include('Axioms/ring+2.ax').
fof(mem_c_2Ecardinal_2E_2B__c,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Ecardinal_2E_2B__c(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),arr(ty_2Esum_2Esum(A_27a,A_27b),bool)))) ) ) )).
fof(mem_c_2Ecardinal_2EHAS__SIZE,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ecardinal_2EHAS__SIZE(A_27a),arr(arr(A_27a,bool),arr(ty_2Enum_2Enum,bool))) ) )).
fof(mem_c_2Ecardinal_2Ebijns,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ecardinal_2Ebijns(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Eoption_2Eoption(A_27a)),bool))) ) )).
fof(mem_c_2Ecardinal_2Ecardeq,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Ecardinal_2Ecardeq(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool))) ) ) )).
fof(mem_c_2Ecardinal_2Ecardgeq,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Ecardinal_2Ecardgeq(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool))) ) ) )).
fof(mem_c_2Ecardinal_2Ecardgt,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Ecardinal_2Ecardgt(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool))) ) ) )).
fof(mem_c_2Ecardinal_2Ecardleq,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Ecardinal_2Ecardleq(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool))) ) ) )).
fof(mem_c_2Ecardinal_2Elist,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ecardinal_2Elist(A_27a),arr(arr(A_27a,bool),arr(ty_2Elist_2Elist(A_27a),bool))) ) )).
fof(mem_c_2Ecardinal_2Eset__exp,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Ecardinal_2Eset__exp(A_27a,A_27b),arr(arr(A_27b,bool),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Eoption_2Eoption(A_27b)),bool)))) ) ) )).
fof(ax_thm_2Ecardinal_2Ecardeq__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2)) <=> ? [V2f] : ( mem(V2f,arr(A_27a,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s1),V1s2)) ) ) ) ) ) ) )).
fof(conj_thm_2Ecardinal_2Ecardeq__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27a),V0s),V0s)) )))).
fof(conj_thm_2Ecardinal_2Ecardeq__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)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) <=> p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27a),V1t),V0s)) ) ) ))))).
fof(conj_thm_2Ecardinal_2Ecardeq__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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27c),V1t),V2u)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27c),V0s),V2u)) ) ) ) )))))).
fof(ax_thm_2Ecardinal_2Ecardleq__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s1),V1s2)) <=> ? [V2f] : ( mem(V2f,arr(A_27a,A_27b)) & p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),V0s1),V1s2)) ) ) ) ) ) ) )).
fof(conj_thm_2Ecardinal_2Ecardleq__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V0s)) )))).
fof(conj_thm_2Ecardinal_2Ecardleq__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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27c),V1t),V2u)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27c),V0s),V2u)) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2Ecardleq__ANTISYM,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARDEQ__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1)) <=> p(ap(c_2Epred__set_2EFINITE(A_27b),V1s2)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARDEQ__CARD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2)) & p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1)) ) => ( ap(c_2Epred__set_2ECARD(A_27a),V0s1) = ap(c_2Epred__set_2ECARD(A_27b),V1s2) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARDEQ__0,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0x),c_2Epred__set_2EEMPTY(A_27b))) <=> ( V0x = c_2Epred__set_2EEMPTY(A_27a) ) ) & ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27c,A_27a),c_2Epred__set_2EEMPTY(A_27c)),V0x)) <=> ( V0x = c_2Epred__set_2EEMPTY(A_27a) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2Ecardeq__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s)),V1s)) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) | ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARDEQ__INSERT__RWT,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),V1s)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s)),V1s)) ) ) )))).
fof(conj_thm_2Ecardinal_2EEMPTY__CARDLEQ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0t] : ( mem(V0t,arr(A_27b,bool)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),c_2Epred__set_2EEMPTY(A_27a)),V0t)) ))))).
fof(conj_thm_2Ecardinal_2EFINITE__CLE__INFINITE,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_2EFINITE(A_27a),V0s)) & ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARDEQ__CROSS,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ! [V2t1] : ( mem(V2t1,arr(A_27c,bool)) => ! [V3t2] : ( mem(V3t2,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27c,A_27d),V2t1),V3t2)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27b,A_27d)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27c),V0s1),V2t1)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27d),V1s2),V3t2))) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2ECARDEQ__CROSS__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)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27a),V1t),V0s))) ) ))))).
fof(conj_thm_2Ecardinal_2ECARDEQ__SUBSET__CARDLEQ,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(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARDEQ__CARDLEQ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ! [V2t1] : ( mem(V2t1,arr(A_27c,bool)) => ! [V3t2] : ( mem(V3t2,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27c,A_27d),V2t1),V3t2)) ) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27c),V0s1),V2t1)) <=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27d),V1s2),V3t2)) ) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2ECARDLEQ__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V1s2)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s1),V1s2)) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1)) ) ) ))))).
fof(conj_thm_2Ecardinal_2EINFINITE__UNIV__INF,lemma,( ! [A_27a]: ( ne(A_27a) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),c_2Epred__set_2EUNIV(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)))) )))).
fof(conj_thm_2Ecardinal_2EIMAGE__cardleq,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(c_2Ecardinal_2Ecardleq(A_27b,A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s)),V1s)) ) ))))).
fof(conj_thm_2Ecardinal_2ECARDLEQ__CROSS__CONG,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0x1] : ( mem(V0x1,arr(A_27a,bool)) => ! [V1x2] : ( mem(V1x2,arr(A_27b,bool)) => ! [V2y1] : ( mem(V2y1,arr(A_27c,bool)) => ! [V3y2] : ( mem(V3y2,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0x1),V1x2)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27d),V2y1),V3y2)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27b,A_27d)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27c),V0x1),V2y1)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27d),V1x2),V3y2))) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2ESUBSET__CARDLEQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ! [V1y] : ( mem(V1y,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0x),V1y)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0x),V1y)) ) ) )))).
fof(conj_thm_2Ecardinal_2EIMAGE__cardleq__rwt,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27c)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V1s),V2t)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27c),V0f),V1s)),V2t)) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2Ecountable__thm,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) <=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,ty_2Enum_2Enum),V0s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) )))).
fof(conj_thm_2Ecardinal_2Ecountable__cardeq,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(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) <=> p(ap(c_2Epred__set_2Ecountable(A_27b),V1t)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2Ecardleq__dichotomy,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) | p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) ) ))))).
fof(conj_thm_2Ecardinal_2Ecardleq__lteq,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s1),V1s2)) <=> ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1s2),V0s1)) ) | p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2Ecardlt__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ~~ ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V0s)) ) ) )))).
fof(conj_thm_2Ecardinal_2Ecardlt__lenoteq,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(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) <=> ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) & ( ~~ p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2Ecardlt__TRANS,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0s] : ( mem(V0s,arr(A_27b,bool)) => ! [V1t] : ( mem(V1t,arr(A_27c,bool)) => ! [V2u] : ( mem(V2u,arr(A_27a,bool)) => ( ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27b),V1t),V0s)) ) & ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27c),V2u),V1t)) ) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V2u),V0s)) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2Ecardlt__leq__trans,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0r] : ( mem(V0r,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ( ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1s),V0r)) ) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27c),V1s),V2t)) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27a),V2t),V0r)) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2Ecardleq__lt__trans,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0r] : ( mem(V0r,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0r),V1s)) & ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27b),V2t),V1s)) ) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27a),V2t),V0r)) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2Ecardleq__empty,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0x),c_2Epred__set_2EEMPTY(A_27b))) <=> ( V0x = c_2Epred__set_2EEMPTY(A_27a) ) ) ))))).
fof(conj_thm_2Ecardinal_2Eset__binomial2,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_2ECROSS(A_27a,A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0A),V1B)),ap(ap(c_2Epred__set_2EUNION(A_27a),V0A),V1B)) = ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27a),V0A),V0A)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27a),V0A),V1B))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27a),V1B),V0A))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27a),V1B),V1B)) ) ) )))).
fof(conj_thm_2Ecardinal_2ESET__SQUARED__CARDEQ__SET,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(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(A_27a,A_27a),A_27a),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27a),V0s),V0s)),V0s)) ) )))).
