ITP001 Axioms: ITP150_5.ax
%------------------------------------------------------------------------------
% File : ITP150_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : probability_2.ax [Gau20]
% : HL4150_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 200 ( 25 unt; 48 typ; 0 def)
% Number of atoms : 6211 ( 135 equ)
% Maximal formula atoms : 18 ( 31 avg)
% Number of connectives : 663 ( 13 ~; 4 |; 157 &)
% ( 15 <=>; 474 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 10 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 5409 (5409 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 168 ( 48 >; 120 *; 0 +; 0 <<)
% Number of predicates : 92 ( 91 usr; 35 prp; 0-3 aty)
% Number of functors : 48 ( 48 usr; 0 con; 1-8 aty)
% Number of variables : 673 ( 673 !; 0 ?; 673 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Eprobability_2Econditional__distribution,type,
c_2Eprobability_2Econditional__distribution: ( del * del * del ) > $i ).
tff(mem_c_2Eprobability_2Econditional__distribution,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Eprobability_2Econditional__distribution(A_27a,A_27b,A_27c),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,A_27b),arr(arr(A_27a,A_27c),arr(arr(A_27b,bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal)))))) ).
tff(tp_c_2Eprobability_2Econditional__expectation,type,
c_2Eprobability_2Econditional__expectation: del > $i ).
tff(mem_c_2Eprobability_2Econditional__expectation,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Econditional__expectation(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(arr(arr(A_27a,bool),bool),arr(A_27a,ty_2Eextreal_2Eextreal))))) ).
tff(tp_c_2Eprobability_2Econditional__prob,type,
c_2Eprobability_2Econditional__prob: del > $i ).
tff(mem_c_2Eprobability_2Econditional__prob,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Econditional__prob(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),arr(A_27a,ty_2Eextreal_2Eextreal))))) ).
tff(tp_c_2Eprobability_2Edistribution,type,
c_2Eprobability_2Edistribution: ( del * del ) > $i ).
tff(mem_c_2Eprobability_2Edistribution,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eprobability_2Edistribution(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),arr(arr(A_27b,A_27a),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) ).
tff(tp_c_2Eprobability_2Eevents,type,
c_2Eprobability_2Eevents: del > $i ).
tff(mem_c_2Eprobability_2Eevents,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Eevents(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Eprobability_2Eexpectation,type,
c_2Eprobability_2Eexpectation: del > $i ).
tff(mem_c_2Eprobability_2Eexpectation,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Eexpectation(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,ty_2Eextreal_2Eextreal),ty_2Eextreal_2Eextreal))) ).
tff(tp_c_2Eprobability_2Eindep,type,
c_2Eprobability_2Eindep: del > $i ).
tff(mem_c_2Eprobability_2Eindep,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Eindep(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))) ).
tff(tp_c_2Eprobability_2Eindep__families,type,
c_2Eprobability_2Eindep__families: del > $i ).
tff(mem_c_2Eprobability_2Eindep__families,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Eindep__families(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(arr(A_27a,bool),bool),arr(arr(arr(A_27a,bool),bool),bool)))) ).
tff(tp_c_2Eprobability_2Eindep__function,type,
c_2Eprobability_2Eindep__function: ( del * del ) > $i ).
tff(mem_c_2Eprobability_2Eindep__function,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eprobability_2Eindep__function(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27a)),bool))) ).
tff(tp_c_2Eprobability_2Eindep__rv,type,
c_2Eprobability_2Eindep__rv: ( del * del * del ) > $i ).
tff(mem_c_2Eprobability_2Eindep__rv,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Eprobability_2Eindep__rv(A_27a,A_27b,A_27c),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,A_27b),arr(arr(A_27a,A_27c),arr(ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),arr(ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)),bool)))))) ).
tff(tp_c_2Eprobability_2Ejoint__distribution,type,
c_2Eprobability_2Ejoint__distribution: ( del * del * del ) > $i ).
tff(mem_c_2Eprobability_2Ejoint__distribution,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Eprobability_2Ejoint__distribution(A_27a,A_27b,A_27c),arr(ty_2Epair_2Eprod(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))),arr(arr(A_27c,A_27a),arr(arr(A_27c,A_27b),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Erealax_2Ereal))))) ).
tff(tp_c_2Eprobability_2Ejoint__distribution3,type,
c_2Eprobability_2Ejoint__distribution3: ( del * del * del * del ) > $i ).
tff(mem_c_2Eprobability_2Ejoint__distribution3,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : mem(c_2Eprobability_2Ejoint__distribution3(A_27a,A_27b,A_27c,A_27d),arr(ty_2Epair_2Eprod(arr(A_27d,bool),ty_2Epair_2Eprod(arr(arr(A_27d,bool),bool),arr(arr(A_27d,bool),ty_2Erealax_2Ereal))),arr(arr(A_27d,A_27a),arr(arr(A_27d,A_27b),arr(arr(A_27d,A_27c),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Erealax_2Ereal)))))) ).
tff(tp_c_2Eprobability_2Ep__space,type,
c_2Eprobability_2Ep__space: del > $i ).
tff(mem_c_2Eprobability_2Ep__space,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Ep__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(A_27a,bool))) ).
tff(tp_c_2Eprobability_2Epossibly,type,
c_2Eprobability_2Epossibly: del > $i ).
tff(mem_c_2Eprobability_2Epossibly,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Epossibly(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Eprobability_2Eprob,type,
c_2Eprobability_2Eprob: del > $i ).
tff(mem_c_2Eprobability_2Eprob,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Eprob(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))) ).
tff(tp_c_2Eprobability_2Eprob__space,type,
c_2Eprobability_2Eprob__space: del > $i ).
tff(mem_c_2Eprobability_2Eprob__space,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Eprob__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ).
tff(tp_c_2Eprobability_2Eprobably,type,
c_2Eprobability_2Eprobably: del > $i ).
tff(mem_c_2Eprobability_2Eprobably,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Eprobably(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Eprobability_2Erandom__variable,type,
c_2Eprobability_2Erandom__variable: ( del * del ) > $i ).
tff(mem_c_2Eprobability_2Erandom__variable,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eprobability_2Erandom__variable(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),bool)))) ).
tff(tp_c_2Eprobability_2Ereal__random__variable,type,
c_2Eprobability_2Ereal__random__variable: del > $i ).
