD(A_27a),V1s))) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFINITE__SURJ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V1f),V0s),V2t)) ) => p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFINITE__SURJ__BIJ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V1f),V0s),V2t)) & ( ap(c_2Epred__set_2ECARD(A_27b),V2t) = ap(c_2Epred__set_2ECARD(A_27a),V0s) ) ) ) => p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V1f),V0s),V2t)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFINITE__COMPLETE__INDUCTION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( ( ! [V1x] : ( mem(V1x,arr(A_27a,bool)) => ( ( ! [V2y] : ( mem(V2y,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V2y),V1x)) => p(ap(V0P,V2y)) ) ) ) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1x)) => p(ap(V0P,V1x)) ) ) ) ) => ! [V3x] : ( mem(V3x,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3x)) => p(ap(V0P,V3x)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINJ__CARD__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1t] : ( mem(V1t,arr(A_27b,bool)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V2s),V1t)) => ( ap(c_2Epred__set_2ECARD(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)) = ap(c_2Epred__set_2ECARD(A_27a),V2s) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__CARD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27a),V1s)),ap(c_2Epred__set_2ECARD(A_27b),V2t))) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EPHP,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) & p(ap(ap(c_2Eprim__rec_2E_3C,ap(c_2Epred__set_2ECARD(A_27b),V2t)),ap(c_2Epred__set_2ECARD(A_27a),V1s))) ) => ( ~~ p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__CARD__IMAGE__EQ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( ap(c_2Epred__set_2ECARD(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s)) = ap(c_2Epred__set_2ECARD(A_27a),V1s) ) ) ) ) ) ))))).
fof(ax_thm_2Epred__set_2Ecount__def,axiom, ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Ecount,V0n) = ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f275(V0n)) ) )).
fof(conj_thm_2Epred__set_2EIN__COUNT,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0m),ap(c_2Epred__set_2Ecount,V1n))) <=> p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n)) ) ) ))).
fof(conj_thm_2Epred__set_2ECOUNT__applied,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Epred__set_2Ecount,V1n),V0m)) <=> p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n)) ) ) ))).
fof(conj_thm_2Epred__set_2ECOUNT__ZERO,lemma,(( ap(c_2Epred__set_2Ecount,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ))).
fof(conj_thm_2Epred__set_2ECOUNT__SUC,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Ecount,ap(c_2Enum_2ESUC,V0n)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0n),ap(c_2Epred__set_2Ecount,V0n)) ) ))).
fof(conj_thm_2Epred__set_2EFINITE__COUNT,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,V0n))) ))).
fof(conj_thm_2Epred__set_2ECARD__COUNT,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,V0n)) = V0n ) ))).
fof(conj_thm_2Epred__set_2ECOUNT__11,lemma,(! [V0n1] : ( mem(V0n1,ty_2Enum_2Enum) => ! [V1n2] : ( mem(V1n2,ty_2Enum_2Enum) => ( ( ap(c_2Epred__set_2Ecount,V0n1) = ap(c_2Epred__set_2Ecount,V1n2) ) <=> ( V0n1 = V1n2 ) ) ) ))).
fof(conj_thm_2Epred__set_2ENOT__IN__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a))) ) <=> ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ? [V1x] : ( mem(V1x,A_27a) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINFINITE__INHAB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0P)) ) => ? [V1x] : ( mem(V1x,A_27a) & p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0P)) ) ) )))).
fof(conj_thm_2Epred__set_2EIMAGE__11__INFINITE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ( ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( ( ap(V0f,V1x) = ap(V0f,V2y) ) => ( V1x = V2y ) ) ) ) ) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V3s)) ) => ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V3s))) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINFINITE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) => ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EIN__INFINITE__NOT__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) => ? [V2x] : ( mem(V2x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINFINITE__INJ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) & ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) => ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EINFINITE__INJ__NOT__SURJ,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(A_27a,A_27a)) & ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27a),V1f),V0s),V0s)) & ( ~~ p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27a),V1f),V0s),V0s)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINFINITE__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a))) ) <=> ? [V0f] : ( mem(V0f,arr(A_27a,A_27a)) & ( ( ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( ( ap(V0f,V1x) = ap(V0f,V2y) ) => ( V1x = V2y ) ) ) ) ) & ? [V3y] : ( mem(V3y,A_27a) & ! [V4x] : ( mem(V4x,A_27a) => ( ~~ ( ap(V0f,V4x) = V3y ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINFINITE__NUM__UNIV,lemma,(( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ))).
fof(conj_thm_2Epred__set_2EFINITE__PSUBSET__INFINITE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) <=> ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) => p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V1t),V0s)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__PSUBSET__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a))) ) <=> ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),c_2Epred__set_2EUNIV(A_27a))) ) ) )))).
fof(conj_thm_2Epred__set_2EINFINITE__DIFF__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) => ( ~~ ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t) = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__BIJ__COUNT__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) <=> ? [V1c] : ( mem(V1c,arr(ty_2Enum_2Enum,A_27a)) & ? [V2n] : ( mem(V2n,ty_2Enum_2Enum) & p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),V1c),ap(c_2Epred__set_2Ecount,V2n)),V0s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__BIJ__COUNT,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)) & ? [V2b] : ( mem(V2b,ty_2Enum_2Enum) & p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),V1f),ap(c_2Epred__set_2Ecount,V2b)),V0s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__ISO__NUM,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)) & ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Eprim__rec_2E_3C,V2n),ap(c_2Epred__set_2ECARD(A_27a),V0s))) & p(ap(ap(c_2Eprim__rec_2E_3C,V3m),ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) => ( ( ap(V1f,V2n) = ap(V1f,V3m) ) => ( V2n = V3m ) ) ) ) ) ) & ( V0s = ap(c_2Epred__set_2EGSPEC(A_27a,ty_2Enum_2Enum),f276(A_27a,V1f,V0s)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__WEAK__ENUMERATE,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)) & ? [V2b] : ( mem(V2b,ty_2Enum_2Enum) & ! [V3e] : ( mem(V3e,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3e),V0s)) <=> ? [V4n] : ( mem(V4n,ty_2Enum_2Enum) & ( p(ap(ap(c_2Eprim__rec_2E_3C,V4n),V2b)) & ( V3e = ap(V1f,V4n) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__WF__noloops,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( p(ap(c_2Erelation_2EWF(A_27a),ap(ap(c_2Epred__set_2EREL__RESTRICT(A_27a),V0R),V1s))) <=> p(ap(c_2Erelation_2Eirreflexive(A_27a),ap(c_2Erelation_2ETC(A_27a),ap(ap(c_2Epred__set_2EREL__RESTRICT(A_27a),V0R),V1s)))) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__StrongOrder__WF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & p(ap(c_2Erelation_2EStrongOrder(A_27a),ap(ap(c_2Epred__set_2EREL__RESTRICT(A_27a),V0R),V1s))) ) => p(ap(c_2Erelation_2EWF(A_27a),ap(ap(c_2Epred__set_2EREL__RESTRICT(A_27a),V0R),V1s))) ) ) )))).
fof(ax_thm_2Epred__set_2EBIGUNION,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( ap(c_2Epred__set_2EBIGUNION(A_27a),V0P) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f278(A_27a,V0P)) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1sos] : ( mem(V1sos,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Epred__set_2EBIGUNION(A_27a),V1sos))) <=> ? [V2s] : ( mem(V2s,arr(A_27a,bool)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2s)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),V1sos)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGUNION__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1sos] : ( mem(V1sos,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(c_2Epred__set_2EBIGUNION(A_27a),V1sos),V0x)) <=> ? [V2s] : ( mem(V2s,arr(A_27a,bool)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2s)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),V1sos)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EIN__BIGUNION__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2y] : ( mem(V2y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V2y),ap(c_2Epred__set_2EBIGUNION(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,arr(A_27b,bool)),V0f),V1s)))) <=> ? [V3x] : ( mem(V3x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27b),V2y),ap(V0f,V3x))) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIGUNION__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ( ap(c_2Epred__set_2EBIGUNION(A_27a),c_2Epred__set_2EEMPTY(arr(A_27a,bool))) = c_2Epred__set_2EEMPTY(A_27a) )))).
fof(conj_thm_2Epred__set_2EBIGUNION__EQ__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( ( ( ap(c_2Epred__set_2EBIGUNION(A_27a),V0P) = c_2Epred__set_2EEMPTY(A_27a) ) <=> ( ( V0P = c_2Epred__set_2EEMPTY(arr(A_27a,bool)) ) | ( V0P = ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),c_2Epred__set_2EEMPTY(arr(A_27a,bool))) ) ) ) & ( ( c_2Epred__set_2EEMPTY(A_27a) = ap(c_2Epred__set_2EBIGUNION(A_27a),V0P) ) <=> ( ( V0P = c_2Epred__set_2EEMPTY(arr(A_27a,bool)) ) | ( V0P = ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),c_2Epred__set_2EEMPTY(arr(A_27a,bool))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGUNION__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V0x),c_2Epred__set_2EEMPTY(arr(A_27a,bool)))) = V0x ) )))).