fof(conj_thm_2Ecardinal_2ESET__SUM__CARDEQ__SET,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)) ) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,ty_2Epair_2Eprod(bool,A_27a)),V0s),ap(ap(c_2Epred__set_2ECROSS(bool,A_27a),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)))),V0s))) & ! [V1A] : ( mem(V1A,arr(A_27b,bool)) => ! [V2B] : ( mem(V2B,arr(A_27b,bool)) => ( ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27b),V1A),V2B)) & ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27a),V1A),V0s)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27a),V2B),V0s)) ) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27a),ap(ap(c_2Epred__set_2EUNION(A_27b),V1A),V2B)),V0s)) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0k] : ( mem(V0k,arr(A_27a,bool)) => ! [V1s1] : ( mem(V1s1,arr(arr(A_27b,bool),bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0k)) ) & ( p(ap(ap(c_2Ecardinal_2Ecardleq(arr(A_27b,bool),A_27a),V1s1),V0k)) & ! [V2e] : ( mem(V2e,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V2e),V1s1)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V2e),V0k)) ) ) ) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),ap(c_2Epred__set_2EBIGUNION(A_27b),V1s1)),V0k)) ) ) ))))).
fof(ax_thm_2Ecardinal_2Eset__exp__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0A] : ( mem(V0A,arr(A_27b,bool)) => ! [V1B] : ( mem(V1B,arr(A_27a,bool)) => ( ap(ap(c_2Ecardinal_2Eset__exp(A_27a,A_27b),V0A),V1B) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,ty_2Eoption_2Eoption(A_27b)),arr(A_27a,ty_2Eoption_2Eoption(A_27b))),f1851(A_27b,A_27a,V1B,V0A)) ) ) ) ) ) )).
fof(conj_thm_2Ecardinal_2EBIJ__functions__agree,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27a,A_27b)) => ! [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),V2s),V3t)) & ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s)) => ( ap(V0f,V4x) = ap(V1g,V4x) ) ) ) ) => p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V1g),V2s),V3t)) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__CARDEQ__I,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1)) & ( p(ap(c_2Epred__set_2EFINITE(A_27b),V1s2)) & ( ap(c_2Epred__set_2ECARD(A_27a),V0s1) = ap(c_2Epred__set_2ECARD(A_27b),V1s2) ) ) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARDEQ__CARD__EQN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1s2)) ) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2)) <=> ( ap(c_2Epred__set_2ECARD(A_27a),V0s1) = ap(c_2Epred__set_2ECARD(A_27b),V1s2) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARDLEQ__CARD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s1] : ( mem(V0s1,arr(A_27a,bool)) => ! [V1s2] : ( mem(V1s2,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1s2)) ) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s1),V1s2)) <=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27a),V0s1)),ap(c_2Epred__set_2ECARD(A_27b),V1s2))) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EEMPTY__set__exp,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0A] : ( mem(V0A,arr(A_27b,bool)) => ! [V1B] : ( mem(V1B,arr(A_27c,bool)) => ( ( ap(ap(c_2Ecardinal_2Eset__exp(A_27a,A_27b),V0A),c_2Epred__set_2EEMPTY(A_27a)) = ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,ty_2Eoption_2Eoption(A_27b))),ap(c_2Ecombin_2EK(ty_2Eoption_2Eoption(A_27b),A_27a),c_2Eoption_2ENONE(A_27b))),c_2Epred__set_2EEMPTY(arr(A_27a,ty_2Eoption_2Eoption(A_27b)))) ) & ( ( ~~ ( V1B = c_2Epred__set_2EEMPTY(A_27c) ) ) => ( ap(ap(c_2Ecardinal_2Eset__exp(A_27c,A_27d),c_2Epred__set_2EEMPTY(A_27d)),V1B) = c_2Epred__set_2EEMPTY(arr(A_27c,ty_2Eoption_2Eoption(A_27d))) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2EEMPTY__set__exp__CARD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0A] : ( mem(V0A,arr(A_27b,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(arr(A_27a,ty_2Eoption_2Eoption(A_27b)),ty_2Enum_2Enum),ap(ap(c_2Ecardinal_2Eset__exp(A_27a,A_27b),V0A),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Epred__set_2Ecount,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ))))).
fof(conj_thm_2Ecardinal_2ESING__set__exp,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0x] : ( mem(V0x,A_27b) => ! [V1B] : ( mem(V1B,arr(A_27a,bool)) => ! [V2A] : ( mem(V2A,arr(A_27c,bool)) => ( ( ap(ap(c_2Ecardinal_2Eset__exp(A_27a,A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27b),V0x),c_2Epred__set_2EEMPTY(A_27b))),V1B) = ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,ty_2Eoption_2Eoption(A_27b))),f1852(A_27a,A_27b,V0x,V1B)),c_2Epred__set_2EEMPTY(arr(A_27a,ty_2Eoption_2Eoption(A_27b)))) ) & ( ap(ap(c_2Ecardinal_2Eset__exp(A_27b,A_27c),V2A),ap(ap(c_2Epred__set_2EINSERT(A_27b),V0x),c_2Epred__set_2EEMPTY(A_27b))) = ap(c_2Epred__set_2EGSPEC(arr(A_27b,ty_2Eoption_2Eoption(A_27c)),A_27c),f1854(A_27b,A_27c,V0x,V2A)) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ESING__set__exp__CARD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0x] : ( mem(V0x,A_27b) => ! [V1B] : ( mem(V1B,arr(A_27a,bool)) => ! [V2A] : ( mem(V2A,arr(A_27c,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(arr(A_27a,ty_2Eoption_2Eoption(A_27b)),ty_2Enum_2Enum),ap(ap(c_2Ecardinal_2Eset__exp(A_27a,A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27b),V0x),c_2Epred__set_2EEMPTY(A_27b))),V1B)),ap(c_2Epred__set_2Ecount,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) & p(ap(ap(c_2Ecardinal_2Ecardeq(arr(A_27b,ty_2Eoption_2Eoption(A_27c)),A_27c),ap(ap(c_2Ecardinal_2Eset__exp(A_27b,A_27c),V2A),ap(ap(c_2Epred__set_2EINSERT(A_27b),V0x),c_2Epred__set_2EEMPTY(A_27b)))),V2A)) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2EPOW__TWO__set__exp,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(arr(A_27a,bool),arr(A_27a,ty_2Eoption_2Eoption(ty_2Enum_2Enum))),ap(c_2Epred__set_2EPOW(A_27a),V0A)),ap(ap(c_2Ecardinal_2Eset__exp(A_27a,ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V0A))) )))).
fof(conj_thm_2Ecardinal_2Eset__exp__count,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => p(ap(ap(c_2Ecardinal_2Ecardeq(arr(ty_2Enum_2Enum,ty_2Eoption_2Eoption(A_27a)),ty_2Elist_2Elist(A_27a)),ap(ap(c_2Ecardinal_2Eset__exp(ty_2Enum_2Enum,A_27a),V0A),ap(c_2Epred__set_2Ecount,V1n))),ap(c_2Epred__set_2EGSPEC(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27a)),f1856(A_27a,V0A,V1n)))) ) )))).