tff(mem_c_2Eprobability_2Ereal__random__variable,axiom,
! [A_27a: del] : mem(c_2Eprobability_2Ereal__random__variable(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool))) ).
tff(tp_c_2Eprobability_2Erv__conditional__expectation,type,
c_2Eprobability_2Erv__conditional__expectation: ( del * del ) > $i ).
tff(mem_c_2Eprobability_2Erv__conditional__expectation,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eprobability_2Erv__conditional__expectation(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(arr(A_27a,A_27b),arr(A_27a,ty_2Eextreal_2Eextreal)))))) ).
tff(tp_c_2Eprobability_2Euniform__distribution,type,
c_2Eprobability_2Euniform__distribution: ( del * del * del * del ) > $i ).
tff(mem_c_2Eprobability_2Euniform__distribution,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : mem(c_2Eprobability_2Euniform__distribution(A_27a,A_27b,A_27c,A_27d),arr(A_27b,arr(A_27c,arr(ty_2Epair_2Eprod(arr(A_27d,bool),arr(arr(A_27d,bool),bool)),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))) ).
tff(ax_thm_2Eprobability_2Ep__space__def,axiom,
! [A_27a: del] : ( c_2Eprobability_2Ep__space(A_27a) = c_2Emeasure_2Em__space(A_27a) ) ).
tff(ax_thm_2Eprobability_2Eevents__def,axiom,
! [A_27a: del] : ( c_2Eprobability_2Eevents(A_27a) = c_2Emeasure_2Emeasurable__sets(A_27a) ) ).
tff(ax_thm_2Eprobability_2Eprob__def,axiom,
! [A_27a: del] : ( c_2Eprobability_2Eprob(A_27a) = c_2Emeasure_2Emeasure(A_27a) ) ).
tff(ax_thm_2Eprobability_2Eprob__space__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
<=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0p))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0p),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Eindep__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1a: $i] :
( mem(V1a,arr(A_27a,bool))
=> ! [V2b: $i] :
( mem(V2b,arr(A_27a,bool))
=> ( p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),V1a),V2b))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1a),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2b),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINTER(A_27a),V1a),V2b))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1a)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2b))) ) ) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Eindep__families__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1q: $i] :
( mem(V1q,arr(arr(A_27a,bool),bool))
=> ! [V2r: $i] :
( mem(V2r,arr(arr(A_27a,bool),bool))
=> ( p(ap(ap(ap(c_2Eprobability_2Eindep__families(A_27a),V0p),V1q),V2r))
<=> ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ! [V4t: $i] :
( mem(V4t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),V1q))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4t),V2r)) )
=> p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),V3s),V4t)) ) ) ) ) ) ) ) ).
tff(lamtp_f4121,type,
f4121: ( del * del * $i ) > $i ).
tff(lameq_f4121,axiom,
! [A_27b: del,A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] : ( ap(f4121(A_27b,A_27a,V0p),V1f) = ap(ap(c_2Epair_2E_2C(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27a)),bool),V1f),ap(ap(ap(c_2Eprobability_2Eindep__families(A_27a),V0p),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,ty_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2EFST(A_27b,A_27a)),V1f))),c_2Epred__set_2EUNIV(arr(A_27b,bool)))),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,A_27a,ty_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2ESND(A_27b,A_27a)),V1f))),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) ) ) ).
tff(ax_thm_2Eprobability_2Eindep__function__def,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ap(c_2Eprobability_2Eindep__function(A_27a,A_27b),V0p) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27a)),arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27a))),f4121(A_27b,A_27a,V0p)) ) ) ).
tff(ax_thm_2Eprobability_2Eprobably__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1e: $i] :
( mem(V1e,arr(A_27a,bool))
=> ( p(ap(ap(c_2Eprobability_2Eprobably(A_27a),V0p),V1e))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1e),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1e)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Epossibly__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1e: $i] :
( mem(V1e,arr(A_27a,bool))
=> ( p(ap(ap(c_2Eprobability_2Epossibly(A_27a),V0p),V1e))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1e),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1e)) != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Erandom__variable__def,axiom,
! [A_27a: del,A_27b: del,V0X: $i] :
( mem(V0X,arr(A_27a,A_27b))
=> ! [V1p: $i] :
( mem(V1p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2s: $i] :
( mem(V2s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ( p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V0X),V1p),V2s))
<=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V1p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0X),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Eprobability_2Ep__space(A_27a),V1p)),ap(c_2Eprobability_2Eevents(A_27a),V1p))),V2s))) ) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Ereal__random__variable__def,axiom,
! [A_27a: del,V0X: $i] :
( mem(V0X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1p: $i] :
( mem(V1p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(ap(c_2Eprobability_2Ereal__random__variable(A_27a),V0X),V1p))
<=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V1p))
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Eprobability_2Ep__space(A_27a),V1p)))
=> ( ( surj__ty_2Eextreal_2Eextreal(ap(V0X,V2x)) != fo__c_2Eextreal_2ENegInf )
& ( surj__ty_2Eextreal_2Eextreal(ap(V0X,V2x)) != fo__c_2Eextreal_2EPosInf ) ) ) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0X),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Eprobability_2Ep__space(A_27a),V1p)),ap(c_2Eprobability_2Eevents(A_27a),V1p))),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(lamtp_f4122,type,
f4122: ( del * del * $i * $i ) > $i ).
tff(lameq_f4122,axiom,
! [A_27b: del,A_27a: del,V1X: $i] :
( mem(V1X,arr(A_27b,A_27a))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2s: $i] : ( ap(f4122(A_27b,A_27a,V1X,V0p),V2s) = ap(ap(c_2Eprobability_2Eprob(A_27b),V0p),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27a),V1X),V2s)),ap(c_2Eprobability_2Ep__space(A_27b),V0p))) ) ) ) ).
tff(ax_thm_2Eprobability_2Edistribution__def,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27b,A_27a))
=> ( ap(ap(c_2Eprobability_2Edistribution(A_27a,A_27b),V0p),V1X) = f4122(A_27b,A_27a,V1X,V0p) ) ) ) ).
tff(lamtp_f4123,type,
f4123: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f4123,axiom,
! [A_27b: del,A_27a: del,A_27c: del,V1X: $i] :
( mem(V1X,arr(A_27c,A_27a))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27c,A_27b))
=> ! [V4x: $i] : ( ap(f4123(A_27b,A_27a,A_27c,V1X,V2Y),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),ap(V1X,V4x)),ap(V2Y,V4x)) ) ) ) ).