fof(conj_thm_2Epred__set_2EBIGUNION__PAIR,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V0s),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V1t),c_2Epred__set_2EEMPTY(arr(A_27a,bool))))) = ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGUNION__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(arr(A_27a,bool),bool)) => ! [V1s2] : ( mem(V1s2,arr(arr(A_27a,bool),bool)) => ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EUNION(arr(A_27a,bool)),V0s1),V1s2)) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s1)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s2)) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ! [V0s] : ( mem(V0s,arr(arr(A_27a,bool),bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s)),V1t)) <=> ! [V2s_27] : ( mem(V2s_27,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s_27),V0s)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V2s_27),V1t)) ) ) ) ) ) ) & ! [V3s] : ( mem(V3s,arr(arr(A_27a,bool),bool)) => ! [V4t] : ( mem(V4t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V4t),ap(c_2Epred__set_2EBIGUNION(A_27a),V3s))) <=> ! [V5s_27] : ( mem(V5s_27,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V5s_27),V3s)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V4t),V5s_27)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGUNION__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1P] : ( mem(V1P,arr(arr(A_27a,bool),bool)) => ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V0s),V1P)) = ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),ap(c_2Epred__set_2EBIGUNION(A_27a),V1P)) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGUNION__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0X] : ( mem(V0X,arr(A_27a,bool)) => ! [V1P] : ( mem(V1P,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V1P)),V0X)) <=> ! [V2Y] : ( mem(V2Y,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2Y),V1P)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2Y),V0X)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGUNION__IMAGE__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V1N] : ( mem(V1N,ty_2Enum_2Enum) => ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1N),V2n)) => ( ap(V0f,V2n) = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) => ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V0f),ap(c_2Epred__set_2Ecount,V1N))) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V0P)) & ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),V0P)) => p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0P))) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__BIGUNION__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0P))) <=> ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V0P)) & ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),V0P)) => p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__BIGUNION__I,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ! [V1P] : ( mem(V1P,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V0x),V1P)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0x),ap(c_2Epred__set_2EBIGUNION(A_27a),V1P))) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__BIGUNION__SAME__SIZED__SETS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s)) & ( ( ! [V2e] : ( mem(V2e,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2e),V1s)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2e)) & ( ap(c_2Epred__set_2ECARD(A_27a),V2e) = V0n ) ) ) ) ) & ! [V3e1] : ( mem(V3e1,arr(A_27a,bool)) => ! [V4e2] : ( mem(V4e2,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3e1),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4e2),V1s)) & ( ~~ ( V3e1 = V4e2 ) ) ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V3e1),V4e2)) ) ) ) ) ) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s)) = ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(arr(A_27a,bool)),V1s)),V0n) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__COUNT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( ~~ ( V1m = V2n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0f,V1m)),ap(V0f,V2n))) ) ) ) ) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0f,V3n)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V0f),ap(c_2Epred__set_2Ecount,V3n))))) ) ) )))).
fof(conj_thm_2Epred__set_2EFORALL__IN__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(arr(A_27a,bool),bool)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s))) => p(ap(V0P,V2x)) ) ) ) <=> ! [V3t] : ( mem(V3t,arr(A_27a,bool)) => ! [V4x] : ( mem(V4x,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3t)) ) => p(ap(V0P,V4x)) ) ) ) ) ) )))).
fof(ax_thm_2Epred__set_2EBIGINTER,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( ap(c_2Epred__set_2EBIGINTER(A_27a),V0P) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f280(A_27a,V0P)) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1B] : ( mem(V1B,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Epred__set_2EBIGINTER(A_27a),V1B))) <=> ! [V2P] : ( mem(V2P,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2P),V1B)) => p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2P)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGINTER__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0B] : ( mem(V0B,arr(arr(A_27a,bool),bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( p(ap(ap(c_2Epred__set_2EBIGINTER(A_27a),V0B),V1x)) <=> ! [V2P] : ( mem(V2P,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2P),V0B)) => p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V2P)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EIN__BIGINTER__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1f] : ( mem(V1f,arr(A_27b,arr(A_27a,bool))) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27b,arr(A_27a,bool)),V1f),V2s)))) <=> ! [V3y] : ( mem(V3y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V2s)) => p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(V1f,V3y))) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EBIGINTER__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1B] : ( mem(V1B,arr(arr(A_27a,bool),bool)) => ( ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V0P),V1B)) = ap(ap(c_2Epred__set_2EINTER(A_27a),V0P),ap(c_2Epred__set_2EBIGINTER(A_27a),V1B)) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGINTER__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ( ap(c_2Epred__set_2EBIGINTER(A_27a),c_2Epred__set_2EEMPTY(arr(A_27a,bool))) = c_2Epred__set_2EUNIV(A_27a) )))).
fof(conj_thm_2Epred__set_2EBIGINTER__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V0P),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V1Q),c_2Epred__set_2EEMPTY(arr(A_27a,bool))))) = ap(ap(c_2Epred__set_2EINTER(A_27a),V0P),V1Q) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGINTER__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V0P),c_2Epred__set_2EEMPTY(arr(A_27a,bool)))) = V0P ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0X] : ( mem(V0X,arr(A_27a,bool)) => ! [V1P] : ( mem(V1P,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),ap(c_2Epred__set_2EBIGINTER(A_27a),V1P))) <=> ! [V2Y] : ( mem(V2Y,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2Y),V1P)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),V2Y)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDISJOINT__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0X] : ( mem(V0X,arr(A_27a,bool)) => ! [V1Y] : ( mem(V1Y,arr(A_27a,bool)) => ! [V2P] : ( mem(V2P,arr(arr(A_27a,bool),bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1Y),V2P)) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1Y),V0X)) ) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0X),ap(c_2Epred__set_2EBIGINTER(A_27a),V2P))) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(c_2Epred__set_2EBIGINTER(A_27a),V2P)),V0X)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGINTER__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(arr(A_27a,bool),bool)) => ! [V1s2] : ( mem(V1s2,arr(arr(A_27a,bool),bool)) => ( ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EUNION(arr(A_27a,bool)),V0s1),V1s2)) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EBIGINTER(A_27a),V0s1)),ap(c_2Epred__set_2EBIGINTER(A_27a),V1s2)) ) ) )))).
fof(conj_thm_2Epred__set_2EBIGINTER__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(arr(A_27a,bool),bool)) => ( ( ( ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),V1s)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V0sp)) ) ) ) & ( ~~ ( V1s = c_2Epred__set_2EEMPTY(arr(A_27a,bool)) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Epred__set_2EBIGINTER(A_27a),V1s)),V0sp)) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__BIGINTER1,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(arr(A_27a,bool),bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),ap(c_2Epred__set_2EBIGINTER(A_27a),V1s)) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27a,bool)),f281(A_27a,V0sp)),V1s)) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(arr(A_27a,bool),bool)) => ( ( ( ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),V1s)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V0sp)) ) ) ) & ( ~~ ( V1s = c_2Epred__set_2EEMPTY(arr(A_27a,bool)) ) ) ) => ( ap(c_2Epred__set_2EBIGINTER(A_27a),V1s) = ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27a,bool)),f282(A_27a,V0sp)),V1s))) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( ( ? [V1s] : ( mem(V1s,arr(A_27a,bool)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),V0P)) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EBIGINTER(A_27a),V0P))) ) )))).
fof(ax_thm_2Epred__set_2ECROSS__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q) = ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,A_27b)),f283(A_27a,A_27b,V1Q,V0P)) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__CROSS,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ! [V2x] : ( mem(V2x,ty_2Epair_2Eprod(A_27a,A_27b)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),V2x),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q))) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),ap(c_2Epair_2EFST(A_27a,A_27b),V2x)),V0P)) & p(ap(ap(c_2Ebool_2EIN(A_27b),ap(c_2Epair_2ESND(A_27a,A_27b),V2x)),V1Q)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECROSS__applied,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ! [V2x] : ( mem(V2x,ty_2Epair_2Eprod(A_27a,A_27b)) => ( p(ap(ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q),V2x)) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),ap(c_2Epair_2EFST(A_27a,A_27b),V2x)),V0P)) & p(ap(ap(c_2Ebool_2EIN(A_27b),ap(c_2Epair_2ESND(A_27a,A_27b),V2x)),V1Q)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECROSS__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ( ( ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),c_2Epred__set_2EEMPTY(A_27b)) = c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27b)) ) & ( ap(ap(c_2Epred__set_2ECROSS(A_27c,A_27a),c_2Epred__set_2EEMPTY(A_27c)),V0P) = c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27c,A_27a)) ) ) )))))).