fof(conj_thm_2Ecardinal_2Eset__exp__card__cong,lemma,( ! [A_27a1]: ( ne(A_27a1) =>  ! [A_27a2]: ( ne(A_27a2) =>  ! [A_27b1]: ( ne(A_27b1) =>  ! [A_27b2]: ( ne(A_27b2) => ! [V0a1] : ( mem(V0a1,arr(A_27a1,bool)) => ! [V1a2] : ( mem(V1a2,arr(A_27a2,bool)) => ! [V2b1] : ( mem(V2b1,arr(A_27b1,bool)) => ! [V3b2] : ( mem(V3b2,arr(A_27b2,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a1,A_27a2),V0a1),V1a2)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b1,A_27b2),V2b1),V3b2)) => p(ap(ap(c_2Ecardinal_2Ecardeq(arr(A_27b1,ty_2Eoption_2Eoption(A_27a1)),arr(A_27b2,ty_2Eoption_2Eoption(A_27a2))),ap(ap(c_2Ecardinal_2Eset__exp(A_27b1,A_27a1),V0a1),V2b1)),ap(ap(c_2Ecardinal_2Eset__exp(A_27b2,A_27a2),V1a2),V3b2))) ) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2Eset__exp__cardle__cong,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0b] : ( mem(V0b,arr(A_27a,bool)) => ! [V1d] : ( mem(V1d,arr(A_27b,bool)) => ! [V2a] : ( mem(V2a,arr(A_27c,bool)) => ! [V3c] : ( mem(V3c,arr(A_27d,bool)) => ( ( ( V0b = c_2Epred__set_2EEMPTY(A_27a) ) => ( V1d = c_2Epred__set_2EEMPTY(A_27b) ) ) => ( ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27a),V2a),V0b)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27d,A_27b),V3c),V1d)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(arr(A_27d,ty_2Eoption_2Eoption(A_27c)),arr(A_27b,ty_2Eoption_2Eoption(A_27a))),ap(ap(c_2Ecardinal_2Eset__exp(A_27d,A_27c),V2a),V3c)),ap(ap(c_2Ecardinal_2Eset__exp(A_27b,A_27a),V0b),V1d))) ) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2Eexp__INSERT__cardeq,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0e] : ( mem(V0e,A_27a) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2A] : ( mem(V2A,arr(A_27b,bool)) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0e),V1s)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(arr(A_27a,ty_2Eoption_2Eoption(A_27b)),ty_2Epair_2Eprod(A_27b,arr(A_27a,ty_2Eoption_2Eoption(A_27b)))),ap(ap(c_2Ecardinal_2Eset__exp(A_27a,A_27b),V2A),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0e),V1s))),ap(ap(c_2Epred__set_2ECROSS(A_27b,arr(A_27a,ty_2Eoption_2Eoption(A_27b))),V2A),ap(ap(c_2Ecardinal_2Eset__exp(A_27a,A_27b),V2A),V1s)))) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2Eexp__count__cardeq,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0A)) ) & p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V1n)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(arr(ty_2Enum_2Enum,ty_2Eoption_2Eoption(A_27a)),A_27a),ap(ap(c_2Ecardinal_2Eset__exp(ty_2Enum_2Enum,A_27a),V0A),ap(c_2Epred__set_2Ecount,V1n))),V0A)) ) ) )))).
fof(conj_thm_2Ecardinal_2EINFINITE__Unum,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0A)) ) <=> p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Enum_2Enum,A_27a),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0A)) ) )))).
fof(conj_thm_2Ecardinal_2Ecardleq__SURJ,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)) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0A),V1B)) <=> ( ( ? [V2f] : ( mem(V2f,arr(A_27b,A_27a)) & p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27b,A_27a),V2f),V1B),V0A)) ) ) | ( V0A = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EINFINITE__cardleq__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27b) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0A)) ) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27b),V1x),V2s)),V0A)) <=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V2s),V0A)) ) ) ) ) ))))).
fof(ax_thm_2Ecardinal_2Elist__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ( ap(c_2Ecardinal_2Elist(A_27a),V0A) = ap(c_2Epred__set_2EGSPEC(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27a)),f1858(A_27a,V0A)) ) ) ) )).
fof(conj_thm_2Ecardinal_2Elist__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ( ap(c_2Ecardinal_2Elist(A_27a),c_2Epred__set_2EEMPTY(A_27a)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Elist_2Elist(A_27a)),c_2Elist_2ENIL(A_27a)),c_2Epred__set_2EEMPTY(ty_2Elist_2Elist(A_27a))) )))).
fof(conj_thm_2Ecardinal_2Elist__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0e] : ( mem(V0e,A_27a) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Elist_2Elist(A_27a),ty_2Enum_2Enum),ap(c_2Ecardinal_2Elist(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0e),c_2Epred__set_2EEMPTY(A_27a)))),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) )))).
fof(conj_thm_2Ecardinal_2EUNIV__list,lemma,( ! [A_27a]: ( ne(A_27a) => ( c_2Epred__set_2EUNIV(ty_2Elist_2Elist(A_27a)) = ap(c_2Ecardinal_2Elist(A_27a),c_2Epred__set_2EUNIV(A_27a)) )))).
fof(conj_thm_2Ecardinal_2Elist__BIGUNION__EXP,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Elist_2Elist(A_27a),arr(ty_2Enum_2Enum,ty_2Eoption_2Eoption(A_27a))),ap(c_2Ecardinal_2Elist(A_27a),V0A)),ap(c_2Epred__set_2EBIGUNION(arr(ty_2Enum_2Enum,ty_2Eoption_2Eoption(A_27a))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(arr(ty_2Enum_2Enum,ty_2Eoption_2Eoption(A_27a)),bool)),f1859(A_27a,V0A)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) )))).
fof(conj_thm_2Ecardinal_2EINFINITE__A__list__BIJ__A,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0A)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Elist_2Elist(A_27a),A_27a),ap(c_2Ecardinal_2Elist(A_27a),V0A)),V0A)) ) )))).
fof(conj_thm_2Ecardinal_2Efinite__subsets__bijection,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0A)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,arr(A_27a,bool)),V0A),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f1860(A_27a,V0A)))) ) )))).
fof(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ_27,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)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V0s)) ) => ( ( ap(V1f,V2x) = ap(V1f,V3y) ) <=> ( V2x = V3y ) ) ) ) ) ) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),V0s))) <=> p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2Ecountable__decomposition,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) => ? [V1A] : ( mem(V1A,arr(arr(A_27a,bool),bool)) & ( ( ap(c_2Epred__set_2EBIGUNION(A_27a),V1A) = V0s ) & ! [V2a] : ( mem(V2a,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2a),V1A)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V2a)) ) & p(ap(c_2Epred__set_2Ecountable(A_27a),V2a)) ) ) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2Edisjoint__countable__decomposition,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) => ? [V1A] : ( mem(V1A,arr(arr(A_27a,bool),bool)) & ( ( ap(c_2Epred__set_2EBIGUNION(A_27a),V1A) = V0s ) & ( ( ! [V2a] : ( mem(V2a,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2a),V1A)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V2a)) ) & p(ap(c_2Epred__set_2Ecountable(A_27a),V2a)) ) ) ) ) & ! [V3a1] : ( mem(V3a1,arr(A_27a,bool)) => ! [V4a2] : ( mem(V4a2,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3a1),V1A)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4a2),V1A)) & ( ~~ ( V3a1 = V4a2 ) ) ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V3a1),V4a2)) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2Ecount__cardle,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1A] : ( mem(V1A,arr(A_27a,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Enum_2Enum,A_27a),ap(c_2Epred__set_2Ecount,V0n)),V1A)) <=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1A)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V0n),ap(c_2Epred__set_2ECARD(A_27a),V1A))) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ECANTOR,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(arr(A_27a,bool),A_27a),ap(c_2Epred__set_2EPOW(A_27a),V0A)),V0A)) ) )))).
fof(conj_thm_2Ecardinal_2Ecardlt__cardle,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0B] : ( mem(V0B,arr(A_27b,bool)) => ! [V1A] : ( mem(V1A,arr(A_27a,bool)) => ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V0B),V1A)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V1A),V0B)) ) ) ))))).
fof(conj_thm_2Ecardinal_2Eset__exp__product,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0A] : ( mem(V0A,arr(A_27c,bool)) => ! [V1B1] : ( mem(V1B1,arr(A_27b,bool)) => ! [V2B2] : ( mem(V2B2,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(arr(A_27a,ty_2Eoption_2Eoption(arr(A_27b,ty_2Eoption_2Eoption(A_27c)))),arr(ty_2Epair_2Eprod(A_27b,A_27a),ty_2Eoption_2Eoption(A_27c))),ap(ap(c_2Ecardinal_2Eset__exp(A_27a,arr(A_27b,ty_2Eoption_2Eoption(A_27c))),ap(ap(c_2Ecardinal_2Eset__exp(A_27b,A_27c),V0A),V1B1)),V2B2)),ap(ap(c_2Ecardinal_2Eset__exp(ty_2Epair_2Eprod(A_27b,A_27a),A_27c),V0A),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27a),V1B1),V2B2)))) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECOUNT__EQ__EMPTY,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ( ap(c_2Epred__set_2Ecount,V0n) = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ) <=> ( V0n = c_2Enum_2E0 ) ) ))).