tff(lamtp_f4124,type,
f4124: ( del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f4124,axiom,
! [A_27c: del,A_27b: del,A_27a: del,V2Y: $i] :
( mem(V2Y,arr(A_27c,A_27b))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27c,A_27a))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))))
=> ! [V3a: $i] : ( ap(f4124(A_27c,A_27b,A_27a,V2Y,V1X,V0p),V3a) = ap(ap(c_2Eprobability_2Eprob(A_27c),V0p),ap(ap(c_2Epred__set_2EINTER(A_27c),ap(ap(c_2Epred__set_2EPREIMAGE(A_27c,ty_2Epair_2Eprod(A_27a,A_27b)),f4123(A_27b,A_27a,A_27c,V1X,V2Y)),V3a)),ap(c_2Eprobability_2Ep__space(A_27c),V0p))) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Ejoint__distribution__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27c,A_27a))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27c,A_27b))
=> ( ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27a,A_27b,A_27c),V0p),V1X),V2Y) = f4124(A_27c,A_27b,A_27a,V2Y,V1X,V0p) ) ) ) ) ).
tff(lamtp_f4125,type,
f4125: ( del * del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f4125,axiom,
! [A_27c: del,A_27b: del,A_27a: del,A_27d: del,V3Z: $i] :
( mem(V3Z,arr(A_27d,A_27c))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27d,A_27b))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27d,A_27a))
=> ! [V5x: $i] : ( ap(f4125(A_27c,A_27b,A_27a,A_27d,V3Z,V2Y,V1X),V5x) = ap(ap(c_2Epair_2E_2C(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),ap(V1X,V5x)),ap(ap(c_2Epair_2E_2C(A_27b,A_27c),ap(V2Y,V5x)),ap(V3Z,V5x))) ) ) ) ) ).
tff(lamtp_f4126,type,
f4126: ( del * del * del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f4126,axiom,
! [A_27d: del,A_27a: del,A_27b: del,A_27c: del,V1X: $i] :
( mem(V1X,arr(A_27d,A_27a))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27d,A_27b))
=> ! [V3Z: $i] :
( mem(V3Z,arr(A_27d,A_27c))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27d,bool),ty_2Epair_2Eprod(arr(arr(A_27d,bool),bool),arr(arr(A_27d,bool),ty_2Erealax_2Ereal))))
=> ! [V4a: $i] : ( ap(f4126(A_27d,A_27a,A_27b,A_27c,V1X,V2Y,V3Z,V0p),V4a) = ap(ap(c_2Eprobability_2Eprob(A_27d),V0p),ap(ap(c_2Epred__set_2EINTER(A_27d),ap(ap(c_2Epred__set_2EPREIMAGE(A_27d,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),f4125(A_27c,A_27b,A_27a,A_27d,V3Z,V2Y,V1X)),V4a)),ap(c_2Eprobability_2Ep__space(A_27d),V0p))) ) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Ejoint__distribution3__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27d,bool),ty_2Epair_2Eprod(arr(arr(A_27d,bool),bool),arr(arr(A_27d,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27d,A_27a))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27d,A_27b))
=> ! [V3Z: $i] :
( mem(V3Z,arr(A_27d,A_27c))
=> ( ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution3(A_27a,A_27b,A_27c,A_27d),V0p),V1X),V2Y),V3Z) = f4126(A_27d,A_27a,A_27b,A_27c,V1X,V2Y,V3Z,V0p) ) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Econditional__distribution__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(A_27b,bool))
=> ! [V4b: $i] :
( mem(V4b,arr(A_27c,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(ap(c_2Eprobability_2Econditional__distribution(A_27a,A_27b,A_27c),V0p),V1X),V2Y),V3a),V4b)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2E_2F,ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3a),V4b))),ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V4b))) ) ) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Eexpectation__def,axiom,
! [A_27a: del] : ( c_2Eprobability_2Eexpectation(A_27a) = c_2Elebesgue_2Eintegral(A_27a) ) ).
tff(lamtp_f4127,type,
f4127: ( del * $i * $i ) > $i ).
tff(lameq_f4127,axiom,
! [A_27a: del,V3f: $i] :
( mem(V3f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4g: $i] :
( mem(V4g,arr(A_27a,bool))
=> ! [V5x: $i] : ( ap(f4127(A_27a,V3f,V4g),V5x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V3f,V5x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V4g),V5x)) ) ) ) ).
tff(lamtp_f4128,type,
f4128: ( del * $i * $i ) > $i ).
tff(lameq_f4128,axiom,
! [A_27a: del,V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4g: $i] :
( mem(V4g,arr(A_27a,bool))
=> ! [V6x: $i] : ( ap(f4128(A_27a,V1X,V4g),V6x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1X,V6x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V4g),V6x)) ) ) ) ).
tff(lamtp_f4129,type,
f4129: ( del * $i * $i * $i * $i ) > $i ).
tff(lameq_f4129,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s: $i] :
( mem(V2s,arr(arr(A_27a,bool),bool))
=> ! [V4g: $i] : ( ap(f4129(A_27a,V0p,V1X,V3f,V2s),V4g) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4g),V2s)),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0p),f4127(A_27a,V3f,V4g))),ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0p),f4128(A_27a,V1X,V4g)))) ) ) ) ) ) ).
tff(lamtp_f4130,type,
f4130: ( del * $i * $i * $i ) > $i ).
tff(lameq_f4130,axiom,
! [A_27a: del,V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s: $i] :
( mem(V2s,arr(arr(A_27a,bool),bool))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3f: $i] : ( ap(f4130(A_27a,V1X,V2s,V0p),V3f) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eprobability_2Ereal__random__variable(A_27a),V3f),V0p)),ap(c_2Ebool_2E_21(arr(A_27a,bool)),f4129(A_27a,V0p,V1X,V3f,V2s))) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Econditional__expectation__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s: $i] :
( mem(V2s,arr(arr(A_27a,bool),bool))
=> ( ap(ap(ap(c_2Eprobability_2Econditional__expectation(A_27a),V0p),V1X),V2s) = ap(c_2Emin_2E_40(arr(A_27a,ty_2Eextreal_2Eextreal)),f4130(A_27a,V1X,V2s,V0p)) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Econditional__prob__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1e1: $i] :
( mem(V1e1,arr(A_27a,bool))
=> ! [V2e2: $i] :
( mem(V2e2,arr(arr(A_27a,bool),bool))
=> ( ap(ap(ap(c_2Eprobability_2Econditional__prob(A_27a),V0p),V1e1),V2e2) = ap(ap(ap(c_2Eprobability_2Econditional__expectation(A_27a),V0p),ap(c_2Emeasure_2Eindicator__fn(A_27a),V1e1)),V2e2) ) ) ) ) ).