fof(conj_thm_2Epred__set_2ECROSS__EMPTY__EQN,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)) => ( ( ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27b)) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) | ( V1t = c_2Epred__set_2EEMPTY(A_27b) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECROSS__INSERT__LEFT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V0P)),V1Q) = ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),V1Q)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECROSS__INSERT__RIGHT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ! [V2x] : ( mem(V2x,A_27b) => ( ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),ap(ap(c_2Epred__set_2EINSERT(A_27b),V2x),V1Q)) = ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),ap(ap(c_2Epred__set_2EINSERT(A_27b),V2x),c_2Epred__set_2EEMPTY(A_27b)))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFINITE__CROSS,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1Q)) ) => p(ap(c_2Epred__set_2EFINITE(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q))) ) ) ))))).
fof(conj_thm_2Epred__set_2ECROSS__SINGS,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27b) => ( ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))),ap(ap(c_2Epred__set_2EINSERT(A_27b),V1y),c_2Epred__set_2EEMPTY(A_27b))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y)),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27b))) ) ) ))))).
fof(conj_thm_2Epred__set_2ECARD__SING__CROSS,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1P] : ( mem(V1P,arr(A_27b,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),V1P)) => ( ap(c_2Epred__set_2ECARD(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))),V1P)) = ap(c_2Epred__set_2ECARD(A_27b),V1P) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECARD__CROSS,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1Q)) ) => ( ap(c_2Epred__set_2ECARD(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q)) = ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V0P)),ap(c_2Epred__set_2ECARD(A_27b),V1Q)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECROSS__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ! [V2P0] : ( mem(V2P0,arr(A_27a,bool)) => ! [V3Q0] : ( mem(V3Q0,arr(A_27b,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V2P0),V3Q0)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q))) <=> ( ( V2P0 = c_2Epred__set_2EEMPTY(A_27a) ) | ( ( V3Q0 = c_2Epred__set_2EEMPTY(A_27b) ) | ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2P0),V0P)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V3Q0),V1Q)) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EFINITE__CROSS__EQ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27b,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q))) <=> ( ( V0P = c_2Epred__set_2EEMPTY(A_27a) ) | ( ( V1Q = c_2Epred__set_2EEMPTY(A_27b) ) | ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1Q)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECROSS__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( c_2Epred__set_2EUNIV(ty_2Epair_2Eprod(A_27a,A_27b)) = ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),c_2Epred__set_2EUNIV(A_27a)),c_2Epred__set_2EUNIV(A_27b)) ))))).
fof(conj_thm_2Epred__set_2EINFINITE__PAIR__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Epair_2Eprod(A_27a,A_27b)),c_2Epred__set_2EUNIV(ty_2Epair_2Eprod(A_27a,A_27b)))) <=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a))) & p(ap(c_2Epred__set_2EFINITE(A_27b),c_2Epred__set_2EUNIV(A_27b))) ) ))))).
fof(conj_thm_2Epred__set_2ESUM__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( c_2Epred__set_2EUNIV(ty_2Esum_2Esum(A_27a,A_27b)) = ap(ap(c_2Epred__set_2EUNION(ty_2Esum_2Esum(A_27a,A_27b)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Esum_2Esum(A_27a,A_27b)),c_2Esum_2EINL(A_27a,A_27b)),c_2Epred__set_2EUNIV(A_27a))),ap(ap(c_2Epred__set_2EIMAGE(A_27b,ty_2Esum_2Esum(A_27a,A_27b)),c_2Esum_2EINR(A_27a,A_27b)),c_2Epred__set_2EUNIV(A_27b))) ))))).
fof(conj_thm_2Epred__set_2EINJ__INL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Esum_2Esum(A_27a,A_27b),bool)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Esum_2Esum(A_27a,A_27b)),ap(c_2Esum_2EINL(A_27a,A_27b),V2x)),V1t)) ) ) ) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,ty_2Esum_2Esum(A_27a,A_27b)),c_2Esum_2EINL(A_27a,A_27b)),V0s),V1t)) ) ) ))))).
fof(conj_thm_2Epred__set_2EINJ__INR,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Esum_2Esum(A_27b,A_27a),bool)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Esum_2Esum(A_27b,A_27a)),ap(c_2Esum_2EINR(A_27b,A_27a),V2x)),V1t)) ) ) ) => p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,ty_2Esum_2Esum(A_27b,A_27a)),c_2Esum_2EINR(A_27b,A_27a)),V0s),V1t)) ) ) ))))).
fof(ax_thm_2Epred__set_2ECOMPL__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ( ap(c_2Epred__set_2ECOMPL(A_27a),V0P) = ap(ap(c_2Epred__set_2EDIFF(A_27a),c_2Epred__set_2EUNIV(A_27a)),V0P) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Epred__set_2ECOMPL(A_27a),V1s))) <=> ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECOMPL__applied,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ECOMPL(A_27a),V1s),V0x)) <=> ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECOMPL__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(c_2Epred__set_2ECOMPL(A_27a),ap(c_2Epred__set_2ECOMPL(A_27a),V0s)) = V0s ) )))).
fof(conj_thm_2Epred__set_2ECOMPL__CLAUSES,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2ECOMPL(A_27a),V0s)),V0s) = c_2Epred__set_2EEMPTY(A_27a) ) & ( ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Epred__set_2ECOMPL(A_27a),V0s)),V0s) = c_2Epred__set_2EUNIV(A_27a) ) ) )))).
fof(conj_thm_2Epred__set_2ECOMPL__SPLITS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,arr(A_27a,bool)) => ! [V1q] : ( mem(V1q,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0p),V1q)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2ECOMPL(A_27a),V0p)),V1q)) = V1q ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__UNION__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t) = ap(c_2Epred__set_2ECOMPL(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Epred__set_2ECOMPL(A_27a),V0s)),ap(c_2Epred__set_2ECOMPL(A_27a),V1t))) ) ) )))).
fof(conj_thm_2Epred__set_2ECOMPL__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ( ap(c_2Epred__set_2ECOMPL(A_27a),c_2Epred__set_2EEMPTY(A_27a)) = c_2Epred__set_2EUNIV(A_27a) )))).
fof(conj_thm_2Epred__set_2ECOMPL__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0x),ap(c_2Epred__set_2ECOMPL(A_27a),V0x)) = c_2Epred__set_2EEMPTY(A_27a) ) & ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2ECOMPL(A_27a),V0x)),V0x) = c_2Epred__set_2EEMPTY(A_27a) ) ) )))).
fof(conj_thm_2Epred__set_2ECOMPL__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(c_2Epred__set_2ECOMPL(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2ECOMPL(A_27a),V0s)),ap(c_2Epred__set_2ECOMPL(A_27a),V1t)) ) ) )))).
fof(conj_thm_2Epred__set_2EITSET__ind,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1P] : ( mem(V1P,arr(arr(A_27a,bool),arr(A_27b,bool))) => ( ( ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3b] : ( mem(V3b,A_27b) => ( ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ~~ ( V2s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) => p(ap(ap(V1P,ap(c_2Epred__set_2EREST(A_27a),V2s)),ap(ap(V0f,ap(c_2Epred__set_2ECHOICE(A_27a),V2s)),V3b))) ) => p(ap(ap(V1P,V2s),V3b)) ) ) ) ) => ! [V4v] : ( mem(V4v,arr(A_27a,bool)) => ! [V5v1] : ( mem(V5v1,A_27b) => p(ap(ap(V1P,V4v),V5v1)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EITSET__def,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,A_27b))) => ! [V2b] : ( mem(V2b,A_27b) => ( ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V1f),V0s),V2b) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(c_2Epred__set_2EFINITE(A_27a),V0s)),ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Emin_2E_3D(arr(A_27a,bool)),V0s),c_2Epred__set_2EEMPTY(A_27a))),V2b),ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V1f),ap(c_2Epred__set_2EREST(A_27a),V0s)),ap(ap(V1f,ap(c_2Epred__set_2ECHOICE(A_27a),V0s)),V2b)))),c_2Ebool_2EARB(A_27b)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EITSET__IND,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1P] : ( mem(V1P,arr(arr(A_27a,bool),arr(A_27b,bool))) => ( ( ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3b] : ( mem(V3b,A_27b) => ( ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ( ~~ ( V2s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) => p(ap(ap(V1P,ap(c_2Epred__set_2EREST(A_27a),V2s)),ap(ap(V0f,ap(c_2Epred__set_2ECHOICE(A_27a),V2s)),V3b))) ) => p(ap(ap(V1P,V2s),V3b)) ) ) ) ) => ! [V4v] : ( mem(V4v,arr(A_27a,bool)) => ! [V5v1] : ( mem(V5v1,A_27b) => p(ap(ap(V1P,V4v),V5v1)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EITSET__THM,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,A_27b))) => ! [V2b] : ( mem(V2b,A_27b) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ( ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V1f),V0s),V2b) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Emin_2E_3D(arr(A_27a,bool)),V0s),c_2Epred__set_2EEMPTY(A_27a))),V2b),ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V1f),ap(c_2Epred__set_2EREST(A_27a),V0s)),ap(ap(V1f,ap(c_2Epred__set_2ECHOICE(A_27a),V0s)),V2b))) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EITSET__EMPTY,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) => ( ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V0f),c_2Epred__set_2EEMPTY(A_27a)),V1b) = V1b ) ) ))))).