fof(conj_thm_2Ecardinal_2EPOW__EQ__X__EXP__X,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0A)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(arr(A_27a,bool),arr(A_27a,ty_2Eoption_2Eoption(A_27a))),ap(c_2Epred__set_2EPOW(A_27a),V0A)),ap(ap(c_2Ecardinal_2Eset__exp(A_27a,A_27a),V0A),V0A))) ) )))).
fof(ax_thm_2Ecardinal_2Ebijns__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ( ap(c_2Ecardinal_2Ebijns(A_27a),V0A) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,ty_2Eoption_2Eoption(A_27a)),arr(A_27a,ty_2Eoption_2Eoption(A_27a))),f1863(A_27a,V0A)) ) ) ) )).
fof(conj_thm_2Ecardinal_2Ecardeq__bijns__cong,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)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0A),V1B)) => p(ap(ap(c_2Ecardinal_2Ecardeq(arr(A_27a,ty_2Eoption_2Eoption(A_27a)),arr(A_27b,ty_2Eoption_2Eoption(A_27b))),ap(c_2Ecardinal_2Ebijns(A_27a),V0A)),ap(c_2Ecardinal_2Ebijns(A_27b),V1B))) ) ) ))))).
fof(conj_thm_2Ecardinal_2Ebijections__cardeq,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(c_2Ecardinal_2Ecardeq(arr(A_27a,ty_2Eoption_2Eoption(A_27a)),arr(A_27a,bool)),ap(c_2Ecardinal_2Ebijns(A_27a),V0s)),ap(c_2Epred__set_2EPOW(A_27a),V0s))) ) )))).
fof(conj_thm_2Ecardinal_2ELEFT__IMP__EXISTS__THM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,bool) => ( ( ( ? [V2x] : ( mem(V2x,A_27a) & p(ap(V0P,V2x)) ) ) => p(V1Q) ) <=> ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(V0P,V3x)) => p(V1Q) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ELEFT__IMP__FORALL__THM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,bool) => ( ( ( ! [V2x] : ( mem(V2x,A_27a) => p(ap(V0P,V2x)) ) ) => p(V1Q) ) <=> ? [V3x] : ( mem(V3x,A_27a) & ( p(ap(V0P,V3x)) => p(V1Q) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ERIGHT__IMP__EXISTS__THM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,bool) => ! [V1Q] : ( mem(V1Q,arr(A_27a,bool)) => ( ( p(V0P) => ? [V2x] : ( mem(V2x,A_27a) & p(ap(V1Q,V2x)) ) ) <=> ? [V3x] : ( mem(V3x,A_27a) & ( p(V0P) => p(ap(V1Q,V3x)) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ERIGHT__IMP__FORALL__THM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,bool) => ! [V1Q] : ( mem(V1Q,arr(A_27a,bool)) => ( ( p(V0P) => ! [V2x] : ( mem(V2x,A_27a) => p(ap(V1Q,V2x)) ) ) <=> ! [V3x] : ( mem(V3x,A_27a) => ( p(V0P) => p(ap(V1Q,V3x)) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2EFINITE__FINITE__BIGUNIONS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(arr(A_27a,bool),bool)) => ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V0s)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s))) <=> ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1t),V0s)) => p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ECONJ__EQ__IMP,lemma,(! [V0r] : ( mem(V0r,bool) => ! [V1p] : ( mem(V1p,bool) => ! [V2q] : ( mem(V2q,bool) => ( ( ( p(V1p) & p(V2q) ) => p(V0r) ) <=> ( p(V1p) => ( p(V2q) => p(V0r) ) ) ) ) ) ))).
fof(conj_thm_2Ecardinal_2EIMP__CONJ__ALT,lemma,(! [V0r] : ( mem(V0r,bool) => ! [V1p] : ( mem(V1p,bool) => ! [V2q] : ( mem(V2q,bool) => ( ( ( p(V1p) & p(V2q) ) => p(V0r) ) <=> ( p(V2q) => ( p(V1p) => p(V0r) ) ) ) ) ) ))).
fof(conj_thm_2Ecardinal_2ELT__SUC__LE,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V0m),ap(c_2Enum_2ESUC,V1n))) <=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) ) ) ))).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__ON__RIGHT__INVERSE,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)) => ( ( ! [V3y] : ( mem(V3y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V2t)) => ? [V4x] : ( mem(V4x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s)) & ( ap(V1f,V4x) = V3y ) ) ) ) ) ) <=> ? [V5g] : ( mem(V5g,arr(A_27b,A_27a)) & ! [V6y] : ( mem(V6y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V6y),V2t)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V5g,V6y)),V0s)) & ( ap(V1f,ap(V5g,V6y)) = V6y ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__RIGHT__INVERSE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27b,A_27a)) => ( ( ! [V1y] : ( mem(V1y,A_27a) => ? [V2x] : ( mem(V2x,A_27b) & ( ap(V0f,V2x) = V1y ) ) ) ) <=> ? [V3g] : ( mem(V3g,arr(A_27a,A_27b)) & ! [V4y] : ( mem(V4y,A_27a) => ( ap(V0f,ap(V3g,V4y)) = V4y ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ__GENERAL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1A] : ( mem(V1A,arr(A_27b,bool)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ( ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s)) & ( ap(V0f,V3x) = ap(V0f,V4y) ) ) ) => ( V3x = V4y ) ) ) ) ) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1A)) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f1864(A_27b,A_27a,V2s,V0f,V1A)))) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1A] : ( mem(V1A,arr(A_27b,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_27b),V1A)) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f1865(A_27b,A_27a,V1A,V0f)))) ) ) ))))).
fof(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ__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)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) & ( ap(V0f,V2x) = ap(V0f,V3y) ) ) ) => ( V2x = V3y ) ) ) ) ) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s))) <=> p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EINFINITE__IMAGE__INJ,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_2Ecardinal_2EINFINITE__NONEMPTY,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) ) ) ) )))).