tff(lamtp_f4131,type,
f4131: ( del * del * $i * $i ) > $i ).
tff(lameq_f4131,axiom,
! [A_27a: del,A_27b: del,V3Y: $i] :
( mem(V3Y,arr(A_27a,A_27b))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V4a: $i] : ( ap(f4131(A_27a,A_27b,V3Y,V0p),V4a) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V3Y),V4a)),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) ) ) ).
tff(ax_thm_2Eprobability_2Erv__conditional__expectation__def,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2X: $i] :
( mem(V2X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3Y: $i] :
( mem(V3Y,arr(A_27a,A_27b))
=> ( ap(ap(ap(ap(c_2Eprobability_2Erv__conditional__expectation(A_27a,A_27b),V0p),V1s),V2X),V3Y) = ap(ap(ap(c_2Eprobability_2Econditional__expectation(A_27a),V0p),V2X),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),f4131(A_27a,A_27b,V3Y,V0p)),ap(c_2Emeasure_2Esubsets(A_27b),V1s))) ) ) ) ) ) ).
tff(ax_thm_2Eprobability_2Eindep__rv__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3s: $i] :
( mem(V3s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V4t: $i] :
( mem(V4t,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)))
=> ( p(ap(ap(ap(ap(ap(c_2Eprobability_2Eindep__rv(A_27a,A_27b,A_27c),V0p),V1X),V2Y),V3s),V4t))
<=> ! [V5A: $i] :
( mem(V5A,arr(A_27b,bool))
=> ! [V6B: $i] :
( mem(V6B,arr(A_27c,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V5A),ap(c_2Emeasure_2Esubsets(A_27b),V3s)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V6B),ap(c_2Emeasure_2Esubsets(A_27c),V4t))) )
=> p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V1X),V5A)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27c),V2Y),V6B))) ) ) ) ) ) ) ) ) ) ).
tff(lamtp_f4132,type,
f4132: ( del * del * $i ) > $i ).
tff(lameq_f4132,axiom,
! [A_27d: del,A_27a: del,V2s: $i] :
( mem(V2s,ty_2Epair_2Eprod(arr(A_27d,bool),arr(arr(A_27d,bool),bool)))
=> ! [V3a: $i] : ( ap(f4132(A_27d,A_27a,V2s),V3a) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V3a))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27d),ap(c_2Emeasure_2Espace(A_27d),V2s)))) ) ) ).
tff(ax_thm_2Eprobability_2Euniform__distribution__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0p: $i] :
( mem(V0p,A_27b)
=> ! [V1X: $i] :
( mem(V1X,A_27c)
=> ! [V2s: $i] :
( mem(V2s,ty_2Epair_2Eprod(arr(A_27d,bool),arr(arr(A_27d,bool),bool)))
=> ( ap(ap(ap(c_2Eprobability_2Euniform__distribution(A_27a,A_27b,A_27c,A_27d),V0p),V1X),V2s) = f4132(A_27d,A_27a,V2s) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPOSITIVE__PROB,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Epositive(A_27a),V0p))
<=> ( ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),c_2Epred__set_2EEMPTY(A_27a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EINCREASING__PROB,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0p))
<=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t))) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EADDITIVE__PROB,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eadditive(A_27a),V0p))
<=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t))) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2ECOUNTABLY__ADDITIVE__PROB,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0p))
<=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Eprobability_2Eevents(A_27a),V0p))))
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n)))) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Eprobability_2Eprob(A_27a),V0p)),V1f)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__SPACE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),ap(c_2Eprobability_2Eevents(A_27a),V0p))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0p))
& p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0p))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__SIGMA__ALGEBRA,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__ALGEBRA,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__EMPTY,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),c_2Epred__set_2EEMPTY(A_27a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__UNIV,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__SPACE__POSITIVE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> p(ap(c_2Emeasure_2Epositive(A_27a),V0p)) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__SPACE__COUNTABLY__ADDITIVE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0p)) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__SPACE__ADDITIVE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> p(ap(c_2Emeasure_2Eadditive(A_27a),V0p)) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__SPACE__INCREASING,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> p(ap(c_2Emeasure_2Eincreasing(A_27a),V0p)) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__POSITIVE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__INCREASING,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__ADDITIVE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u: $i] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t))
& ( V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V3u)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t))) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__COUNTABLY__ADDITIVE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Eprobability_2Eevents(A_27a),V0p))))
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3m))),ap(V2f,inj__ty_2Enum_2Enum(V4n)))) )
& ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) )
=> p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Eprobability_2Eprob(A_27a),V0p)),V2f)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__COMPL,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__INDEP,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u: $i] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),V1s),V2t))
& ( V3u = ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V3u)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t))) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPSPACE,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,arr(A_27a,bool))
=> ! [V1b: $i] :
( mem(V1b,arr(arr(A_27a,bool),bool))
=> ! [V2c: $i] :
( mem(V2c,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ap(c_2Eprobability_2Ep__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0a),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1b),V2c))) = V0a ) ) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,arr(A_27a,bool))
=> ! [V1b: $i] :
( mem(V1b,arr(arr(A_27a,bool),bool))
=> ! [V2c: $i] :
( mem(V2c,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ap(c_2Eprobability_2Eevents(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0a),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1b),V2c))) = V1b ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,arr(A_27a,bool))
=> ! [V1b: $i] :
( mem(V1b,arr(arr(A_27a,bool),bool))
=> ! [V2c: $i] :
( mem(V2c,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ap(c_2Eprobability_2Eprob(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0a),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1b),V2c))) = V2c ) ) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__INTER,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__EMPTY,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__SPACE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__UNION,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EINDEP__EMPTY,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),c_2Epred__set_2EEMPTY(A_27a)),V1s)) ) ) ) ).
tff(conj_thm_2Eprobability_2EINTER__PSPACE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s) = V1s ) ) ) ) ).