fof(conj_thm_2Epred__set_2EITSET__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,A_27b))) => ! [V2x] : ( mem(V2x,A_27a) => ! [V3b] : ( mem(V3b,A_27b) => ( ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V1f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V0s)),V3b) = ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V1f),ap(c_2Epred__set_2EREST(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V0s))),ap(ap(V1f,ap(c_2Epred__set_2ECHOICE(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V0s))),V3b)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECOMMUTING__ITSET__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ! [V4z] : ( mem(V4z,A_27b) => ( ap(ap(V0f,V2x),ap(ap(V0f,V3y),V4z)) = ap(ap(V0f,V3y),ap(ap(V0f,V2x),V4z)) ) ) ) ) ) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) => ! [V5x] : ( mem(V5x,A_27a) => ! [V6b] : ( mem(V6b,A_27b) => ( ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V5x),V1s)),V6b) = ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V5x)),ap(ap(V0f,V5x),V6b)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECOMMUTING__ITSET__RECURSES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27b))) => ! [V1e] : ( mem(V1e,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3b] : ( mem(V3b,A_27b) => ( ( ( ! [V4x] : ( mem(V4x,A_27a) => ! [V5y] : ( mem(V5y,A_27a) => ! [V6z] : ( mem(V6z,A_27b) => ( ap(ap(V0f,V4x),ap(ap(V0f,V5y),V6z)) = ap(ap(V0f,V5y),ap(ap(V0f,V4x),V6z)) ) ) ) ) ) & p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) ) => ( ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s)),V3b) = ap(ap(V0f,V1e),ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V1e)),V3b)) ) ) ) ) ) ))))).
fof(ax_thm_2Epred__set_2ESUM__IMAGE__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s) = ap(ap(ap(c_2Epred__set_2EITSET(A_27a,ty_2Enum_2Enum),f285(A_27a,V0f)),V1s),c_2Enum_2E0) ) ) ) ) )).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__THM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ( ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),c_2Epred__set_2EEMPTY(A_27a)) = c_2Enum_2E0 ) & ! [V1e] : ( mem(V1e,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) => ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s)) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,V1e)),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V1e))) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1e] : ( mem(V1e,A_27a) => ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),c_2Epred__set_2EEMPTY(A_27a))) = ap(V0f,V1e) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__SUBSET__LE,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)) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V2t)),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__IN__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2e] : ( mem(V2e,A_27a) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V2e),V1s)) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V0f,V2e)),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__DELETE,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)) => ! [V2e] : ( mem(V2e,A_27a) => ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2e)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(A_27a),V2e),V1s)),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s)),ap(V0f,V2e))),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__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)) ) => ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s)),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V2t))),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__lower__bound,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)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V2n),ap(V0f,V3x))) ) ) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V1s)),V2n)),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__upper__bound,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)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V0f,V3x)),V2n)) ) ) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s)),ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V1s)),V2n))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__SAME__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Enum_2Enum)) => ! [V2p] : ( mem(V2p,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2p),V0P)) & ! [V3q] : ( mem(V3q,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3q),V0P)) => ( ap(V1f,V2p) = ap(V1f,V3q) ) ) ) ) => ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V1f),V0P) = ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V0P)),ap(V1f,V2p)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__CONG,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27a,bool)) => ! [V2f1] : ( mem(V2f1,arr(A_27a,ty_2Enum_2Enum)) => ! [V3f2] : ( mem(V3f2,arr(A_27a,ty_2Enum_2Enum)) => ( ( ( V0s1 = V1s2 ) & ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1s2)) => ( ap(V2f1,V4x) = ap(V3f2,V4x) ) ) ) ) => ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V2f1),V0s1) = ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V3f2),V1s2) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__ZERO,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_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s) = 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_2Epred__set_2EABS__DIFF__SUM__IMAGE,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)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Earithmetic_2EABS__DIFF,ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V2s)),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V1g),V2s))),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),f286(A_27a,V0f,V1g)),V2s))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__MONO__LESS__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)) => ( 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_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V2s)),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V1g),V2s))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__MONO__LESS,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_2Eprim__rec_2E_3C,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_2Earithmetic_2E_3C_3D,ap(V0f,V4x)),ap(V1g,V4x))) ) ) ) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V2s)),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V1g),V2s))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__INJ__o,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27b,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),V0s)) => ! [V1g] : ( mem(V1g,arr(A_27b,A_27a)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27b,A_27a),V1g),V0s),c_2Epred__set_2EUNIV(A_27a))) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Enum_2Enum)) => ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V2f),ap(ap(c_2Epred__set_2EIMAGE(A_27b,A_27a),V1g),V0s)) = ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27b),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Enum_2Enum,A_27a),V2f),V1g)),V0s) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__PERMUTES,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => ! [V1g] : ( mem(V1g,arr(A_27a,A_27a)) => ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27a),V1g),V0s),V0s)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Enum_2Enum)) => ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27a),V2f),V1g)),V0s) = ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V2f),V0s) ) ) ) ) ) )))).
fof(ax_thm_2Epred__set_2ESUM__SET__DEF,axiom, ( c_2Epred__set_2ESUM__SET = ap(c_2Epred__set_2ESUM__IMAGE(ty_2Enum_2Enum),c_2Ecombin_2EI(ty_2Enum_2Enum)) )).
fof(conj_thm_2Epred__set_2ESUM__SET__THM,lemma,(( ( ap(c_2Epred__set_2ESUM__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = c_2Enum_2E0 ) & ! [V0x] : ( mem(V0x,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) => ( ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0x),V1s)) = ap(ap(c_2Earithmetic_2E_2B,V0x),ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EDELETE(ty_2Enum_2Enum),V1s),V0x))) ) ) ) ) ))).
fof(conj_thm_2Epred__set_2ESUM__SET__EMPTY,lemma,(( ap(c_2Epred__set_2ESUM__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = c_2Enum_2E0 ))).
fof(conj_thm_2Epred__set_2ESUM__SET__SING,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0n),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))) = V0n ) ))).
fof(conj_thm_2Epred__set_2ESUM__SET__SUBSET__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V0s),V1t)) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ESUM__SET,V0s)),ap(c_2Epred__set_2ESUM__SET,V1t))) ) ) ))).
fof(conj_thm_2Epred__set_2ESUM__SET__IN__LE,lemma,(! [V0x] : ( mem(V0x,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0x),V1s)) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V0x),ap(c_2Epred__set_2ESUM__SET,V1s))) ) ) ))).
fof(conj_thm_2Epred__set_2ESUM__SET__DELETE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s)) => ! [V1e] : ( mem(V1e,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EDELETE(ty_2Enum_2Enum),V0s),V1e)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1e),V0s)),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Epred__set_2ESUM__SET,V0s)),V1e)),ap(c_2Epred__set_2ESUM__SET,V0s)) ) ) ) ))).
fof(conj_thm_2Epred__set_2ESUM__SET__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s)) & p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1t)) ) => ( ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),V0s),V1t)) = ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epred__set_2ESUM__SET,V0s)),ap(c_2Epred__set_2ESUM__SET,V1t))),ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V0s),V1t))) ) ) ) ))).
fof(ax_thm_2Epred__set_2EPROD__IMAGE__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EPROD__IMAGE(A_27a),V0f),V1s) = ap(ap(ap(c_2Epred__set_2EITSET(A_27a,ty_2Enum_2Enum),f288(A_27a,V0f)),V1s),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EPROD__IMAGE__THM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ( ( ap(ap(c_2Epred__set_2EPROD__IMAGE(A_27a),V0f),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ) & ! [V1e] : ( mem(V1e,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) => ( ap(ap(c_2Epred__set_2EPROD__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s)) = ap(ap(c_2Earithmetic_2E_2A,ap(V0f,V1e)),ap(ap(c_2Epred__set_2EPROD__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V1e))) ) ) ) ) ) )))).
fof(ax_thm_2Epred__set_2EPROD__SET__DEF,axiom, ( c_2Epred__set_2EPROD__SET = ap(c_2Epred__set_2EPROD__IMAGE(ty_2Enum_2Enum),c_2Ecombin_2EI(ty_2Enum_2Enum)) )).
fof(conj_thm_2Epred__set_2EPROD__SET__THM,lemma,(( ( ap(c_2Epred__set_2EPROD__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ) & ! [V0x] : ( mem(V0x,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) => ( ap(c_2Epred__set_2EPROD__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0x),V1s)) = ap(ap(c_2Earithmetic_2E_2A,V0x),ap(c_2Epred__set_2EPROD__SET,ap(ap(c_2Epred__set_2EDELETE(ty_2Enum_2Enum),V1s),V0x))) ) ) ) ) ))).
fof(conj_thm_2Epred__set_2EPROD__SET__EMPTY,lemma,(( ap(c_2Epred__set_2EPROD__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ))).
