include('Axioms/lebesgue+2.ax').
fof(mem_c_2Eprobability_2Econditional__distribution,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => 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)))))) ) ) ) )).
fof(mem_c_2Eprobability_2Econditional__expectation,axiom,( ! [A_27a] : ( ne(A_27a) => 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))))) ) )).
fof(mem_c_2Eprobability_2Econditional__prob,axiom,( ! [A_27a] : ( ne(A_27a) => 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))))) ) )).
fof(mem_c_2Eprobability_2Edistribution,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => 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)))) ) ) )).
fof(mem_c_2Eprobability_2Eevents,axiom,( ! [A_27a] : ( ne(A_27a) => 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))) ) )).
fof(mem_c_2Eprobability_2Eexpectation,axiom,( ! [A_27a] : ( ne(A_27a) => 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))) ) )).
fof(mem_c_2Eprobability_2Eindep,axiom,( ! [A_27a] : ( ne(A_27a) => 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)))) ) )).
fof(mem_c_2Eprobability_2Eindep__families,axiom,( ! [A_27a] : ( ne(A_27a) => 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)))) ) )).
fof(mem_c_2Eprobability_2Eindep__function,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => 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))) ) ) )).
fof(mem_c_2Eprobability_2Eindep__rv,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => 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)))))) ) ) ) )).
fof(mem_c_2Eprobability_2Ejoint__distribution,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => 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))))) ) ) ) )).
fof(mem_c_2Eprobability_2Ejoint__distribution3,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => ! [A_27d] : ( ne(A_27d) => 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)))))) ) ) ) ) )).
fof(mem_c_2Eprobability_2Ep__space,axiom,( ! [A_27a] : ( ne(A_27a) => 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))) ) )).
fof(mem_c_2Eprobability_2Epossibly,axiom,( ! [A_27a] : ( ne(A_27a) => 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))) ) )).
fof(mem_c_2Eprobability_2Eprob,axiom,( ! [A_27a] : ( ne(A_27a) => 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))) ) )).
fof(mem_c_2Eprobability_2Eprob__space,axiom,( ! [A_27a] : ( ne(A_27a) => 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)) ) )).
fof(mem_c_2Eprobability_2Eprobably,axiom,( ! [A_27a] : ( ne(A_27a) => 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))) ) )).
fof(mem_c_2Eprobability_2Erandom__variable,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => 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)))) ) ) )).
fof(mem_c_2Eprobability_2Ereal__random__variable,axiom,( ! [A_27a] : ( ne(A_27a) => 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))) ) )).
fof(mem_c_2Eprobability_2Erv__conditional__expectation,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => 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)))))) ) ) )).
fof(mem_c_2Eprobability_2Euniform__distribution,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => ! [A_27d] : ( ne(A_27d) => 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))))) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Ep__space__def,axiom,( ! [A_27a] : ( ne(A_27a) => ( c_2Eprobability_2Ep__space(A_27a) = c_2Emeasure_2Em__space(A_27a) ) ) )).
fof(ax_thm_2Eprobability_2Eevents__def,axiom,( ! [A_27a] : ( ne(A_27a) => ( c_2Eprobability_2Eevents(A_27a) = c_2Emeasure_2Emeasurable__sets(A_27a) ) ) )).
fof(ax_thm_2Eprobability_2Eprob__def,axiom,( ! [A_27a] : ( ne(A_27a) => ( c_2Eprobability_2Eprob(A_27a) = c_2Emeasure_2Emeasure(A_27a) ) ) )).
fof(ax_thm_2Eprobability_2Eprob__space__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( 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)) & ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0p),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Eindep__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1a,arr(A_27a,bool)) => ! [V2b] : ( 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))) & ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINTER(A_27a),V1a),V2b)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1a)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2b)) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Eindep__families__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1q,arr(arr(A_27a,bool),bool)) => ! [V2r] : ( mem(V2r,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(ap(c_2Eprobability_2Eindep__families(A_27a),V0p),V1q),V2r)) <=> ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ! [V4t] : ( 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)) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Eindep__function__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0p] : ( 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)) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Eprobably__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( 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] : ( 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))) & ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1e) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Epossibly__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( 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] : ( 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))) & ( ~~ ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1e) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Erandom__variable__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0X] : ( mem(V0X,arr(A_27a,A_27b)) => ! [V1p] : ( 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] : ( 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))) ) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Ereal__random__variable__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0X] : ( mem(V0X,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1p] : ( 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] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Eprobability_2Ep__space(A_27a),V1p))) => ( ( ~~ ( ap(V0X,V2x) = c_2Eextreal_2ENegInf ) ) & ( ~~ ( ap(V0X,V2x) = 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))) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Edistribution__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0p] : ( 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] : ( 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) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Ejoint__distribution__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27c,A_27a)) => ! [V2Y] : ( 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) ) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Ejoint__distribution3__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => ! [A_27d] : ( ne(A_27d) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27d,A_27a)) => ! [V2Y] : ( mem(V2Y,arr(A_27d,A_27b)) => ! [V3Z] : ( 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) ) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Econditional__distribution__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3a] : ( mem(V3a,arr(A_27b,bool)) => ! [V4b] : ( mem(V4b,arr(A_27c,bool)) => ( ap(ap(ap(ap(ap(c_2Eprobability_2Econditional__distribution(A_27a,A_27b,A_27c),V0p),V1X),V2Y),V3a),V4b) = 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)) ) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Eexpectation__def,axiom,( ! [A_27a] : ( ne(A_27a) => ( c_2Eprobability_2Eexpectation(A_27a) = c_2Elebesgue_2Eintegral(A_27a) ) ) )).