tff(conj_thm_2Eprobability_2EINDEP__SPACE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s)) ) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__DIFF,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__COMPL,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s)),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__COUNTABLE__UNION,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1c: $i] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__ZERO__UNION,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__EQ__COMPL,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V2t))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__ONE__INTER,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EEVENTS__COUNTABLE__INTER,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1c: $i] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c))
& ( V1c != c_2Epred__set_2EEMPTY(arr(A_27a,bool)) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),V1c)),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ) ).
tff(conj_thm_2Eprobability_2EABS__PROB,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Eabs,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__COMPL__LE1,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2r: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s))),inj__ty_2Erealax_2Ereal(V2r)))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V2r))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__LE__1,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__EQ__BIGUNION__IMAGE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2p: $i] :
( mem(V2p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V2p))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V0f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Eprobability_2Eevents(A_27a),V2p))))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1g),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Eprobability_2Eevents(A_27a),V2p))))
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0f,inj__ty_2Enum_2Enum(V3m))),ap(V0f,inj__ty_2Enum_2Enum(V4n)))) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( V5m != V6n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1g,inj__ty_2Enum_2Enum(V5m))),ap(V1g,inj__ty_2Enum_2Enum(V6n)))) )
& ! [V7n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V2p),ap(V0f,inj__ty_2Enum_2Enum(V7n)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V2p),ap(V1g,inj__ty_2Enum_2Enum(V7n)))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V2p),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V2p),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1g),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__FINITELY__ADDITIVE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V3n))),ap(c_2Eprobability_2Eevents(A_27a),V0p))))
& ! [V4a: tp__ty_2Enum_2Enum,V5b: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V4a)),inj__ty_2Enum_2Enum(V3n)))
& p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V5b)),inj__ty_2Enum_2Enum(V3n)))
& ( V4a != V5b ) )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V4a))),ap(V2f,inj__ty_2Enum_2Enum(V5b)))) )
& ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V3n)))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V3n))),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Eprobability_2Eprob(A_27a),V0p)),V2f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EABS__1__MINUS__PROB,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__INCREASING__UNION,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Eprobability_2Eevents(A_27a),V0p))))
& ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3n))),ap(V2f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))))
& ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Eprobability_2Eprob(A_27a),V0p)),V2f)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__SUBADDITIVE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u: $i] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3u),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( V1s = ap(ap(c_2Epred__set_2EUNION(A_27a),V2t),V3u) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V3u)))) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__COUNTABLY__SUBADDITIVE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Eprobability_2Eprob(A_27a),V0p)),V1f))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(c_2Eseq_2Esuminf,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Eprobability_2Eprob(A_27a),V0p)),V1f)))) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__COUNTABLY__ZERO,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1c: $i] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ! [V2x: $i] :
( mem(V2x,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1c))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EINDEP__SYM,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1a: $i] :
( mem(V1a,arr(A_27a,bool))
=> ! [V2b: $i] :
( mem(V2b,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),V1a),V2b)) )
=> p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),V2b),V1a)) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EINDEP__REFL,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1a: $i] :
( mem(V1a,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1a),ap(c_2Eprobability_2Eevents(A_27a),V0p))) )
=> ( p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),V1a),V1a))
<=> ( ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1a)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
| ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1a)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ) ) ) ).
tff(lamtp_f4133,type,
f4133: ( del * $i ) > $i ).
tff(lameq_f4133,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3x: $i] : ( ap(f4133(A_27a,V0p),V3x) = ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V3x),c_2Epred__set_2EEMPTY(A_27a))) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__REAL__SUM__IMAGE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f4133(A_27a,V0p)),V1s)) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__EQUIPROBABLE__FINITE__UNIONS,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V1s)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V3x),c_2Epred__set_2EEMPTY(A_27a)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V4y),c_2Epred__set_2EEMPTY(A_27a)))) ) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V1s))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),ap(c_2Epred__set_2ECHOICE(A_27a),V1s)),c_2Epred__set_2EEMPTY(A_27a))))) ) ) ) ) ).
tff(lamtp_f4134,type,
f4134: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f4134,axiom,
! [A_27a: del,A_27b: del,V2e: $i] :
( mem(V2e,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,arr(A_27a,bool)))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V7x: $i] : ( ap(f4134(A_27a,A_27b,V2e,V1f,V0p),V7x) = ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINTER(A_27a),V2e),ap(V1f,V7x))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__REAL__SUM__IMAGE__FN,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,arr(A_27a,bool)))
=> ! [V2e: $i] :
( mem(V2e,arr(A_27a,bool))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2e),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ! [V4x: $i] :
( mem(V4x,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4x),V3s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V2e),ap(V1f,V4x))),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27b),V3s))
& ! [V5x: $i] :
( mem(V5x,A_27b)
=> ! [V6y: $i] :
( mem(V6y,A_27b)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5x),V3s))
& p(ap(ap(c_2Ebool_2EIN(A_27b),V6y),V3s))
& ( V5x != V6y ) )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V5x)),ap(V1f,V6y))) ) ) )
& ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27b,arr(A_27a,bool)),V1f),V3s))),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) = ap(c_2Eprobability_2Ep__space(A_27a),V0p) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2e)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),f4134(A_27a,A_27b,V2e,V1f,V0p)),V3s)) ) ) ) ) ) ) ).
tff(lamtp_f4135,type,
f4135: ( del * $i ) > $i ).
tff(lameq_f4135,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2x: $i] : ( ap(f4135(A_27a,V0p),V2x) = ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__REAL__SUM__IMAGE__SPACE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f4135(A_27a,V0p)),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__DISCRETE__EVENTS,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) )
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2s),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2EPROB__DISCRETE__EVENTS__COUNTABLE,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(c_2Epred__set_2Ecountable(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) )
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2s),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Emarginal__joint__zero,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ! [V4t: $i] :
( mem(V4t,arr(A_27c,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) )
& ( ( surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V3s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
| ( surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V4t)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3s),V4t))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Emarginal__joint__zero3,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3Z: $i] :
( mem(V3Z,arr(A_27a,A_27d))
=> ! [V4s: $i] :
( mem(V4s,arr(A_27b,bool))
=> ! [V5t: $i] :
( mem(V5t,arr(A_27c,bool))
=> ! [V6u: $i] :
( mem(V6u,arr(A_27d,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) )
& ( ( surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V4s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
| ( surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V5t)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
| ( surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27d,A_27a),V0p),V3Z),V6u)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution3(A_27b,A_27c,A_27d,A_27a),V0p),V1X),V2Y),V3Z),ap(ap(c_2Epred__set_2ECROSS(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),V4s),ap(ap(c_2Epred__set_2ECROSS(A_27c,A_27d),V5t),V6u)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Edistribution__prob__space,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ( p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V1X),V0p),V2s))
=> p(ap(c_2Eprobability_2Eprob__space(A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27b),V2s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27b),V2s)),ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X))))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Euniform__distribution__prob__space,axiom,
! [A_27a: del,A_27b: del,V0X: $i] :
( mem(V0X,arr(A_27a,A_27b))
=> ! [V1p: $i] :
( mem(V1p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2s: $i] :
( mem(V2s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V1p)))
& p(ap(c_2Epred__set_2EFINITE(A_27b),ap(c_2Emeasure_2Espace(A_27b),V2s)))
& p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V0X),V1p),V2s)) )
=> p(ap(c_2Eprobability_2Eprob__space(A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27b),V2s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27b),V2s)),ap(ap(ap(c_2Eprobability_2Euniform__distribution(A_27b,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(A_27a,A_27b),A_27b),V1p),V0X),V2s))))) ) ) ) ) ).