fof(conj_thm_2Epred__set_2EPROD__SET__IMAGE__REDUCTION,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(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Enum_2Enum),V0f),V1s))) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(V0f,V2x)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Enum_2Enum),V0f),V1s))) ) ) => ( ap(c_2Epred__set_2EPROD__SET,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Enum_2Enum),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V1s))) = ap(ap(c_2Earithmetic_2E_2A,ap(V0f,V2x)),ap(c_2Epred__set_2EPROD__SET,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Enum_2Enum),V0f),V1s))) ) ) ) ) )))).
fof(ax_thm_2Epred__set_2EMAX__SET__DEF,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s)) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) ) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(c_2Epred__set_2EMAX__SET,V0s)),V0s)) & ! [V1y] : ( mem(V1y,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1y),V0s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V1y),ap(c_2Epred__set_2EMAX__SET,V0s))) ) ) ) ) & ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) => ( ap(c_2Epred__set_2EMAX__SET,V0s) = c_2Enum_2E0 ) ) ) ) )).
fof(conj_thm_2Epred__set_2EMAX__SET__THM,lemma,(( ( ap(c_2Epred__set_2EMAX__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = c_2Enum_2E0 ) & ! [V0e] : ( mem(V0e,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) => ( ap(c_2Epred__set_2EMAX__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0e),V1s)) = ap(ap(c_2Earithmetic_2EMAX,V0e),ap(c_2Epred__set_2EMAX__SET,V1s)) ) ) ) ) ))).
fof(conj_thm_2Epred__set_2EMAX__SET__REWRITES,lemma,(! [V0e] : ( mem(V0e,ty_2Enum_2Enum) => ( ( ap(c_2Epred__set_2EMAX__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = c_2Enum_2E0 ) & ( ap(c_2Epred__set_2EMAX__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0e),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))) = V0e ) ) ))).
fof(conj_thm_2Epred__set_2EMAX__SET__ELIM,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,bool)) => ! [V1Q] : ( mem(V1Q,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0P)) & ( ( ( V0P = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) => p(ap(V1Q,c_2Enum_2E0)) ) & ! [V2x] : ( mem(V2x,ty_2Enum_2Enum) => ( ( ( ! [V3y] : ( mem(V3y,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3y),V0P)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V3y),V2x)) ) ) ) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2x),V0P)) ) => p(ap(V1Q,V2x)) ) ) ) ) => p(ap(V1Q,ap(c_2Epred__set_2EMAX__SET,V0P))) ) ) ))).
fof(ax_thm_2Epred__set_2EMIN__SET__DEF,axiom, ( c_2Epred__set_2EMIN__SET = c_2Ewhile_2ELEAST )).
fof(conj_thm_2Epred__set_2EMIN__SET__ELIM,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,bool)) => ! [V1Q] : ( mem(V1Q,arr(ty_2Enum_2Enum,bool)) => ( ( ( ~~ ( V0P = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) ) & ! [V2x] : ( mem(V2x,ty_2Enum_2Enum) => ( ( ( ! [V3y] : ( mem(V3y,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3y),V0P)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V2x),V3y)) ) ) ) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2x),V0P)) ) => p(ap(V1Q,V2x)) ) ) ) => p(ap(V1Q,ap(c_2Epred__set_2EMIN__SET,V0P))) ) ) ))).
fof(conj_thm_2Epred__set_2EMIN__SET__THM,lemma,(( ( ! [V0e] : ( mem(V0e,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2EMIN__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0e),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))) = V0e ) ) ) & ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2e1] : ( mem(V2e1,ty_2Enum_2Enum) => ! [V3e2] : ( mem(V3e2,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2EMIN__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V2e1),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V3e2),V1s))) = ap(ap(c_2Earithmetic_2EMIN,V2e1),ap(c_2Epred__set_2EMIN__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V3e2),V1s))) ) ) ) ) ))).
fof(conj_thm_2Epred__set_2EMIN__SET__LEM,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) ) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(c_2Epred__set_2EMIN__SET,V0s)),V0s)) & ! [V1x] : ( mem(V1x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1x),V0s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2EMIN__SET,V0s)),V1x)) ) ) ) ) ))).
fof(conj_thm_2Epred__set_2ESUBSET__MIN__SET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0I] : ( mem(V0I,arr(ty_2Enum_2Enum,bool)) => ! [V1J] : ( mem(V1J,arr(ty_2Enum_2Enum,bool)) => ! [V2n] : ( mem(V2n,A_27a) => ( ( ( ~~ ( V0I = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) ) & ( ( ~~ ( V1J = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V0I),V1J)) ) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2EMIN__SET,V1J)),ap(c_2Epred__set_2EMIN__SET,V0I))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__MAX__SET,lemma,(! [V0I] : ( mem(V0I,arr(ty_2Enum_2Enum,bool)) => ! [V1J] : ( mem(V1J,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0I)) & ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1J)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V0I),V1J)) ) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2EMAX__SET,V0I)),ap(c_2Epred__set_2EMAX__SET,V1J))) ) ) ))).
fof(conj_thm_2Epred__set_2EMIN__SET__LEQ__MAX__SET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) ) & p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s)) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2EMIN__SET,V0s)),ap(c_2Epred__set_2EMAX__SET,V0s))) ) ))).
fof(conj_thm_2Epred__set_2EMIN__SET__UNION,lemma,(! [V0A] : ( mem(V0A,arr(ty_2Enum_2Enum,bool)) => ! [V1B] : ( mem(V1B,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0A)) & ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1B)) & ( ( ~~ ( V0A = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) ) & ( ~~ ( V1B = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) ) ) ) ) => ( ap(c_2Epred__set_2EMIN__SET,ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),V0A),V1B)) = ap(ap(c_2Earithmetic_2EMIN,ap(c_2Epred__set_2EMIN__SET,V0A)),ap(c_2Epred__set_2EMIN__SET,V1B)) ) ) ) ))).
fof(conj_thm_2Epred__set_2EMAX__SET__UNION,lemma,(! [V0A] : ( mem(V0A,arr(ty_2Enum_2Enum,bool)) => ! [V1B] : ( mem(V1B,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0A)) & p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1B)) ) => ( ap(c_2Epred__set_2EMAX__SET,ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),V0A),V1B)) = ap(ap(c_2Earithmetic_2EMAX,ap(c_2Epred__set_2EMAX__SET,V0A)),ap(c_2Epred__set_2EMAX__SET,V1B)) ) ) ) ))).
fof(ax_thm_2Epred__set_2EPOW__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0set] : ( mem(V0set,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EPOW(A_27a),V0set) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f289(A_27a,V0set)) ) ) ) )).
fof(conj_thm_2Epred__set_2EIN__POW,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0set] : ( mem(V0set,arr(A_27a,bool)) => ! [V1e] : ( mem(V1e,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1e),ap(c_2Epred__set_2EPOW(A_27a),V0set))) <=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1e),V0set)) ) ) )))).
fof(conj_thm_2Epred__set_2EUNIV__FUN__TO__BOOL,lemma,( ! [A_27a]: ( ne(A_27a) => ( c_2Epred__set_2EUNIV(arr(A_27a,bool)) = ap(c_2Epred__set_2EPOW(A_27a),c_2Epred__set_2EUNIV(A_27a)) )))).
fof(conj_thm_2Epred__set_2ESUBSET__POW,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s1),V1s2)) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Epred__set_2EPOW(A_27a),V0s1)),ap(c_2Epred__set_2EPOW(A_27a),V1s2))) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__INSERT__RIGHT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0e] : ( mem(V0e,A_27a) => ! [V1s1] : ( mem(V1s1,arr(A_27a,bool)) => ! [V2s2] : ( mem(V2s2,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s1),V2s2)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s1),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0e),V2s2))) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__DELETE__BOTH,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s1),V1s2)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s1),V2x)),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s2),V2x))) ) ) ) )))).
fof(conj_thm_2Epred__set_2EPOW__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ~~ ( ap(c_2Epred__set_2EPOW(A_27a),V0s) = c_2Epred__set_2EEMPTY(arr(A_27a,bool)) ) ) )))).
fof(conj_thm_2Epred__set_2EPOW__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0e] : ( mem(V0e,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EPOW(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0e),V1s)) = ap(ap(c_2Epred__set_2EUNION(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EINSERT(A_27a),V0e)),ap(c_2Epred__set_2EPOW(A_27a),V1s))),ap(c_2Epred__set_2EPOW(A_27a),V1s)) ) ) )))).
fof(conj_thm_2Epred__set_2EPOW__EQNS,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ap(c_2Epred__set_2EPOW(A_27a),c_2Epred__set_2EEMPTY(A_27a)) = ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),c_2Epred__set_2EEMPTY(arr(A_27a,bool))) ) & ! [V0e] : ( mem(V0e,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EPOW(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0e),V1s)) = ap(ap(c_2Ebool_2ELET(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool)),f290(A_27a,V0e)),ap(c_2Epred__set_2EPOW(A_27a),V1s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__POW,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),ap(c_2Epred__set_2EPOW(A_27a),V0s))) ) )))).
