include('Axioms/lebesgue_2.ax').
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(ax_thm_2Eprobability_2Eindep__function__def,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (ap(c_2Eprobability_2Eindep__function(A_27a,A_27b),V0p) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27a)),arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27a))),f4208(A_27b,A_27a,V0p)))))))).
tff(ax_thm_2Eprobability_2Eprobably__def,axiom,( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V2x,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Eprobability_2Ep__space(A_27a),V1p))) => ((~~ (surj__ty_2Eextreal_2Eextreal(ap(V0X,V2x)) = fo__c_2Eextreal_2ENegInf)) & (~~ (surj__ty_2Eextreal_2Eextreal(ap(V0X,V2x)) = fo__c_2Eextreal_2EPosInf)))))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0X),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Eprobability_2Ep__space(A_27a),V1p)),ap(c_2Eprobability_2Eevents(A_27a),V1p))),c_2Emeasure_2EBorel)))))))))))).
tff(ax_thm_2Eprobability_2Edistribution__def,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0p:$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:$i] : (mem(V1X,arr(A_27b,A_27a)) => (ap(ap(c_2Eprobability_2Edistribution(A_27a,A_27b),V0p),V1X) = f4209(A_27a,A_27b,V1X,V0p))))))))).
tff(ax_thm_2Eprobability_2Ejoint__distribution__def,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0p:$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) = f4211(A_27a,A_27b,A_27c,V2Y,V1X,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) = f4213(A_27c,A_27b,A_27a,A_27d,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:$i] : (mem(V4b,arr(A_27c,bool)) => (surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(ap(c_2Eprobability_2Econditional__distribution(A_27a,A_27b,A_27c),V0p),V1X),V2Y),V3a),V4b)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2E_2F,ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3a),V4b))),ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V4b)))))))))))))))))).
tff(ax_thm_2Eprobability_2Eexpectation__def,axiom,( ! [A_27a:del]: (c_2Eprobability_2Eexpectation(A_27a) = c_2Elebesgue_2Eintegral(A_27a)))).
tff(ax_thm_2Eprobability_2Econditional__expectation__def,axiom,( ! [A_27a:del]: ( ! [V0p:$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)),f4217(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:$i] : (mem(V2e2,arr(arr(A_27a,bool),bool)) => (ap(ap(ap(c_2Eprobability_2Econditional__prob(A_27a),V0p),V1e1),V2e2) = ap(ap(ap(c_2Eprobability_2Econditional__expectation(A_27a),V0p),ap(c_2Emeasure_2Eindicator__fn(A_27a),V1e1)),V2e2)))))))))).
tff(ax_thm_2Eprobability_2Erv__conditional__expectation__def,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0p:$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:$i] : (mem(V3Y,arr(A_27a,A_27b)) => (ap(ap(ap(ap(c_2Eprobability_2Erv__conditional__expectation(A_27a,A_27b),V0p),V1s),V2X),V3Y) = ap(ap(ap(c_2Eprobability_2Econditional__expectation(A_27a),V0p),V2X),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),f4218(A_27b,A_27a,V3Y,V0p)),ap(c_2Emeasure_2Esubsets(A_27b),V1s))))))))))))))).
tff(ax_thm_2Eprobability_2Eindep__rv__def,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0p:$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(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:$i] : (mem(V2s,ty_2Epair_2Eprod(arr(A_27d,bool),arr(arr(A_27d,bool),bool))) => (ap(ap(ap(c_2Eprobability_2Euniform__distribution(A_27a,A_27b,A_27c,A_27d),V0p),V1X),V2s) = f4219(A_27a,A_27d,V2s))))))))))))).
tff(conj_thm_2Eprobability_2EPOSITIVE__PROB,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))))))))))))).
tff(conj_thm_2Eprobability_2EINCREASING__PROB,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [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:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t))))))))))))))).
tff(conj_thm_2Eprobability_2ECOUNTABLY__ADDITIVE__PROB,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) & (( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((~~ (V2m = V3n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n))))))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Eprobability_2Eevents(A_27a),V0p))))) => p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Eprobability_2Eprob(A_27a),V0p)),V1f)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))))))))))))).
tff(conj_thm_2Eprobability_2EPROB__SPACE,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) => p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),ap(c_2Eprobability_2Eevents(A_27a),V0p)))))))))).
tff(conj_thm_2Eprobability_2EEVENTS__ALGEBRA,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),c_2Epred__set_2EEMPTY(A_27a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))).
tff(conj_thm_2Eprobability_2EPROB__UNIV,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) => p(ap(c_2Emeasure_2Epositive(A_27a),V0p)))))))).
tff(conj_thm_2Eprobability_2EPROB__SPACE__COUNTABLY__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) => p(ap(c_2Emeasure_2Eadditive(A_27a),V0p)))))))).