tff(lamtp_f4136,type,
f4136: ( del * $i * $i ) > $i ).
tff(lameq_f4136,axiom,
! [A_27a: del,V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3x: tp__ty_2Eextreal_2Eextreal] : ( ap(f4136(A_27a,V1X,V0p),inj__ty_2Eextreal_2Eextreal(V3x)) = ap(ap(ap(c_2Eprobability_2Edistribution(ty_2Eextreal_2Eextreal,A_27a),V0p),V1X),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V3x)),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal))) ) ) ) ).
tff(conj_thm_2Eprobability_2Edistribution__partition,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,ty_2Eextreal_2Eextreal),V1X),V0p),c_2Emeasure_2EBorel)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Eextreal_2Eextreal),f4136(A_27a,V1X,V0p)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Edistribution__lebesgue__thm1,axiom,
! [A_27a: del,A_27b: del,V0X: $i] :
( mem(V0X,arr(A_27a,A_27b))
=> ! [V1p: $i] :
( mem(V1p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2s: $i] :
( mem(V2s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V3A: $i] :
( mem(V3A,arr(A_27b,bool))
=> ( ( p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V0X),V1p),V2s))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3A),ap(c_2Emeasure_2Esubsets(A_27b),V2s))) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2ENormal,ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V1p),V0X),V3A))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Elebesgue_2Eintegral(A_27a),V1p),ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0X),V3A)),ap(c_2Eprobability_2Ep__space(A_27a),V1p))))) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Edistribution__lebesgue__thm2,axiom,
! [A_27a: del,A_27b: del,V0X: $i] :
( mem(V0X,arr(A_27a,A_27b))
=> ! [V1p: $i] :
( mem(V1p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2s: $i] :
( mem(V2s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V3A: $i] :
( mem(V3A,arr(A_27b,bool))
=> ( ( p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V0X),V1p),V2s))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3A),ap(c_2Emeasure_2Esubsets(A_27b),V2s))) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2ENormal,ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V1p),V0X),V3A))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Elebesgue_2Eintegral(A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27b),V2s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27b),V2s)),ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V1p),V0X)))),ap(c_2Emeasure_2Eindicator__fn(A_27b),V3A))) ) ) ) ) ) ) ).
tff(lamtp_f4137,type,
f4137: ( del * $i * $i ) > $i ).
tff(lameq_f4137,axiom,
! [A_27a: del,V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2r: tp__ty_2Eextreal_2Eextreal] : ( ap(f4137(A_27a,V1X,V0p),inj__ty_2Eextreal_2Eextreal(V2r)) = ap(ap(c_2Eextreal_2Eextreal__mul,inj__ty_2Eextreal_2Eextreal(V2r)),ap(c_2Eextreal_2ENormal,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1X),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V2r)),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal)))),ap(c_2Eprobability_2Ep__space(A_27a),V0p))))) ) ) ) ).
tff(conj_thm_2Eprobability_2Efinite__expectation1,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& p(ap(ap(c_2Eprobability_2Ereal__random__variable(A_27a),V1X),V0p)) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eprobability_2Eexpectation(A_27a),V0p),V1X)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Eextreal_2Eextreal),f4137(A_27a,V1X,V0p)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Efinite__expectation2,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Eextreal_2Eextreal),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p))))
& p(ap(ap(c_2Eprobability_2Ereal__random__variable(A_27a),V1X),V0p)) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eprobability_2Eexpectation(A_27a),V0p),V1X)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Eextreal_2Eextreal),f4137(A_27a,V1X,V0p)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) ) ) ) ) ).
tff(lamtp_f4138,type,
f4138: ( del * $i * $i ) > $i ).
tff(lameq_f4138,axiom,
! [A_27a: del,V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2r: tp__ty_2Eextreal_2Eextreal] : ( ap(f4138(A_27a,V1X,V0p),inj__ty_2Eextreal_2Eextreal(V2r)) = ap(ap(c_2Eextreal_2Eextreal__mul,inj__ty_2Eextreal_2Eextreal(V2r)),ap(c_2Eextreal_2ENormal,ap(ap(ap(c_2Eprobability_2Edistribution(ty_2Eextreal_2Eextreal,A_27a),V0p),V1X),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V2r)),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal))))) ) ) ) ).
tff(conj_thm_2Eprobability_2Efinite__expectation,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& p(ap(ap(c_2Eprobability_2Ereal__random__variable(A_27a),V1X),V0p)) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eprobability_2Eexpectation(A_27a),V0p),V1X)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Eextreal_2Eextreal),f4138(A_27a,V1X,V0p)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Efinite__support__expectation,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Eextreal_2Eextreal),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p))))
& p(ap(ap(c_2Eprobability_2Ereal__random__variable(A_27a),V1X),V0p)) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eprobability_2Eexpectation(A_27a),V0p),V1X)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Eextreal_2Eextreal),f4138(A_27a,V1X,V0p)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) ) ) ) ) ).
tff(lamtp_f4139,type,
f4139: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f4139,axiom,
! [A_27c: del,A_27b: del,A_27a: del,V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V4x: $i] : ( ap(f4139(A_27c,A_27b,A_27a,V1X,V2Y),V4x) = ap(ap(c_2Epair_2E_2C(A_27b,A_27c),ap(V1X,V4x)),ap(V2Y,V4x)) ) ) ) ).