fof(conj_thm_2Epred__set_2ECARD__POW,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(arr(A_27a,bool)),ap(c_2Epred__set_2EPOW(A_27a),V0s)) = ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),ap(c_2Epred__set_2ECARD(A_27a),V0s)) ) ) )))).
fof(conj_thm_2Epred__set_2EGSPEC__F,lemma,( ! [A_27a]: ( ne(A_27a) => ( ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f291(A_27a)) = c_2Epred__set_2EEMPTY(A_27a) )))).
fof(conj_thm_2Epred__set_2EGSPEC__T,lemma,( ! [A_27a]: ( ne(A_27a) => ( ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f292(A_27a)) = c_2Epred__set_2EUNIV(A_27a) )))).
fof(conj_thm_2Epred__set_2EGSPEC__ID,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0y] : ( mem(V0y,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f293(A_27a,V0y)) = V0y ) )))).
fof(conj_thm_2Epred__set_2EGSPEC__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0y] : ( mem(V0y,A_27a) => ( ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f294(A_27a,V0y)) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0y),c_2Epred__set_2EEMPTY(A_27a)) ) )))).
fof(conj_thm_2Epred__set_2EGSPEC__EQ2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0y] : ( mem(V0y,A_27a) => ( ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f295(A_27a,V0y)) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0y),c_2Epred__set_2EEMPTY(A_27a)) ) )))).
fof(conj_thm_2Epred__set_2EGSPEC__F__COND,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Epair_2Eprod(A_27b,bool))) => ( ( ! [V1x] : ( mem(V1x,A_27a) => ( ~~ p(ap(c_2Epair_2ESND(A_27b,bool),ap(V0f,V1x))) ) ) ) => ( ap(c_2Epred__set_2EGSPEC(A_27b,A_27a),V0f) = c_2Epred__set_2EEMPTY(A_27b) ) ) ))))).
fof(conj_thm_2Epred__set_2EGSPEC__AND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f296(A_27a,V1Q,V0P)) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f297(A_27a,V0P))),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f298(A_27a,V1Q))) ) ) )))).
fof(conj_thm_2Epred__set_2EGSPEC__OR,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,bool)) => ( ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f299(A_27a,V1Q,V0P)) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f297(A_27a,V0P))),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f298(A_27a,V1Q))) ) ) )))).
fof(ax_thm_2Epred__set_2Eequiv__on__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2Eequiv__on(A_27a),V0R),V1s)) <=> ( ( ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) => p(ap(ap(V0R,V2x),V2x)) ) ) ) & ( ( ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V1s)) ) => ( p(ap(ap(V0R,V3x),V4y)) <=> p(ap(ap(V0R,V4y),V3x)) ) ) ) ) ) & ! [V5x] : ( mem(V5x,A_27a) => ! [V6y] : ( mem(V6y,A_27a) => ! [V7z] : ( mem(V7z,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6y),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7z),V1s)) & ( p(ap(ap(V0R,V5x),V6y)) & p(ap(ap(V0R,V6y),V7z)) ) ) ) ) => p(ap(ap(V0R,V5x),V7z)) ) ) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Epred__set_2Epartition__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f302(A_27a,V0R,V1s)) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EBIGUNION__partition,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2Eequiv__on(A_27a),V0R),V1s)) => ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s)) = V1s ) ) ) )))).
fof(conj_thm_2Epred__set_2EEMPTY__NOT__IN__partition,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2Eequiv__on(A_27a),V0R),V1s)) => ( ~~ p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s))) ) ) ) )))).
fof(conj_thm_2Epred__set_2Epartition__elements__disjoint,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2Eequiv__on(A_27a),V0R),V1s)) => ! [V2t1] : ( mem(V2t1,arr(A_27a,bool)) => ! [V3t2] : ( mem(V3t2,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t1),ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t2),ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s))) & ( ~~ ( V2t1 = V3t2 ) ) ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V2t1),V3t2)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2Epartition__elements__interrelate,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2Eequiv__on(A_27a),V0R),V1s)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s))) => ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2t)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2t)) ) => p(ap(ap(V0R,V3x),V4y)) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2Epartition__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s))) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V1s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__partition,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s))) & ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s))) => p(ap(c_2Epred__set_2EFINITE(A_27a),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2Epartition__CARD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2Eequiv__on(A_27a),V0R),V1s)) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) => ( ap(c_2Epred__set_2ECARD(A_27a),V1s) = ap(ap(c_2Epred__set_2ESUM__IMAGE(arr(A_27a,bool)),c_2Epred__set_2ECARD(A_27a)),ap(ap(c_2Epred__set_2Epartition(A_27a),V0R),V1s)) ) ) ) )))).
fof(ax_thm_2Epred__set_2Epairwise__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2Epairwise(A_27a),V0P),V1s)) <=> ! [V2e1] : ( mem(V2e1,A_27a) => ! [V3e2] : ( mem(V3e2,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2e1),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V3e2),V1s)) ) => p(ap(ap(V0P,V2e1),V3e2)) ) ) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2Epairwise__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s1] : ( mem(V1s1,arr(A_27a,bool)) => ! [V2s2] : ( mem(V2s2,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2Epairwise(A_27a),V0R),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s1),V2s2))) <=> ( p(ap(ap(c_2Epred__set_2Epairwise(A_27a),V0R),V1s1)) & ( p(ap(ap(c_2Epred__set_2Epairwise(A_27a),V0R),V2s2)) & ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s1)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s2)) ) => ( p(ap(ap(V0R,V3x),V4y)) & p(ap(ap(V0R,V4y),V3x)) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2Epairwise__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2Epairwise(A_27a),V0R),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) ) => p(ap(ap(c_2Epred__set_2Epairwise(A_27a),V0R),V1s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EKoenigsLemma,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ( ( ! [V1x] : ( mem(V1x,A_27a) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f303(A_27a,V0R,V1x)))) ) ) => ! [V3x] : ( mem(V3x,A_27a) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f304(A_27a,V0R,V3x)))) ) => ? [V5f] : ( mem(V5f,arr(ty_2Enum_2Enum,A_27a)) & ( ( ap(V5f,c_2Enum_2E0) = V3x ) & ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => p(ap(ap(V0R,ap(V5f,V6n)),ap(V5f,ap(c_2Enum_2ESUC,V6n)))) ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EKoenigsLemma__WF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27a,bool))) => ( ( ( ! [V1x] : ( mem(V1x,A_27a) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f303(A_27a,V0R,V1x)))) ) ) & p(ap(c_2Erelation_2EWF(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R))) ) => ! [V3x] : ( mem(V3x,A_27a) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f304(A_27a,V0R,V3x)))) ) ) )))).
fof(conj_thm_2Epred__set_2ESET__EQ__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27a,bool)) => ( ( V0s1 = V1s2 ) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s1),V1s2)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s2),V0s1)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EPSUBSET__EQN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s1),V1s2)) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s1),V1s2)) & ( ~~ p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s2),V0s1)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EPSUBSET__SUBSET__TRANS,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(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V2u)) ) => p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V2u)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__PSUBSET__TRANS,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(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) & p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V1t),V2u)) ) => p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V2u)) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECROSS__EQNS,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,A_27a) => ! [V1s1] : ( mem(V1s1,arr(A_27a,bool)) => ! [V2s2] : ( mem(V2s2,arr(A_27b,bool)) => ( ( ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),c_2Epred__set_2EEMPTY(A_27a)),V2s2) = c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27b)) ) & ( ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0a),V1s1)),V2s2) = ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2EIMAGE(A_27b,ty_2Epair_2Eprod(A_27a,A_27b)),f305(A_27a,A_27b,V0a)),V2s2)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V1s1),V2s2)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2Ecount__EQN,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Ecount,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_2Ebool_2ELET(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool)),f306),ap(c_2Eprim__rec_2EPRE,V0n))) ) ))).
fof(conj_thm_2Epred__set_2EUNIQUE__MEMBER__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) & ! [V2y] : ( mem(V2y,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),V1s)) => ( V0x = V2y ) ) ) ) <=> ( V1s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a)) ) ) ) )))).
fof(conj_thm_2Epred__set_2Einj__surj,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) => ( ( V1s = c_2Epred__set_2EEMPTY(A_27a) ) | ? [V3f_27] : ( mem(V3f_27,arr(A_27b,A_27a)) & p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27b,A_27a),V3f_27),V2t),V1s)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2Einfinite__rest,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,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_2EREST(A_27a),V0s))) ) ) )))).