tff(conj_thm_2Eprobability_2EPROB__SPACE__INCREASING,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [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,lemma,(( ! [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:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t))))))))))))).
tff(conj_thm_2Eprobability_2EPROB__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V3u,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) & (V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V3u)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)))))))))))))))).
tff(conj_thm_2Eprobability_2EPROB__COUNTABLY__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [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,lemma,(( ! [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,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,arr(A_27a,bool)) => ( ! [V1b:$i] : (mem(V1b,arr(arr(A_27a,bool),bool)) => ( ! [V2c:$i] : (mem(V2c,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => (ap(c_2Eprobability_2Ep__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0a),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1b),V2c))) = V0a)))))))))).
tff(conj_thm_2Eprobability_2EEVENTS,lemma,(( ! [A_27a:del]: ( ! [V0a:$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,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,arr(A_27a,bool)) => ( ! [V1b:$i] : (mem(V1b,arr(arr(A_27a,bool),bool)) => ( ! [V2c:$i] : (mem(V2c,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => (ap(c_2Eprobability_2Eprob(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0a),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1b),V2c))) = V2c)))))))))).
tff(conj_thm_2Eprobability_2EEVENTS__INTER,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Eprobability_2Eevents(A_27a),V0p))))))))).
tff(conj_thm_2Eprobability_2EEVENTS__SPACE,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [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:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Eprobability_2Eevents(A_27a),V0p))))))))))))).
tff(conj_thm_2Eprobability_2EINDEP__EMPTY,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) => p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),c_2Epred__set_2EEMPTY(A_27a)),V1s)))))))))).
tff(conj_thm_2Eprobability_2EINTER__PSPACE,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) => (ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s) = V1s))))))))).
tff(conj_thm_2Eprobability_2EINDEP__SPACE,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) => p(ap(ap(ap(c_2Eprobability_2Eindep(A_27a),V0p),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s)))))))))).
tff(conj_thm_2Eprobability_2EEVENTS__DIFF,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [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,lemma,(( ! [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,lemma,(( ! [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:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))))))))))))).
tff(conj_thm_2Eprobability_2EPROB__EQ__COMPL,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [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:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))))))))))))).
tff(conj_thm_2Eprobability_2EEVENTS__COUNTABLE__INTER,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1c,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c)) & (~~ (V1c = c_2Epred__set_2EEMPTY(arr(A_27a,bool))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),V1c)),ap(c_2Eprobability_2Eevents(A_27a),V0p))))))))))).
tff(conj_thm_2Eprobability_2EABS__PROB,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) => (surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Eabs,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))))))))))).
tff(conj_thm_2Eprobability_2EPROB__COMPL__LE1,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2r:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) => (p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s))),inj__ty_2Erealax_2Ereal(V2r))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V2r))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))))))))))))).
tff(conj_thm_2Eprobability_2EPROB__LE__1,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))))))))))).
tff(conj_thm_2Eprobability_2EPROB__EQ__BIGUNION__IMAGE,lemma,(( ! [A_27a:del]: ( ! [V0f:$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,lemma,(( ! [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:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ! [V3n:tp__ty_2Enum_2Enum] : ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V3n))),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) & (( ! [V4a:tp__ty_2Enum_2Enum] : ( ! [V5b:tp__ty_2Enum_2Enum] : ((p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V4a)),inj__ty_2Enum_2Enum(V3n))) & (p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V5b)),inj__ty_2Enum_2Enum(V3n))) & (~~ (V4a = V5b)))) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V4a))),ap(V2f,inj__ty_2Enum_2Enum(V5b))))))) & (V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V3n)))))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V3n))),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Eprobability_2Eprob(A_27a),V0p)),V2f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)))))))))))))).
tff(conj_thm_2Eprobability_2EABS__1__MINUS__PROB,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Eprobability_2Eevents(A_27a),V0p))) & (~~ (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))))))))))).
tff(conj_thm_2Eprobability_2EPROB__INCREASING__UNION,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [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,lemma,(( ! [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,lemma,(( ! [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:$i] : (mem(V2x,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1c)) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))))).
tff(conj_thm_2Eprobability_2EINDEP__SYM,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [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(conj_thm_2Eprobability_2EPROB__REAL__SUM__IMAGE,lemma,(( ! [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:$i] : (mem(V2x,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p)))))) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f4220(A_27a,V0p)),V1s))))))))))).