fof(conj_thm_2Ecardinal_2EFINITE__PRODUCT__DEPENDENT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,A_27c))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,arr(A_27b,bool))) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(c_2Epred__set_2EFINITE(A_27b),ap(V2t,V3x))) ) ) ) => p(ap(c_2Epred__set_2EFINITE(A_27c),ap(c_2Epred__set_2EGSPEC(A_27c,ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Epair_2Eprod(A_27c,bool)),f1867(A_27c,A_27b,A_27a,V1s,V2t,V0f))))) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2EFINITE__PRODUCT,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_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) => p(ap(c_2Epred__set_2EFINITE(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27a,A_27b),bool)),f1869(A_27a,A_27b,V1t,V0s))))) ) ) ))))).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__IMAGE__THM,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ( ( ! [V1y] : ( mem(V1y,A_27b) => ? [V2x] : ( mem(V2x,A_27a) & ( ap(V0f,V2x) = V1y ) ) ) ) <=> ! [V3P] : ( mem(V3P,arr(A_27b,bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f1870(A_27b,A_27a,V3P,V0f))) = ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f1871(A_27b,V3P)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__ON__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ! [V2v] : ( mem(V2v,arr(A_27b,bool)) => ( ( ! [V3t] : ( mem(V3t,arr(A_27b,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V3t),V2v)) => ? [V4s] : ( mem(V4s,arr(A_27a,bool)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V4s),V1u)) & ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V4s) = V3t ) ) ) ) ) ) <=> ! [V5y] : ( mem(V5y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5y),V2v)) => ? [V6x] : ( mem(V6x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V1u)) & ( ap(V0f,V6x) = V5y ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__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)) & ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s) = V1t ) ) ) ) <=> ! [V3y] : ( mem(V3y,A_27b) => ? [V4x] : ( mem(V4x,A_27a) & ( ap(V0f,V4x) = V3y ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__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)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27b),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)),V1t)) & ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V0s)) & ( ap(V2f,V3x) = ap(V2f,V4y) ) ) ) => ( V3x = V4y ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__IMAGE__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)) => ( ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) & ( 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_2Ecardinal_2ECARD__IMAGE__LE,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_2Ecardinal_2ESURJECTIVE__IFF__INJECTIVE__GEN,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(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( 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) ) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f),V0s)),V1t)) ) ) ) => ( ( ! [V3y] : ( mem(V3y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V1t)) => ? [V4x] : ( mem(V4x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s)) & ( ap(V2f,V4x) = V3y ) ) ) ) ) ) <=> ! [V5x] : ( mem(V5x,A_27a) => ! [V6y] : ( mem(V6y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6y),V0s)) & ( ap(V2f,V5x) = ap(V2f,V6y) ) ) ) => ( V5x = V6y ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__IFF__INJECTIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27a)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),V0s)),V0s)) ) => ( ( ! [V2y] : ( mem(V2y,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),V0s)) => ? [V3x] : ( mem(V3x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) & ( ap(V1f,V3x) = V2y ) ) ) ) ) ) <=> ! [V4x] : ( mem(V4x,A_27a) => ! [V5y] : ( mem(V5y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V0s)) & ( ap(V1f,V4x) = ap(V1f,V5y) ) ) ) => ( V4x = V5y ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__BIJECTION,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_2EFINITE(A_27a),V0s)) & ( 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) ) ) ) => ? [V2f] : ( mem(V2f,arr(A_27a,A_27b)) & ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V2f,V3x)),V1t)) ) ) ) & ( ( ! [V4y] : ( mem(V4y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V1t)) => ? [V5x] : ( mem(V5x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V0s)) & ( ap(V2f,V5x) = V4y ) ) ) ) ) ) & ! [V6x] : ( mem(V6x,A_27a) => ! [V7y] : ( mem(V7y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7y),V0s)) & ( ap(V2f,V6x) = ap(V2f,V7y) ) ) ) => ( V6x = V7y ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__BIJECTIONS,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_2EFINITE(A_27a),V0s)) & ( 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) ) ) ) => ? [V2f] : ( mem(V2f,arr(A_27a,A_27b)) & ? [V3g] : ( mem(V3g,arr(A_27b,A_27a)) & ( ( ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s)) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V2f,V4x)),V1t)) & ( ap(V3g,ap(V2f,V4x)) = V4x ) ) ) ) ) & ! [V5y] : ( mem(V5y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5y),V1t)) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V3g,V5y)),V0s)) & ( ap(V2f,ap(V3g,V5y)) = V5y ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ESING__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),V0s)) <=> p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) ) ) )))).
fof(conj_thm_2Ecardinal_2EINJECTIVE__ON__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ( ( ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2s),V1u)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3t),V1u)) & ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s) = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V3t) ) ) ) => ( V2s = V3t ) ) ) ) ) <=> ! [V4x] : ( mem(V4x,A_27a) => ! [V5y] : ( mem(V5y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1u)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V1u)) & ( ap(V0f,V4x) = ap(V0f,V5y) ) ) ) => ( V4x = V5y ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EINJECTIVE__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_27a,bool)) => ( ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s) = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2t) ) => ( V1s = V2t ) ) ) ) ) <=> ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( ap(V0f,V3x) = ap(V0f,V4y) ) => ( V3x = V4y ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EFINITE__FINITE__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(arr(A_27a,bool),bool)) => ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V0s)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s))) <=> ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1t),V0s)) => p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2Enum__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s)) <=> ? [V1a] : ( mem(V1a,ty_2Enum_2Enum) & ! [V2x] : ( mem(V2x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2x),V0s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V2x),V1a)) ) ) ) ) ))).
fof(conj_thm_2Ecardinal_2Enum__FINITE__AVOID,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s)) => ? [V1a] : ( mem(V1a,ty_2Enum_2Enum) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1a),V0s)) ) ) ) ))).
fof(conj_thm_2Ecardinal_2Enum__INFINITE,lemma,(( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ))).
fof(ax_thm_2Ecardinal_2EHAS__SIZE,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),V1n)) <=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & ( ap(c_2Epred__set_2ECARD(A_27a),V0s) = V1n ) ) ) ) ) ) )).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__CARD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),V1n)) => ( ap(c_2Epred__set_2ECARD(A_27a),V0s) = V1n ) ) ) )))).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__0,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),c_2Enum_2E0)) <=> ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) )))).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__SUC,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),ap(c_2Enum_2ESUC,V1n))) <=> ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ! [V2a] : ( mem(V2a,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V0s)) => p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V2a)),V1n)) ) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2EFINITE__HAS__SIZE,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(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) )))).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__CLAUSES,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V1s),c_2Enum_2E0)) <=> ( V1s = c_2Epred__set_2EEMPTY(A_27a) ) ) & ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V1s),ap(c_2Enum_2ESUC,V0n))) <=> ? [V2a] : ( mem(V2a,A_27a) & ? [V3t] : ( mem(V3t,arr(A_27a,bool)) & ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V3t),V0n)) & ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V3t)) ) & ( V1s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V2a),V3t) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__SUBSET__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,arr(A_27a,bool)) => ! [V1b] : ( mem(V1b,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1b)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0a),V1b)) & ( ap(c_2Epred__set_2ECARD(A_27a),V0a) = ap(c_2Epred__set_2ECARD(A_27a),V1b) ) ) ) => ( V0a = V1b ) ) ) )))).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__INDEX,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),V1n)) => ? [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,A_27a)) & ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V3m),V1n)) => p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V2f,V3m)),V0s)) ) ) ) & ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s)) => p(ap(c_2Ebool_2E_3F_21(ty_2Enum_2Enum),f1872(A_27a,V4x,V2f,V1n))) ) ) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__BIGUNION__LE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,arr(A_27b,bool))) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),V2m)) & ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s)) => ( p(ap(c_2Epred__set_2EFINITE(A_27b),ap(V1t,V4x))) & p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27b),ap(V1t,V4x))),V3n)) ) ) ) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27b),ap(c_2Epred__set_2EBIGUNION(A_27b),ap(c_2Epred__set_2EGSPEC(arr(A_27b,bool),A_27a),f1873(A_27b,A_27a,V1t,V0s))))),ap(ap(c_2Earithmetic_2E_2A,V2m),V3n))) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__BOOL,lemma,(p(ap(ap(c_2Ecardinal_2EHAS__SIZE(bool),c_2Epred__set_2EUNIV(bool)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))))).
fof(conj_thm_2Ecardinal_2ECARD__BOOL,lemma,(( ap(c_2Epred__set_2ECARD(bool),c_2Epred__set_2EUNIV(bool)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)) ))).
fof(conj_thm_2Ecardinal_2EFINITE__BOOL,lemma,(p(ap(c_2Epred__set_2EFINITE(bool),c_2Epred__set_2EUNIV(bool))))).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__CART__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),c_2Epred__set_2EUNIV(A_27a)),V0m)) => p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),c_2Epred__set_2EUNIV(A_27a)),ap(ap(c_2Earithmetic_2EEXP,V0m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__CART__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a))) => ( ap(c_2Epred__set_2ECARD(A_27a),c_2Epred__set_2EUNIV(A_27a)) = ap(ap(c_2Earithmetic_2EEXP,ap(c_2Epred__set_2ECARD(A_27a),c_2Epred__set_2EUNIV(A_27a))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) )))).
fof(conj_thm_2Ecardinal_2EFINITE__CART__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a))) => p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a))) )))).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__NUMSEG__LT,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => p(ap(ap(c_2Ecardinal_2EHAS__SIZE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f275(V0n))),V0n)) ))).
fof(conj_thm_2Ecardinal_2EFINITE__NUMSEG__LT,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f275(V0n)))) ))).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__NUMSEG__LE,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => p(ap(ap(c_2Ecardinal_2EHAS__SIZE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f1874(V0n))),ap(ap(c_2Earithmetic_2E_2B,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ))).
fof(conj_thm_2Ecardinal_2EFINITE__NUMSEG__LE,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f1874(V0n)))) ))).
fof(conj_thm_2Ecardinal_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)) ) => ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t))) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ELE__CASES,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) | p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V0m)) ) ) ))).
fof(conj_thm_2Ecardinal_2ELT__CASES,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n)) | ( p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V0m)) | ( V0m = V1n ) ) ) ) ))).