fof(ax_thm_2Epred__set_2Echooser__def,axiom,( ! [A_27a] : ( ne(A_27a) => ( ( ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2Echooser(A_27a),V0s),c_2Enum_2E0) = ap(c_2Epred__set_2ECHOICE(A_27a),V0s) ) ) ) & ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ap(ap(c_2Epred__set_2Echooser(A_27a),V1s),ap(c_2Enum_2ESUC,V2n)) = ap(ap(c_2Epred__set_2Echooser(A_27a),ap(c_2Epred__set_2EREST(A_27a),V1s)),V2n) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2Echooser__def__compute,lemma,( ! [A_27a]: ( ne(A_27a) => ( ( ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2Echooser(A_27a),V0s),c_2Enum_2E0) = ap(c_2Epred__set_2ECHOICE(A_27a),V0s) ) ) ) & ( ( ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ap(ap(c_2Epred__set_2Echooser(A_27a),V1s),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V2n))) = ap(ap(c_2Epred__set_2Echooser(A_27a),ap(c_2Epred__set_2EREST(A_27a),V1s)),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V2n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ) ) ) ) & ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ap(ap(c_2Epred__set_2Echooser(A_27a),V3s),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V4n))) = ap(ap(c_2Epred__set_2Echooser(A_27a),ap(c_2Epred__set_2EREST(A_27a),V3s)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V4n))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2Einfinite__num__inj,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)) & p(ap(ap(ap(c_2Epred__set_2EINJ(ty_2Enum_2Enum,A_27a),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) ) )))).
fof(ax_thm_2Epred__set_2Ecountable__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) <=> ? [V1f] : ( mem(V1f,arr(A_27a,ty_2Enum_2Enum)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,ty_2Enum_2Enum),V1f),V0s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ) )).
fof(conj_thm_2Epred__set_2Ecountable__image__nats,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,A_27a)) => p(ap(c_2Epred__set_2Ecountable(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,A_27a),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) )))).
fof(conj_thm_2Epred__set_2Ecountable__surj,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) <=> ( ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) | ? [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,A_27a)) & p(ap(ap(ap(c_2Epred__set_2ESURJ(ty_2Enum_2Enum,A_27a),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) ) ) )))).
fof(conj_thm_2Epred__set_2Enum__countable,lemma,(p(ap(c_2Epred__set_2Ecountable(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))).
fof(conj_thm_2Epred__set_2Esubset__countable,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V0s)) ) => p(ap(c_2Epred__set_2Ecountable(A_27a),V1t)) ) ) )))).
fof(conj_thm_2Epred__set_2Eimage__countable,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V1s)) => p(ap(c_2Epred__set_2Ecountable(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s))) ) ) ))))).
fof(conj_thm_2Epred__set_2ECOUNTABLE__ALT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) <=> ? [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,A_27a)) & ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) => ? [V3n] : ( mem(V3n,ty_2Enum_2Enum) & ( ap(V1f,V3n) = V2x ) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECOUNTABLE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) & p(ap(c_2Epred__set_2Ecountable(A_27a),V1t)) ) => p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) ) ) )))).
fof(conj_thm_2Epred__set_2Efinite__countable,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) => p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) ) )))).
fof(conj_thm_2Epred__set_2ECOUNTABLE__COUNT,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => p(ap(c_2Epred__set_2Ecountable(ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,V0n))) ))).
fof(conj_thm_2Epred__set_2ECOUNTABLE__NUM,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => p(ap(c_2Epred__set_2Ecountable(ty_2Enum_2Enum),V0s)) ))).
fof(conj_thm_2Epred__set_2ECOUNTABLE__IMAGE__NUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,A_27a)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => p(ap(c_2Epred__set_2Ecountable(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,A_27a),V0f),V1s))) ) )))).
fof(ax_thm_2Epred__set_2Enum__to__pair__def,axiom, ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Enum__to__pair,V0n) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(c_2Enumpair_2Enfst,V0n)),ap(c_2Enumpair_2Ensnd,V0n)) ) )).
fof(ax_thm_2Epred__set_2Epair__to__num__def,axiom, ! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Epair__to__num,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V0m),V1n)) = ap(ap(c_2Enumpair_2Enpair,V0m),V1n) ) ) )).
fof(conj_thm_2Epred__set_2Epair__to__num__formula,lemma,(! [V0x] : ( mem(V0x,ty_2Enum_2Enum) => ! [V1y] : ( mem(V1y,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Epair__to__num,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V0x),V1y)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2EDIV,ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2B,V0x),V1y)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Earithmetic_2E_2B,V0x),V1y))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V1y) ) ) ))).
fof(conj_thm_2Epred__set_2Epair__to__num__inv,lemma,(( ( ! [V0x] : ( mem(V0x,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Epair__to__num,ap(c_2Epred__set_2Enum__to__pair,V0x)) = V0x ) ) ) & ! [V1x] : ( mem(V1x,ty_2Enum_2Enum) => ! [V2y] : ( mem(V2y,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Enum__to__pair,ap(c_2Epred__set_2Epair__to__num,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V1x),V2y))) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V1x),V2y) ) ) ) ))).
fof(conj_thm_2Epred__set_2Ecross__countable,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)) => ( ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) & p(ap(c_2Epred__set_2Ecountable(A_27b),V1t)) ) => p(ap(c_2Epred__set_2Ecountable(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t))) ) ) ))))).
fof(conj_thm_2Epred__set_2Einter__countable,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) | p(ap(c_2Epred__set_2Ecountable(A_27a),V1t)) ) => p(ap(c_2Epred__set_2Ecountable(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t))) ) ) )))).
fof(conj_thm_2Epred__set_2Einj__countable,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2Ecountable(A_27b),V2t)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t)) ) => p(ap(c_2Epred__set_2Ecountable(A_27a),V1s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2Ebigunion__countable,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V0s)) & ! [V1x] : ( mem(V1x,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1x),V0s)) => p(ap(c_2Epred__set_2Ecountable(A_27a),V1x)) ) ) ) => p(ap(c_2Epred__set_2Ecountable(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s))) ) )))).
fof(conj_thm_2Epred__set_2Eunion__countable,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) & p(ap(c_2Epred__set_2Ecountable(A_27a),V1t)) ) => p(ap(c_2Epred__set_2Ecountable(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t))) ) ) )))).
fof(conj_thm_2Epred__set_2Eunion__countable__IFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t))) <=> ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) & p(ap(c_2Epred__set_2Ecountable(A_27a),V1t)) ) ) ) )))).
fof(conj_thm_2Epred__set_2Einj__image__countable__IFF,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s))) => ( p(ap(c_2Epred__set_2Ecountable(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s))) <=> p(ap(c_2Epred__set_2Ecountable(A_27a),V1s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2Epow__no__surj,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ~~ ? [V1f] : ( mem(V1f,arr(A_27a,arr(A_27a,bool))) & p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,arr(A_27a,bool)),V1f),V0s),ap(c_2Epred__set_2EPOW(A_27a),V0s))) ) ) )))).
fof(conj_thm_2Epred__set_2Einfinite__pow__uncountable,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) => ( ~~ p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),ap(c_2Epred__set_2EPOW(A_27a),V0s))) ) ) )))).
fof(conj_thm_2Epred__set_2Ecountable__Usum,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( p(ap(c_2Epred__set_2Ecountable(ty_2Esum_2Esum(A_27a,A_27b)),c_2Epred__set_2EUNIV(ty_2Esum_2Esum(A_27a,A_27b)))) <=> ( p(ap(c_2Epred__set_2Ecountable(A_27a),c_2Epred__set_2EUNIV(A_27a))) & p(ap(c_2Epred__set_2Ecountable(A_27b),c_2Epred__set_2EUNIV(A_27b))) ) ))))).
fof(conj_thm_2Epred__set_2Ecountable__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => p(ap(c_2Epred__set_2Ecountable(A_27a),c_2Epred__set_2EEMPTY(A_27a)))))).
fof(conj_thm_2Epred__set_2Ecountable__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s))) <=> p(ap(c_2Epred__set_2Ecountable(A_27a),V1s)) ) ) )))).
fof(conj_thm_2Epred__set_2Ecross__countable__IFF,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)) => ( p(ap(c_2Epred__set_2Ecountable(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t))) <=> ( ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) | ( ( V1t = c_2Epred__set_2EEMPTY(A_27b) ) | ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) & p(ap(c_2Epred__set_2Ecountable(A_27b),V1t)) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2Ecountable__Uprod,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( p(ap(c_2Epred__set_2Ecountable(ty_2Epair_2Eprod(A_27a,A_27b)),c_2Epred__set_2EUNIV(ty_2Epair_2Eprod(A_27a,A_27b)))) <=> ( p(ap(c_2Epred__set_2Ecountable(A_27a),c_2Epred__set_2EUNIV(A_27a))) & p(ap(c_2Epred__set_2Ecountable(A_27b),c_2Epred__set_2EUNIV(A_27b))) ) ))))).
fof(conj_thm_2Epred__set_2EEXPLICIT__ENUMERATE__MONO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(ap(c_2Earithmetic_2EFUNPOW(arr(A_27a,bool)),c_2Epred__set_2EREST(A_27a)),V0n),V1s)),V1s)) ) )))).