tff(lamtp_f4140,type,
f4140: ( del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f4140,axiom,
! [A_27a: del,A_27c: del,A_27b: del,V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3a: $i] : ( ap(f4140(A_27a,A_27c,A_27b,V2Y,V1X,V0p),V3a) = ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),f4139(A_27c,A_27b,A_27a,V1X,V2Y)),V3a)),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Efinite__marginal__product__space__POW,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ( ( ( ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) = ap(c_2Eprobability_2Eevents(A_27a),V0p) )
& p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V1X),V0p),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p))))))
& p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27c),V2Y),V0p),ap(ap(c_2Epair_2E_2C(arr(A_27c,bool),arr(arr(A_27c,bool),bool)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27c),V2Y),ap(c_2Eprobability_2Ep__space(A_27a),V0p))),ap(c_2Epred__set_2EPOW(A_27c),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27c),V2Y),ap(c_2Eprobability_2Ep__space(A_27a),V0p))))))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) )
=> p(ap(c_2Emeasure_2Emeasure__space(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p))),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27c),V2Y),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))),ap(ap(c_2Epair_2E_2C(arr(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p))),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27c),V2Y),ap(c_2Eprobability_2Ep__space(A_27a),V0p))))),f4140(A_27a,A_27c,A_27b,V2Y,V1X,V0p))))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Efinite__marginal__product__space__POW2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s1: $i] :
( mem(V1s1,arr(A_27b,bool))
=> ! [V2s2: $i] :
( mem(V2s2,arr(A_27c,bool))
=> ! [V3X: $i] :
( mem(V3X,arr(A_27a,A_27b))
=> ! [V4Y: $i] :
( mem(V4Y,arr(A_27a,A_27c))
=> ( ( ( ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) = ap(c_2Eprobability_2Eevents(A_27a),V0p) )
& p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V3X),V0p),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),V1s1),ap(c_2Epred__set_2EPOW(A_27b),V1s1))))
& p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27c),V4Y),V0p),ap(ap(c_2Epair_2E_2C(arr(A_27c,bool),arr(arr(A_27c,bool),bool)),V2s2),ap(c_2Epred__set_2EPOW(A_27c),V2s2))))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1s1))
& p(ap(c_2Epred__set_2EFINITE(A_27c),V2s2)) )
=> p(ap(c_2Emeasure_2Emeasure__space(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V1s1),V2s2)),ap(ap(c_2Epair_2E_2C(arr(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27b,A_27c),bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V1s1),V2s2))),ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V3X),V4Y))))) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Efinite__marginal__product__space__POW3,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s1: $i] :
( mem(V1s1,arr(A_27b,bool))
=> ! [V2s2: $i] :
( mem(V2s2,arr(A_27c,bool))
=> ! [V3s3: $i] :
( mem(V3s3,arr(A_27d,bool))
=> ! [V4X: $i] :
( mem(V4X,arr(A_27a,A_27b))
=> ! [V5Y: $i] :
( mem(V5Y,arr(A_27a,A_27c))
=> ! [V6Z: $i] :
( mem(V6Z,arr(A_27a,A_27d))
=> ( ( ( ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) = ap(c_2Eprobability_2Eevents(A_27a),V0p) )
& p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V4X),V0p),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),V1s1),ap(c_2Epred__set_2EPOW(A_27b),V1s1))))
& p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27c),V5Y),V0p),ap(ap(c_2Epair_2E_2C(arr(A_27c,bool),arr(arr(A_27c,bool),bool)),V2s2),ap(c_2Epred__set_2EPOW(A_27c),V2s2))))
& p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27d),V6Z),V0p),ap(ap(c_2Epair_2E_2C(arr(A_27d,bool),arr(arr(A_27d,bool),bool)),V3s3),ap(c_2Epred__set_2EPOW(A_27d),V3s3))))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1s1))
& p(ap(c_2Epred__set_2EFINITE(A_27c),V2s2))
& p(ap(c_2Epred__set_2EFINITE(A_27d),V3s3)) )
=> p(ap(c_2Emeasure_2Emeasure__space(ty_2Epair_2Eprod(A_27b,ty_2Epair_2Eprod(A_27c,A_27d))),ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),bool),ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2ECROSS(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),V1s1),ap(ap(c_2Epred__set_2ECROSS(A_27c,A_27d),V2s2),V3s3))),ap(ap(c_2Epair_2E_2C(arr(arr(ty_2Epair_2Eprod(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(ty_2Epair_2Eprod(A_27b,ty_2Epair_2Eprod(A_27c,A_27d))),ap(ap(c_2Epred__set_2ECROSS(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),V1s1),ap(ap(c_2Epred__set_2ECROSS(A_27c,A_27d),V2s2),V3s3)))),ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution3(A_27b,A_27c,A_27d,A_27a),V0p),V4X),V5Y),V6Z))))) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Eprob__x__eq__1__imp__prob__y__eq__0,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) )
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2y),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p)))
& ( V2y != V1x ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2y),c_2Epred__set_2EEMPTY(A_27a)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Edistribution__x__eq__1__imp__distribution__y__eq__0,axiom,
! [A_27a: del,A_27b: del,V0X: $i] :
( mem(V0X,arr(A_27a,A_27b))
=> ! [V1p: $i] :
( mem(V1p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2x: $i] :
( mem(V2x,A_27b)
=> ( ( p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V0X),V1p),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0X),ap(c_2Eprobability_2Ep__space(A_27a),V1p))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0X),ap(c_2Eprobability_2Ep__space(A_27a),V1p))))))
& ( surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V1p),V0X),ap(ap(c_2Epred__set_2EINSERT(A_27b),V2x),c_2Epred__set_2EEMPTY(A_27b)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) )
=> ! [V3y: $i] :
( mem(V3y,A_27b)
=> ( ( V3y != V2x )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V1p),V0X),ap(ap(c_2Epred__set_2EINSERT(A_27b),V3y),c_2Epred__set_2EEMPTY(A_27b)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Ejoint__distribution__sym,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(A_27b,bool))
=> ! [V4b: $i] :
( mem(V4b,arr(A_27c,bool))
=> ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3a),V4b))) = surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27c,A_27b,A_27a),V0p),V2Y),V1X),ap(ap(c_2Epred__set_2ECROSS(A_27c,A_27b),V4b),V3a))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Ejoint__distribution__pos,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(ty_2Epair_2Eprod(A_27b,A_27c),bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),V3a))) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Ejoint__distribution__le__1,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(ty_2Epair_2Eprod(A_27b,A_27c),bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),V3a)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Ejoint__distribution__le,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(A_27b,bool))
=> ! [V4b: $i] :
( mem(V4b,arr(A_27c,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3a),V4b))),ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V3a))) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Ejoint__distribution__le2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(A_27b,bool))
=> ! [V4b: $i] :
( mem(V4b,arr(A_27c,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3a),V4b))),ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V4b))) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Ejoint__conditional,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(A_27b,bool))
=> ! [V4b: $i] :
( mem(V4b,arr(A_27c,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3a),V4b))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(ap(ap(ap(c_2Eprobability_2Econditional__distribution(A_27a,A_27c,A_27b),V0p),V2Y),V1X),V4b),V3a)),ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V3a))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Edistribution__pos,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2a: $i] :
( mem(V2a,arr(A_27b,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V2a))) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Econditional__distribution__pos,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(A_27b,bool))
=> ! [V4b: $i] :
( mem(V4b,arr(A_27c,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(ap(ap(ap(c_2Eprobability_2Econditional__distribution(A_27a,A_27b,A_27c),V0p),V1X),V2Y),V3a),V4b))) ) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Econditional__distribution__le__1,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(A_27b,bool))
=> ! [V4b: $i] :
( mem(V4b,arr(A_27c,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(ap(ap(ap(c_2Eprobability_2Econditional__distribution(A_27a,A_27b,A_27c),V0p),V1X),V2Y),V3a),V4b)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ) ) ) ) ).