fof(ax_thm_2Eprobability_2Econditional__expectation__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V2s] : ( 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)) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Econditional__prob__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1e1,arr(A_27a,bool)) => ! [V2e2] : ( 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) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Erv__conditional__expectation__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0p] : ( 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] : ( mem(V1s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V2X] : ( mem(V2X,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V3Y] : ( 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))) ) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Eindep__rv__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3s] : ( mem(V3s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V4t] : ( 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] : ( mem(V5A,arr(A_27b,bool)) => ! [V6B] : ( 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))) ) ) ) ) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Eprobability_2Euniform__distribution__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [A_27c] : ( ne(A_27c) => ! [A_27d] : ( ne(A_27d) => ! [V0p] : ( mem(V0p,A_27b) => ! [V1X] : ( mem(V1X,A_27c) => ! [V2s] : ( 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) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Eprobability_2EPOSITIVE__PROB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)) <=> ( ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) & ! [V1s] : ( 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,c_2Enum_2E0)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EINCREASING__PROB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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))) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EADDITIVE__PROB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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)) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2ECOUNTABLY__ADDITIVE__PROB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ~~ ( V2m = V3n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V2m)),ap(V1f,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)))))) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__SPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)) & ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__SIGMA__ALGEBRA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)))) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__ALGEBRA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)))) ) )))).
fof(conj_thm_2Eprobability_2EPROB__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__SPACE__POSITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)) ) )))).
fof(conj_thm_2Eprobability_2EPROB__SPACE__COUNTABLY__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)) ) )))).
fof(conj_thm_2Eprobability_2EPROB__SPACE__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)) ) )))).
fof(conj_thm_2Eprobability_2EPROB__SPACE__INCREASING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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)) ) )))).
fof(conj_thm_2Eprobability_2EPROB__POSITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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,c_2Enum_2E0)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__INCREASING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( 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) ) ) ) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V3u) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__COUNTABLY__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( 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] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( ~~ ( V3m = V4n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,V3m)),ap(V2f,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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Eprobability_2Ep__space(A_27a),V0p)),V1s)) = ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__INDEP,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( 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) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V3u) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPSPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,arr(A_27a,bool)) => ! [V1b] : ( mem(V1b,arr(arr(A_27a,bool),bool)) => ! [V2c] : ( 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 ) ) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,arr(A_27a,bool)) => ! [V1b] : ( mem(V1b,arr(arr(A_27a,bool),bool)) => ! [V2c] : ( 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 ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,arr(A_27a,bool)) => ! [V1b] : ( mem(V1b,arr(arr(A_27a,bool),bool)) => ! [V2c] : ( 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 ) ) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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))) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__SPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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))) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2EINDEP__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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)) ) ) )))).
fof(conj_thm_2Eprobability_2EINTER__PSPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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 ) ) ) )))).
fof(conj_thm_2Eprobability_2EINDEP__SPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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)) ) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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))) ) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__COUNTABLE__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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))) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__ZERO__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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))) & ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__EQ__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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))) & ( 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)) = 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)) ) ) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s) = ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__ONE__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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))) & ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2t) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)) = ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EEVENTS__COUNTABLE__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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))) ) ) )))).
fof(conj_thm_2Eprobability_2EABS__PROB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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(c_2Ereal_2Eabs,ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s)) = ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__COMPL__LE1,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2r] : ( mem(V2r,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))),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,c_2Earithmetic_2EZERO)))),V2r)),ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s))) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__LE__1,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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,c_2Earithmetic_2EZERO))))) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__EQ__BIGUNION__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V2p] : ( 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] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( ~~ ( V3m = V4n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0f,V3m)),ap(V0f,V4n))) ) ) ) ) & ( ( ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( ( ~~ ( V5m = V6n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1g,V5m)),ap(V1g,V6n))) ) ) ) ) & ! [V7n] : ( mem(V7n,ty_2Enum_2Enum) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V2p),ap(V0f,V7n)) = ap(ap(c_2Eprobability_2Eprob(A_27a),V2p),ap(V1g,V7n)) ) ) ) ) ) ) ) => ( 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)))) = 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)))) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__FINITELY__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V3n] : ( mem(V3n,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,V3n)),ap(c_2Eprobability_2Eevents(A_27a),V0p)))) & ( ( ! [V4a] : ( mem(V4a,ty_2Enum_2Enum) => ! [V5b] : ( mem(V5b,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Eprim__rec_2E_3C,V4a),V3n)) & ( p(ap(ap(c_2Eprim__rec_2E_3C,V5b),V3n)) & ( ~~ ( V4a = V5b ) ) ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,V4a)),ap(V2f,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,V3n))) ) ) ) ) => ( ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V3n)),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) ) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EABS__1__MINUS__PROB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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_2Eprobability_2Eprob(A_27a),V0p),V1s) = ap(c_2Ereal_2Ereal__of__num,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,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,c_2Earithmetic_2EZERO))))) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__INCREASING__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( 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] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,V3n)),ap(V2f,ap(c_2Enum_2ESUC,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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__SUBADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( 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)))) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__COUNTABLY__SUBADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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)))) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__COUNTABLY__ZERO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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] : ( mem(V2x,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1c)) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) )))).