fof(conj_thm_2Ecardinal_2ELT,lemma,(( ( ! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V0m),c_2Enum_2E0)) <=> $false ) ) ) & ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V1m),ap(c_2Enum_2ESUC,V2n))) <=> ( ( V1m = V2n ) | p(ap(ap(c_2Eprim__rec_2E_3C,V1m),V2n)) ) ) ) ) ))).
fof(conj_thm_2Ecardinal_2ELT__LE,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n)) <=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) & ( ~~ ( V0m = V1n ) ) ) ) ) ))).
fof(conj_thm_2Ecardinal_2EGE,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V1m),V0n)) <=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0n),V1m)) ) ) ))).
fof(conj_thm_2Ecardinal_2ELE__SUC__LT,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Enum_2ESUC,V0m)),V1n)) <=> p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n)) ) ) ))).
fof(conj_thm_2Ecardinal_2EINJECTIVE__ON__LEFT__INVERSE,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) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) & ( ap(V0f,V2x) = ap(V0f,V3y) ) ) ) => ( V2x = V3y ) ) ) ) ) <=> ? [V4g] : ( mem(V4g,arr(A_27b,A_27a)) & ! [V5x] : ( mem(V5x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V1s)) => ( ap(V4g,ap(V0f,V5x)) = V5x ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EINJECTIVE__LEFT__INVERSE,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 ) ) ) ) ) <=> ? [V3g] : ( mem(V3g,arr(A_27b,A_27a)) & ! [V4x] : ( mem(V4x,A_27a) => ( ap(V3g,ap(V0f,V4x)) = V4x ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EINTER__ACI,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0r] : ( mem(V0r,arr(A_27a,bool)) => ! [V1p] : ( mem(V1p,arr(A_27a,bool)) => ! [V2q] : ( mem(V2q,arr(A_27a,bool)) => ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),V1p),V2q) = ap(ap(c_2Epred__set_2EINTER(A_27a),V2q),V1p) ) & ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V1p),V2q)),V0r) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V1p),V2q)),V0r) ) & ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V1p),V2q)),V0r) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V2q),V1p)),V0r) ) & ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),V1p),V1p) = V1p ) & ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V1p),V1p)),V2q) = ap(ap(c_2Epred__set_2EINTER(A_27a),V1p),V2q) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ECONJ__ACI,lemma,(! [V0r] : ( mem(V0r,bool) => ! [V1p] : ( mem(V1p,bool) => ! [V2q] : ( mem(V2q,bool) => ( ( ( p(V1p) & p(V2q) ) <=> ( p(V2q) & p(V1p) ) ) & ( ( ( ( p(V1p) & p(V2q) ) & p(V0r) ) <=> ( p(V1p) & ( p(V2q) & p(V0r) ) ) ) & ( ( ( p(V1p) & ( p(V2q) & p(V0r) ) ) <=> ( p(V2q) & ( p(V1p) & p(V0r) ) ) ) & ( ( ( p(V1p) & p(V1p) ) <=> p(V1p) ) & ( ( p(V1p) & ( p(V1p) & p(V2q) ) ) <=> ( p(V1p) & p(V2q) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ecardinal_2EUNION__ACI,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0r] : ( mem(V0r,arr(A_27a,bool)) => ! [V1p] : ( mem(V1p,arr(A_27a,bool)) => ! [V2q] : ( mem(V2q,arr(A_27a,bool)) => ( ( ap(ap(c_2Epred__set_2EUNION(A_27a),V1p),V2q) = ap(ap(c_2Epred__set_2EUNION(A_27a),V2q),V1p) ) & ( ( ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1p),V2q)),V0r) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1p),V2q)),V0r) ) & ( ( ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1p),V2q)),V0r) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V2q),V1p)),V0r) ) & ( ( ap(ap(c_2Epred__set_2EUNION(A_27a),V1p),V1p) = V1p ) & ( ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1p),V1p)),V2q) = ap(ap(c_2Epred__set_2EUNION(A_27a),V1p),V2q) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2ELT__NZ,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0n)) <=> ( ~~ ( V0n = c_2Enum_2E0 ) ) ) ))).
fof(conj_thm_2Ecardinal_2ELE__1,lemma,(( ( ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ( ~~ ( V0n = c_2Enum_2E0 ) ) => p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0n)) ) ) ) & ( ( ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ( ~~ ( V1n = c_2Enum_2E0 ) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V1n)) ) ) ) & ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V2n)) => ( ~~ ( V2n = c_2Enum_2E0 ) ) ) ) ) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V3n)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V3n)) ) ) ) & ( ( ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V4n)) => p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V4n)) ) ) ) & ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V5n)) => ( ~~ ( V5n = c_2Enum_2E0 ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ecardinal_2EOR__EXISTS__THM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,bool)) => ( ( ( ? [V2x] : ( mem(V2x,A_27a) & p(ap(V0P,V2x)) ) ) | ? [V3x] : ( mem(V3x,A_27a) & p(ap(V1Q,V3x)) ) ) <=> ? [V4x] : ( mem(V4x,A_27a) & ( p(ap(V0P,V4x)) | p(ap(V1Q,V4x)) ) ) ) ) )))).
fof(conj_thm_2Ecardinal_2EBIJECTIVE__INJECTIVE__SURJECTIVE,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)) => ( ( ( ! [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)) ) ) ) & ! [V4y] : ( mem(V4y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V2t)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f1875(A_27b,A_27a,V4y,V0f,V1s))) ) ) ) <=> ( ( ! [V6x] : ( mem(V6x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V1s)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V6x)),V2t)) ) ) ) & ( ( ! [V7x] : ( mem(V7x,A_27a) => ! [V8y] : ( mem(V8y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8y),V1s)) & ( ap(V0f,V7x) = ap(V0f,V8y) ) ) ) => ( V7x = V8y ) ) ) ) ) & ! [V9y] : ( mem(V9y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V9y),V2t)) => ? [V10x] : ( mem(V10x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V10x),V1s)) & ( ap(V0f,V10x) = V9y ) ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EBIJECTIVE__INVERSES,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)) => ( ( ( ! [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)) ) ) ) & ! [V4y] : ( mem(V4y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V2t)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f1875(A_27b,A_27a,V4y,V0f,V1s))) ) ) ) <=> ( ( ! [V6x] : ( mem(V6x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V1s)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V6x)),V2t)) ) ) ) & ? [V7g] : ( mem(V7g,arr(A_27b,A_27a)) & ( ( ! [V8y] : ( mem(V8y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V8y),V2t)) => p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V7g,V8y)),V1s)) ) ) ) & ( ( ! [V9y] : ( mem(V9y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V9y),V2t)) => ( ap(V0f,ap(V7g,V9y)) = V9y ) ) ) ) & ! [V10x] : ( mem(V10x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V10x),V1s)) => ( ap(V7g,ap(V0f,V10x)) = V10x ) ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2Ele__c,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) <=> ? [V2f] : ( mem(V2f,arr(A_27a,A_27b)) & ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V2f,V3x)),V1t)) ) ) ) & ! [V4x] : ( mem(V4x,A_27a) => ! [V5y] : ( mem(V5y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V0s)) & ( ap(V2f,V4x) = ap(V2f,V5y) ) ) ) => ( V4x = V5y ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2Elt__c,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(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) <=> ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) & ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2Eeq__c,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(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) <=> ? [V2f] : ( mem(V2f,arr(A_27a,A_27b)) & ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V2f,V3x)),V1t)) ) ) ) & ! [V4y] : ( mem(V4y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V1t)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f1876(A_27b,A_27a,V4y,V2f,V0s))) ) ) ) ) ) ) ))))).
fof(ax_thm_2Ecardinal_2Ecardgeq__def,axiom,( ! [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(ap(c_2Ecardinal_2Ecardgeq(A_27a,A_27b),V0s),V1t)) <=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) ) ) ) ) )).
fof(conj_thm_2Ecardinal_2Ege__c,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(ap(c_2Ecardinal_2Ecardgeq(A_27a,A_27b),V0s),V1t)) <=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) ) ))))).
fof(ax_thm_2Ecardinal_2Ecardgt__def,axiom,( ! [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(ap(c_2Ecardinal_2Ecardgt(A_27a,A_27b),V0s),V1t)) <=> ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) ) ) ) ) )).