tff(conj_thm_2Eprobability_2EPROB__EQUIPROBABLE__FINITE__UNIONS,lemma,(( ! [A_27a:del]: ( ! [V0p:$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(conj_thm_2Eprobability_2EPROB__REAL__SUM__IMAGE__FN,lemma,(( ! [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:$i] : (mem(V6y,A_27b) => ((p(ap(ap(c_2Ebool_2EIN(A_27b),V5x),V3s)) & (p(ap(ap(c_2Ebool_2EIN(A_27b),V6y),V3s)) & (~~ (V5x = V6y)))) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V5x)),ap(V1f,V6y)))))))) & (ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27b,arr(A_27a,bool)),V1f),V3s))),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) = ap(c_2Eprobability_2Ep__space(A_27a),V0p))))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2e)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),f4221(A_27b,A_27a,V2e,V1f,V0p)),V3s)))))))))))))))).
tff(conj_thm_2Eprobability_2EPROB__REAL__SUM__IMAGE__SPACE,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V1x,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Eprobability_2Eevents(A_27a),V0p)))))) & p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f4222(A_27a,V0p)),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))))))))).
tff(conj_thm_2Eprobability_2EPROB__DISCRETE__EVENTS,lemma,(( ! [A_27a:del]: ( ! [V0p:$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,lemma,(( ! [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:$i] : (mem(V2s,arr(A_27a,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2s),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Eprobability_2Eevents(A_27a),V0p)))))))))))).
tff(conj_thm_2Eprobability_2Emarginal__joint__zero,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0p:$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,lemma,(( ! [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:$i] : (mem(V6u,arr(A_27d,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & ((ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p))) & ((surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V4s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) | ((surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V5t)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) | (surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Eprobability_2Edistribution(A_27d,A_27a),V0p),V3Z),V6u)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))) => (surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution3(A_27b,A_27c,A_27d,A_27a),V0p),V1X),V2Y),V3Z),ap(ap(c_2Epred__set_2ECROSS(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),V4s),ap(ap(c_2Epred__set_2ECROSS(A_27c,A_27d),V5t),V6u)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))))))))))))))))))))).
tff(conj_thm_2Eprobability_2Edistribution__prob__space,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0p:$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,lemma,(( ! [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:$i] : (mem(V2s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ((p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V1p))) & (p(ap(c_2Epred__set_2EFINITE(A_27b),ap(c_2Emeasure_2Espace(A_27b),V2s))) & p(ap(ap(ap(c_2Eprobability_2Erandom__variable(A_27a,A_27b),V0X),V1p),V2s)))) => p(ap(c_2Eprobability_2Eprob__space(A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27b),V2s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27b),V2s)),ap(ap(ap(c_2Eprobability_2Euniform__distribution(A_27b,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(A_27a,A_27b),A_27b),V1p),V0X),V2s)))))))))))))))).
tff(conj_thm_2Eprobability_2Edistribution__partition,lemma,(( ! [A_27a:del]: ( ! [V0p:$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),f4223(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,lemma,(( ! [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,lemma,(( ! [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(conj_thm_2Eprobability_2Efinite__expectation1,lemma,(( ! [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),f4224(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,lemma,(( ! [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),f4224(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__expectation,lemma,(( ! [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),f4225(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,lemma,(( ! [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),f4225(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__marginal__product__space__POW,lemma,(( ! [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))))),f4227(A_27b,A_27c,A_27a,V2Y,V1X,V0p))))))))))))))))).
tff(conj_thm_2Eprobability_2Efinite__marginal__product__space__POW2,lemma,(( ! [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,lemma,(( ! [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,lemma,(( ! [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,lemma,(( ! [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,lemma,(( ! [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:$i] : (mem(V4b,arr(A_27c,bool)) => (p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) => (surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3a),V4b))) = surj__ty_2Erealax_2Ereal(ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27c,A_27b,A_27a),V0p),V2Y),V1X),ap(ap(c_2Epred__set_2ECROSS(A_27c,A_27b),V4b),V3a)))))))))))))))))))).
tff(conj_thm_2Eprobability_2Ejoint__distribution__pos,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0p:$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:$i] : (mem(V3a,arr(ty_2Epair_2Eprod(A_27b,A_27c),bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),V3a))))))))))))))))).
tff(conj_thm_2Eprobability_2Ejoint__distribution__le__1,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0p:$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:$i] : (mem(V3a,arr(ty_2Epair_2Eprod(A_27b,A_27c),bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),V3a)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))))))))))))))))).
tff(conj_thm_2Eprobability_2Ejoint__distribution__le,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0p:$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,lemma,(( ! [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:$i] : (mem(V4b,arr(A_27c,bool)) => ((p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & (ap(c_2Eprobability_2Eevents(A_27a),V0p) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V3a),V4b))),ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V4b))))))))))))))))))).
tff(conj_thm_2Eprobability_2Ejoint__conditional,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0p:$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,conjecture,(( ! [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)))))))))))))).