tff(lamtp_f4141,type,
f4141: ( del * del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f4141,axiom,
! [A_27b: del,A_27a: del,A_27c: del,V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V3a: $i] :
( mem(V3a,arr(A_27b,bool))
=> ! [V4x: $i] : ( ap(f4141(A_27b,A_27a,A_27c,V2Y,V0p,V1X,V3a),V4x) = ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3a),ap(ap(c_2Epred__set_2EINSERT(A_27c),V4x),c_2Epred__set_2EEMPTY(A_27c)))) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Emarginal__distribution1,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3a: $i] :
( mem(V3a,arr(A_27b,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V3a)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27c),f4141(A_27b,A_27a,A_27c,V2Y,V0p,V1X,V3a)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27c),V2Y),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) ) ) ) ) ) ) ).
tff(lamtp_f4142,type,
f4142: ( del * del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f4142,axiom,
! [A_27c: del,A_27a: del,A_27b: del,V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V3b: $i] :
( mem(V3b,arr(A_27c,bool))
=> ! [V4x: $i] : ( ap(f4142(A_27c,A_27a,A_27b,V2Y,V0p,V1X,V3b),V4x) = ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(ap(c_2Epred__set_2EINSERT(A_27b),V4x),c_2Epred__set_2EEMPTY(A_27b))),V3b)) ) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Emarginal__distribution2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3b: $i] :
( mem(V3b,arr(A_27c,bool))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V3b)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),f4142(A_27c,A_27a,A_27b,V2Y,V0p,V1X,V3b)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) ) ) ) ) ) ) ).
tff(lamtp_f4143,type,
f4143: ( del * del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f4143,axiom,
! [A_27b: del,A_27a: del,A_27c: del,V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V3x: $i] :
( mem(V3x,A_27b)
=> ! [V4y: $i] : ( ap(f4143(A_27b,A_27a,A_27c,V2Y,V0p,V1X,V3x),V4y) = ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2EINSERT(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epair_2E_2C(A_27b,A_27c),V3x),V4y)),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27b,A_27c)))) ) ) ) ) ) ).
tff(lamtp_f4144,type,
f4144: ( del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f4144,axiom,
! [A_27c: del,A_27a: del,A_27b: del,V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V3x: $i] : ( ap(f4144(A_27c,A_27a,A_27b,V2Y,V0p,V1X),V3x) = f4143(A_27b,A_27a,A_27c,V2Y,V0p,V1X,V3x) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Ejoint__distribution__sums__1,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27b,A_27c)),ap(c_2Epair_2EUNCURRY(A_27b,A_27c,ty_2Erealax_2Ereal),f4144(A_27c,A_27a,A_27b,V2Y,V0p,V1X))),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p))),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27c),V2Y),ap(c_2Eprobability_2Ep__space(A_27a),V0p))))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ) ) ).
tff(lamtp_f4145,type,
f4145: ( del * del * del * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f4145,axiom,
! [A_27b: del,A_27a: del,A_27c: del,V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V4x: $i] :
( mem(V4x,A_27b)
=> ! [V5y: $i] : ( ap(f4145(A_27b,A_27a,A_27c,V2Y,V0p,V1X,V3f,V4x),V5y) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2EINSERT(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epair_2E_2C(A_27b,A_27c),V4x),V5y)),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27b,A_27c))))),ap(V3f,V4x)) ) ) ) ) ) ) ).
tff(lamtp_f4146,type,
f4146: ( del * del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f4146,axiom,
! [A_27c: del,A_27a: del,A_27b: del,V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V4x: $i] : ( ap(f4146(A_27c,A_27a,A_27b,V2Y,V0p,V1X,V3f),V4x) = f4145(A_27b,A_27a,A_27c,V2Y,V0p,V1X,V3f,V4x) ) ) ) ) ) ).
tff(lamtp_f4147,type,
f4147: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f4147,axiom,
! [A_27a: del,A_27b: del,V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V6x: $i] : ( ap(f4147(A_27a,A_27b,V1X,V0p,V3f),V6x) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),ap(ap(c_2Epred__set_2EINSERT(A_27b),V6x),c_2Epred__set_2EEMPTY(A_27b)))),ap(V3f,V6x)) ) ) ) ) ).
tff(conj_thm_2Eprobability_2Ejoint__distribution__sum__mul1,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,A_27b))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,A_27c))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27b,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))
& ( ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27b,A_27c)),ap(c_2Epair_2EUNCURRY(A_27b,A_27c,ty_2Erealax_2Ereal),f4146(A_27c,A_27a,A_27b,V2Y,V0p,V1X,V3f))),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p))),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27c),V2Y),ap(c_2Eprobability_2Ep__space(A_27a),V0p))))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),f4147(A_27a,A_27b,V1X,V0p,V3f)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1X),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------