fof(conj_thm_2Ecardinal_2Egt__c,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(ap(c_2Ecardinal_2Ecardgt(A_27a,A_27b),V0s),V1t)) <=> ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ELE__C,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) <=> ? [V2g] : ( mem(V2g,arr(A_27b,A_27a)) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) => ? [V4y] : ( mem(V4y,A_27b) & ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V1t)) & ( ap(V2g,V4y) = V3x ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EGE__C,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(ap(c_2Ecardinal_2Ecardgeq(A_27a,A_27b),V0s),V1t)) <=> ? [V2f] : ( mem(V2f,arr(A_27a,A_27b)) & ! [V3y] : ( mem(V3y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V1t)) => ? [V4x] : ( mem(V4x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s)) & ( V3y = ap(V2f,V4x) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECOUNTABLE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0t] : ( mem(V0t,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0t)) <=> p(ap(ap(c_2Ecardinal_2Ecardgeq(ty_2Enum_2Enum,A_27a),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0t)) ) )))).
fof(conj_thm_2Ecardinal_2EEQ__C,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(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) <=> ? [V2R] : ( mem(V2R,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool)) & ( ( ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27b) => ( p(ap(V2R,ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V3x),V4y))) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)) & p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V1t)) ) ) ) ) ) & ( ( ! [V5x] : ( mem(V5x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V0s)) => p(ap(c_2Ebool_2E_3F_21(A_27b),f1877(A_27a,A_27b,V2R,V5x,V1t))) ) ) ) & ! [V7y] : ( mem(V7y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V7y),V1t)) => p(ap(c_2Ebool_2E_3F_21(A_27a),f1878(A_27b,A_27a,V2R,V7y,V0s))) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V0s)) )))).
fof(conj_thm_2Ecardinal_2ECARD__LE__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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27c),V1t),V2u)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27c),V0s),V2u)) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__LT__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ~~ ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V0s)) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__LET__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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) & ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27b),V2u),V1t)) ) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27a),V2u),V0s)) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__LTE__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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27c),V1t),V2u)) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27a),V2u),V0s)) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__LT__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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) & ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27b),V2u),V1t)) ) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27a),V2u),V0s)) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27a),V0s),V0s)) )))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__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)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) <=> p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27a),V1t),V0s)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__IMP__LE,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(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LT__IMP__LE,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(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__RELATIONAL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0R] : ( mem(V0R,arr(A_27a,arr(A_27b,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27b) => ! [V4y_27] : ( mem(V4y_27,A_27b) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) & ( p(ap(ap(V0R,V2x),V3y)) & p(ap(ap(V0R,V2x),V4y_27)) ) ) => ( V3y = V4y_27 ) ) ) ) ) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f1880(A_27a,A_27b,V0R,V1s))),V1s)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),c_2Epred__set_2EEMPTY(A_27b))) <=> ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),c_2Epred__set_2EEMPTY(A_27b))) <=> ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__ANTISYM,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) <=> p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__TOTAL,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) | p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LET__TOTAL,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) | ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LTE__TOTAL,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(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) | p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LT__TOTAL,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(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) | ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) | ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__NOT__LE,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) <=> ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__NOT__LT,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(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) ) <=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LT__LE,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(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) <=> ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) & ( ~~ p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__LT,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) <=> ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) | p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__CONG,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1s_27] : ( mem(V1s_27,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1s_27)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27c,A_27d),V2t),V3t_27)) ) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27c),V0s),V2t)) <=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27d),V1s_27),V3t_27)) ) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2ECARD__LT__CONG,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1s_27] : ( mem(V1s_27,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1s_27)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27c,A_27d),V2t),V3t_27)) ) => ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27a),V2t),V0s)) ) <=> ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27d,A_27b),V3t_27),V1s_27)) ) ) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27c),V1t),V2u)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27c),V0s),V2u)) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__CONG,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1s_27] : ( mem(V1s_27,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1s_27)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27c,A_27d),V2t),V3t_27)) ) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27c),V0s),V2t)) <=> p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27d),V1s_27),V3t_27)) ) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2EINFINITE__CARD__LE,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(c_2Ecardinal_2Ecardleq(ty_2Enum_2Enum,A_27a),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) )))).
fof(conj_thm_2Ecardinal_2EFINITE__CARD__LT,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(c_2Ecardinal_2Ecardleq(ty_2Enum_2Enum,A_27a),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__LE__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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V1t)) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__LE__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),c_2Epred__set_2EUNIV(A_27a))) )))).
fof(conj_thm_2Ecardinal_2ECARD__LE__EQ__SUBSET,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(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) <=> ? [V2u] : ( mem(V2u,arr(A_27b,bool)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V2u),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V2u)) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__INFINITE__CONG,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(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) => ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) <=> ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__FINITE__CONG,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(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) <=> p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__FINITE,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_2EFINITE(A_27b),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) => p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__FINITE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27b,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27a),V0s),V1t)) ) => p(ap(c_2Epred__set_2EFINITE(A_27b),V0s)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__INFINITE,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_2EFINITE(A_27a),V0s)) ) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) => ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LT__FINITE__INFINITE,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_2EFINITE(A_27a),V0s)) & ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__CARD__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)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27b),V1t))) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__CARD__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)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) ) => ( ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(c_2Epred__set_2ECARD(A_27b),V1t) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__CARD,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_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) => ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) <=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27b),V1t))) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__CARD,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_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) <=> ( ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(c_2Epred__set_2ECARD(A_27b),V1t) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LT__CARD,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_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) => ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1t),V0s)) ) <=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27b),V1t))) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__HAS__SIZE__CONG,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)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),V2n)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t)) ) => p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27b),V1t),V2n)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__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(c_2Ecardinal_2Ecardleq(A_27b,A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s)),V1s)) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__IMAGE__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)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V2t),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s))) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V2t),V1s)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__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) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) & ( ap(V0f,V2x) = ap(V0f,V3y) ) ) ) => ( V2x = V3y ) ) ) ) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27b,A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s)),V1s)) ) ) ))))).
fof(ax_thm_2Ecardinal_2Eadd__c,axiom,( ! [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_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t) = ap(ap(c_2Epred__set_2EUNION(ty_2Esum_2Esum(A_27a,A_27b)),ap(c_2Epred__set_2EGSPEC(ty_2Esum_2Esum(A_27a,A_27b),A_27a),f1881(A_27b,A_27a,V0s))),ap(c_2Epred__set_2EGSPEC(ty_2Esum_2Esum(A_27a,A_27b),A_27b),f1882(A_27a,A_27b,V1t))) ) ) ) ) ) )).
fof(conj_thm_2Ecardinal_2Emul__c,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) = 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)),f1869(A_27a,A_27b,V1t,V0s))) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__ADD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1s_27] : ( mem(V1s_27,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1s_27)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27d),V2t),V3t_27)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Esum_2Esum(A_27a,A_27c),ty_2Esum_2Esum(A_27b,A_27d)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27c),V0s),V2t)),ap(ap(c_2Ecardinal_2E_2B__c(A_27b,A_27d),V1s_27),V3t_27))) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__MUL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1s_27] : ( mem(V1s_27,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1s_27)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27d),V2t),V3t_27)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27b,A_27d)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27c),V0s),V2t)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27d),V1s_27),V3t_27))) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2ECARD__ADD__CONG,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1s_27] : ( mem(V1s_27,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1s_27)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27c,A_27d),V2t),V3t_27)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Esum_2Esum(A_27a,A_27c),ty_2Esum_2Esum(A_27b,A_27d)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27c),V0s),V2t)),ap(ap(c_2Ecardinal_2E_2B__c(A_27b,A_27d),V1s_27),V3t_27))) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2ECARD__MUL__CONG,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1s_27] : ( mem(V1s_27,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(A_27d,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1s_27)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27c,A_27d),V2t),V3t_27)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27b,A_27d)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27c),V0s),V2t)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27d),V1s_27),V3t_27))) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2EIN__CARD__ADD,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)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Esum_2Esum(A_27a,A_27b)),ap(c_2Esum_2EINL(A_27a,A_27b),V2x)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t))) <=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) ) ) ) & ! [V3y] : ( mem(V3y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Esum_2Esum(A_27a,A_27b)),ap(c_2Esum_2EINR(A_27a,A_27b),V3y)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t))) <=> p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V1t)) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2EIN__CARD__MUL,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)) => ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2x),V3y)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t))) <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V1t)) ) ) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__SQUARE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,ty_2Epair_2Eprod(A_27a,A_27a)),V0s),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27a),V0s),V0s))) )))).