fof(conj_thm_2Epred__set_2EEXPLICIT__ENUMERATE__NOT__EMPTY,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)) ) => ( ~~ ( ap(ap(ap(c_2Earithmetic_2EFUNPOW(arr(A_27a,bool)),c_2Epred__set_2EREST(A_27a)),V0n),V1s) = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINFINITE__EXPLICIT__ENUMERATE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) => p(ap(ap(ap(c_2Epred__set_2EINJ(ty_2Enum_2Enum,A_27a),f307(A_27a,V0s)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) )))).
fof(conj_thm_2Epred__set_2EBIJ__NUM__COUNTABLE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ? [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,A_27a)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) ) => p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) ) )))).
fof(ax_thm_2Epred__set_2Eenumerate__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(c_2Epred__set_2Eenumerate(A_27a),V0s) = ap(c_2Emin_2E_40(arr(ty_2Enum_2Enum,A_27a)),f308(A_27a,V0s)) ) ) ) )).
fof(conj_thm_2Epred__set_2EENUMERATE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ? [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,A_27a)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) ) <=> p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),ap(c_2Epred__set_2Eenumerate(A_27a),V0s)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) )))).
fof(conj_thm_2Epred__set_2ECOUNTABLE__ALT__BIJ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) <=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) | p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),ap(c_2Epred__set_2Eenumerate(A_27a),V0s)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) ) )))).
fof(conj_thm_2Epred__set_2ECOUNTABLE__ENUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0c] : ( mem(V0c,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0c)) <=> ( ( V0c = c_2Epred__set_2EEMPTY(A_27a) ) | ? [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,A_27a)) & ( V0c = ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,A_27a),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EIMAGE__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1M] : ( mem(V1M,arr(arr(A_27a,bool),bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Epred__set_2EBIGUNION(A_27a),V1M)) = ap(c_2Epred__set_2EBIGUNION(A_27b),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27b,bool)),ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f)),V1M)) ) ) ))))).
fof(conj_thm_2Epred__set_2ESUBSET__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27a,bool)) => ! [V2s3] : ( mem(V2s3,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s1),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s2),V2s3))) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s1),V1s2)) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s1),V2s3)) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__SUBSET__EQN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ! [V1B] : ( mem(V1B,arr(A_27a,bool)) => ( ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0A),V1B) = V0A ) <=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0A),V1B)) ) & ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0A),V1B) = V1B ) <=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1B),V0A)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EPSUBSET__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,A_27a) => ! [V1x] : ( mem(V1x,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V1x),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0s),c_2Epred__set_2EEMPTY(A_27a)))) <=> ( V1x = c_2Epred__set_2EEMPTY(A_27a) ) ) ) )))).
fof(conj_thm_2Epred__set_2EINTER__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ! [V1B] : ( mem(V1B,arr(A_27a,bool)) => ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0A),V1B)),V0A) = V0A ) & ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1B),V0A)),V0A) = V0A ) & ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0A),ap(ap(c_2Epred__set_2EUNION(A_27a),V0A),V1B)) = V0A ) & ( ap(ap(c_2Epred__set_2EINTER(A_27a),V0A),ap(ap(c_2Epred__set_2EUNION(A_27a),V1B),V0A)) = V0A ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EUNION__DELETE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ! [V1B] : ( mem(V1B,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0A),V1B)),V2x) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0A),V2x)),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1B),V2x)) ) ) ) )))).
fof(conj_thm_2Epred__set_2EDELETE__SUBSET__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1e] : ( mem(V1e,A_27a) => ! [V2s2] : ( mem(V2s2,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V1e)),V2s2)) <=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s2))) ) ) ) )))).
fof(conj_thm_2Epred__set_2EIN__INSERT__EXPAND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1y] : ( mem(V1y,A_27a) => ! [V2P] : ( mem(V2P,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1y),V2P))) <=> ( ( V0x = V1y ) | ( ( ~~ ( V0x = V1y ) ) & p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2P)) ) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EFINITE__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1)) | p(ap(c_2Epred__set_2EFINITE(A_27a),V1s2)) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s1),V1s2))) ) ) )))).
fof(conj_thm_2Epred__set_2EINSERT__EQ__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( ( ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V0s) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V2y),c_2Epred__set_2EEMPTY(A_27a)) ) <=> ( ( V1x = V2y ) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2y),c_2Epred__set_2EEMPTY(A_27a)))) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2ECARD__UNION__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t))),ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27a),V1t)))) ) ) )))).
fof(conj_thm_2Epred__set_2EIMAGE__SUBSET__gen,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2u] : ( mem(V2u,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2u)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2u)),V3t)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s)),V3t)) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2ECARD__REST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) => ( ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EREST(A_27a),V0s)) = ap(ap(c_2Earithmetic_2E_2D,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) )))).
fof(conj_thm_2Epred__set_2ESUBSET__DIFF__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t) = c_2Epred__set_2EEMPTY(A_27a) ) <=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) ) ) )))).
fof(conj_thm_2Epred__set_2EDIFF__INTER__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0r] : ( mem(V0r,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0r),V1s)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),V0r),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)) = ap(ap(c_2Epred__set_2EDIFF(A_27a),V0r),V2t) ) ) ) ) )))).
fof(conj_thm_2Epred__set_2EUNION__DIFF__2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t)) = V0s ) ) )))).
fof(conj_thm_2Epred__set_2Ecount__add,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Ecount,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)) = ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,V0n)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(c_2Earithmetic_2E_2B,V0n)),ap(c_2Epred__set_2Ecount,V1m))) ) ) ))).
fof(conj_thm_2Epred__set_2EIMAGE__EQ__SING,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27b,A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2z] : ( mem(V2z,A_27a) => ( ( ap(ap(c_2Epred__set_2EIMAGE(A_27b,A_27a),V0f),V1s) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V2z),c_2Epred__set_2EEMPTY(A_27a)) ) <=> ( ( ~~ ( V1s = c_2Epred__set_2EEMPTY(A_27b) ) ) & ! [V3x] : ( mem(V3x,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3x),V1s)) => ( ap(V0f,V3x) = V2z ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Epred__set_2Ecount__add1,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Epred__set_2Ecount,ap(ap(c_2Earithmetic_2E_2B,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0n),ap(c_2Epred__set_2Ecount,V0n)) ) ))).
fof(conj_thm_2Epred__set_2Ecompl__insert,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( ap(c_2Epred__set_2ECOMPL(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V0s)) = ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(c_2Epred__set_2ECOMPL(A_27a),V0s)),V1x) ) ) )))).
fof(conj_thm_2Epred__set_2Ein__max__set,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s)) => ! [V1x] : ( mem(V1x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1x),V0s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V1x),ap(c_2Epred__set_2EMAX__SET,V0s))) ) ) ) ))).
fof(ax_thm_2Epred__set_2EPREIMAGE__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V1s) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f309(A_27b,A_27a,V1s,V0f)) ) ) ) ) ) )).
fof(conj_thm_2Epred__set_2EPREIMAGE__ALT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V1s) = ap(ap(c_2Ecombin_2Eo(A_27a,bool,A_27b),V1s),V0f) ) ) ))))).
fof(conj_thm_2Epred__set_2EIN__PREIMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V1s))) <=> p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V2x)),V1s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__applied,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V1s),V2x)) <=> p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V2x)),V1s)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),c_2Epred__set_2EEMPTY(A_27b)) = c_2Epred__set_2EEMPTY(A_27a) ) ))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),c_2Epred__set_2EUNIV(A_27b)) = c_2Epred__set_2EUNIV(A_27a) ) ))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),ap(c_2Epred__set_2ECOMPL(A_27b),V1s)) = ap(c_2Epred__set_2ECOMPL(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V1s)) ) ) ))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__UNION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EUNION(A_27b),V1s),V2t)) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V1s)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V2t)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__INTER,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINTER(A_27b),V1s),V2t)) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V1s)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V2t)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(arr(A_27b,bool),bool)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),ap(c_2Epred__set_2EBIGUNION(A_27b),V1s)) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f)),V1s)) ) ) ))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__COMP,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27b,A_27c)) => ! [V2s] : ( mem(V2s,arr(A_27c,bool)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27c),V1g),V2s)) = ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27c),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),V2s) ) ) ) )))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EDIFF(A_27b),V1s),V2t)) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V1s)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V2t)) ) ) ) ))))).
fof(conj_thm_2Epred__set_2EPREIMAGE__I,lemma,( ! [A_27a]: ( ne(A_27a) => ( ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27a),c_2Ecombin_2EI(A_27a)) = c_2Ecombin_2EI(arr(A_27a,bool)) )))).
fof(conj_thm_2Epred__set_2EPREIMAGE__K,conjecture,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27a),ap(c_2Ecombin_2EK(A_27a,A_27b),V0x)),V1s) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27b,bool)),ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)),c_2Epred__set_2EUNIV(A_27b)),c_2Epred__set_2EEMPTY(A_27b)) ) ) ))))).