fof(conj_thm_2Ecardinal_2ECARD__SQUARE__NUM,lemma,(p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))).
fof(conj_thm_2Ecardinal_2EUNION__LE__ADD__C,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,ty_2Esum_2Esum(A_27a,A_27a)),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27a),V0s),V1t))) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__ADD__C,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_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) => ( ap(c_2Epred__set_2ECARD(ty_2Esum_2Esum(A_27a,A_27b)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27b),V1t)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__ADD__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)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Esum_2Esum(A_27a,A_27b),ty_2Esum_2Esum(A_27b,A_27a)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t)),ap(ap(c_2Ecardinal_2E_2B__c(A_27b,A_27a),V1t),V0s))) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__ADD__ASSOC,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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Esum_2Esum(A_27a,ty_2Esum_2Esum(A_27b,A_27c)),ty_2Esum_2Esum(ty_2Esum_2Esum(A_27a,A_27b),A_27c)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,ty_2Esum_2Esum(A_27b,A_27c)),V0s),ap(ap(c_2Ecardinal_2E_2B__c(A_27b,A_27c),V1t),V2u))),ap(ap(c_2Ecardinal_2E_2B__c(ty_2Esum_2Esum(A_27a,A_27b),A_27c),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t)),V2u))) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__MUL__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)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27a),V1t),V0s))) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__MUL__ASSOC,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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27a,A_27b),A_27c)),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V0s),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V1t),V2u))),ap(ap(c_2Epred__set_2ECROSS(ty_2Epair_2Eprod(A_27a,A_27b),A_27c),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t)),V2u))) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__LDISTRIB,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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(A_27a,ty_2Esum_2Esum(A_27b,A_27c)),ty_2Esum_2Esum(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,A_27c))),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Esum_2Esum(A_27b,A_27c)),V0s),ap(ap(c_2Ecardinal_2E_2B__c(A_27b,A_27c),V1t),V2u))),ap(ap(c_2Ecardinal_2E_2B__c(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,A_27c)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27c),V0s),V2u)))) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__RDISTRIB,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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(ty_2Esum_2Esum(A_27a,A_27b),A_27c),ty_2Esum_2Esum(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27b,A_27c))),ap(ap(c_2Epred__set_2ECROSS(ty_2Esum_2Esum(A_27a,A_27b),A_27c),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t)),V2u)),ap(ap(c_2Ecardinal_2E_2B__c(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27c),V0s),V2u)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V1t),V2u)))) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__ADDR,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(ap(c_2Ecardinal_2Ecardleq(A_27a,ty_2Esum_2Esum(A_27a,A_27b)),V0s),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t))) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__LE__ADDL,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(ap(c_2Ecardinal_2Ecardleq(A_27b,ty_2Esum_2Esum(A_27a,A_27b)),V1t),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t))) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__ADD__LE__MUL__INFINITE,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(c_2Ecardinal_2Ecardleq(ty_2Esum_2Esum(A_27a,A_27a),ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27a),V0s),V0s)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27a),V0s),V0s))) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__DISJOINT__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_2EINTER(A_27a),V0s),V1t) = c_2Epred__set_2EEMPTY(A_27a) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,ty_2Esum_2Esum(A_27a,A_27a)),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27a),V0s),V1t))) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__SQUARE__INFINITE,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(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(A_27a,A_27a),A_27a),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27a),V0s),V0s)),V0s)) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__ADD__FINITE,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_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) => p(ap(c_2Epred__set_2EFINITE(ty_2Esum_2Esum(A_27a,A_27b)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t))) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__ADD__FINITE__EQ,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_2EFINITE(ty_2Esum_2Esum(A_27a,A_27b)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t))) <=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__MUL__FINITE,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_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)) ) => p(ap(c_2Epred__set_2EFINITE(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t))) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__MUL__ABSORB__LE,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_2EFINITE(A_27b),V1t)) ) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Epair_2Eprod(A_27a,A_27b),A_27b),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t)),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__MUL2__ABSORB__LE,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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27c),V2u)) ) & ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27c),V0s),V2u)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27c),V1t),V2u)) ) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Epair_2Eprod(A_27a,A_27b),A_27c),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t)),V2u)) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__ADD__ABSORB__LE,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_2EFINITE(A_27b),V1t)) ) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Esum_2Esum(A_27a,A_27b),A_27b),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t)),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__ADD2__ABSORB__LE,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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27c),V2u)) ) & ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27c),V0s),V2u)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27c),V1t),V2u)) ) ) => p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Esum_2Esum(A_27a,A_27b),A_27c),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t)),V2u)) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__MUL__ABSORB,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_2EFINITE(A_27b),V1t)) ) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Epair_2Eprod(A_27a,A_27b),A_27b),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t)),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__ADD__ABSORB,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_2EFINITE(A_27b),V1t)) ) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Esum_2Esum(A_27a,A_27b),A_27b),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t)),V1t)) ) ) ))))).
fof(conj_thm_2Ecardinal_2ECARD__ADD2__ABSORB__LT,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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27c),V2u)) ) & ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27a),V2u),V0s)) ) & ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27b),V2u),V1t)) ) ) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,ty_2Esum_2Esum(A_27a,A_27b)),V2u),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27b),V0s),V1t))) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__LT__ADD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1s_27] : ( mem(V1s_27,arr(A_27b,bool)) => ! [V2t] : ( mem(V2t,arr(A_27c,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(A_27d,bool)) => ( ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,A_27a),V1s_27),V0s)) ) & ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27d,A_27c),V3t_27),V2t)) ) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Esum_2Esum(A_27b,A_27d),ty_2Esum_2Esum(A_27a,A_27c)),ap(ap(c_2Ecardinal_2E_2B__c(A_27b,A_27d),V1s_27),V3t_27)),ap(ap(c_2Ecardinal_2E_2B__c(A_27a,A_27c),V0s),V2t))) ) ) ) ) ) ))))))).
fof(conj_thm_2Ecardinal_2ECARD__MUL__LT__LEMMA,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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t)) & ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27b),V2u),V1t)) ) & ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27c),V2u)) ) ) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,ty_2Epair_2Eprod(A_27a,A_27b)),V2u),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t))) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECARD__MUL__LT__INFINITE,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_27b,bool)) => ! [V2u] : ( mem(V2u,arr(A_27c,bool)) => ( ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27a),V2u),V0s)) ) & ( ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,A_27b),V2u),V1t)) ) & ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27c),V2u)) ) ) ) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27c,ty_2Epair_2Eprod(A_27a,A_27b)),V2u),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t))) ) ) ) ) )))))).
fof(conj_thm_2Ecardinal_2ECANTOR__THM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(arr(A_27a,bool),A_27a),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f1883(A_27a,V0s))),V0s)) ) )))).
fof(conj_thm_2Ecardinal_2ECANTOR__THM__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ( ~~ p(ap(ap(c_2Ecardinal_2Ecardleq(arr(A_27a,bool),A_27a),c_2Epred__set_2EUNIV(arr(A_27a,bool))),c_2Epred__set_2EUNIV(A_27a))) )))).
fof(conj_thm_2Ecardinal_2ENUM__COUNTABLE,lemma,(p(ap(c_2Epred__set_2Ecountable(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))).
fof(conj_thm_2Ecardinal_2ECOUNTABLE__ALT__cardleq,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) <=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,ty_2Enum_2Enum),V0s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) )))).
fof(conj_thm_2Ecardinal_2ECOUNTABLE__CASES,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(c_2Ecardinal_2Ecardeq(A_27a,ty_2Enum_2Enum),V0s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__LE__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),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V1t)) ) => p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) ) ) )))).
fof(conj_thm_2Ecardinal_2ECARD__EQ__COUNTABLE,conjecture,( ! [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),V1t)) & p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27a),V0s),V1t)) ) => p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) ) ) )))).