fof(conj_thm_2Eprobability_2EINDEP__SYM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1a,arr(A_27a,bool)) => ! [V2b] : ( 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)) ) ) ) )))).
fof(conj_thm_2Eprobability_2EINDEP__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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)) <=> ( ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1a) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) | ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1a) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__REAL__SUM__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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] : ( 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)) ) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f4220(A_27a,V0p)),V1s) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__EQUIPROBABLE__FINITE__UNIONS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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] : ( 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] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V1s)) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V3x),c_2Epred__set_2EEMPTY(A_27a))) = ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V4y),c_2Epred__set_2EEMPTY(A_27a))) ) ) ) ) ) ) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V1s) = 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)))) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__REAL__SUM__IMAGE__FN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0p] : ( 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] : ( mem(V1f,arr(A_27b,arr(A_27a,bool))) => ! [V2e] : ( mem(V2e,arr(A_27a,bool)) => ! [V3s] : ( 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] : ( 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] : ( mem(V5x,A_27b) => ! [V6y] : ( 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) ) ) ) ) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),V2e) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),f4221(A_27b,A_27a,V2e,V1f,V0p)),V3s) ) ) ) ) ) ))))).
fof(conj_thm_2Eprobability_2EPROB__REAL__SUM__IMAGE__SPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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))) ) ) => ( ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f4222(A_27a,V0p)),ap(c_2Eprobability_2Ep__space(A_27a),V0p)) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__DISCRETE__EVENTS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2EPROB__DISCRETE__EVENTS__COUNTABLE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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] : ( 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2Emarginal__joint__zero,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ! [V4t] : ( 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)) ) & ( ( ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V3s) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) | ( ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V4t) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) => ( 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)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) )))))).
fof(conj_thm_2Eprobability_2Emarginal__joint__zero3,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3Z] : ( mem(V3Z,arr(A_27a,A_27d)) => ! [V4s] : ( mem(V4s,arr(A_27b,bool)) => ! [V5t] : ( mem(V5t,arr(A_27c,bool)) => ! [V6u] : ( 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)) ) & ( ( ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V4s) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) | ( ( ap(ap(ap(c_2Eprobability_2Edistribution(A_27c,A_27a),V0p),V2Y),V5t) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) | ( ap(ap(ap(c_2Eprobability_2Edistribution(A_27d,A_27a),V0p),V3Z),V6u) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) => ( 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))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ) ) ))))))).
fof(conj_thm_2Eprobability_2Edistribution__prob__space,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2s] : ( 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))))) ) ) ) ))))).
fof(conj_thm_2Eprobability_2Euniform__distribution__prob__space,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0X] : ( mem(V0X,arr(A_27a,A_27b)) => ! [V1p] : ( 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] : ( 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))))) ) ) ) ))))).
fof(conj_thm_2Eprobability_2Edistribution__partition,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) & ( ( ! [V2x] : ( 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)) ) ) ) => ( 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))) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) )))).
fof(conj_thm_2Eprobability_2Edistribution__lebesgue__thm1,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0X] : ( mem(V0X,arr(A_27a,A_27b)) => ! [V1p] : ( 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] : ( mem(V2s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V3A] : ( 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))) ) => ( ap(c_2Eextreal_2ENormal,ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V1p),V0X),V3A)) = 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)))) ) ) ) ) ) ))))).
fof(conj_thm_2Eprobability_2Edistribution__lebesgue__thm2,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0X] : ( mem(V0X,arr(A_27a,A_27b)) => ! [V1p] : ( 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] : ( mem(V2s,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V3A] : ( 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))) ) => ( ap(c_2Eextreal_2ENormal,ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V1p),V0X),V3A)) = 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)) ) ) ) ) ) ))))).
fof(conj_thm_2Eprobability_2Efinite__expectation1,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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)) ) => ( ap(ap(c_2Eprobability_2Eexpectation(A_27a),V0p),V1X) = 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2Efinite__expectation2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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)) ) => ( ap(ap(c_2Eprobability_2Eexpectation(A_27a),V0p),V1X) = 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2Efinite__expectation,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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)) ) => ( ap(ap(c_2Eprobability_2Eexpectation(A_27a),V0p),V1X) = 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2Efinite__support__expectation,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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)) ) => ( ap(ap(c_2Eprobability_2Eexpectation(A_27a),V0p),V1X) = 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))) ) ) ) )))).
fof(conj_thm_2Eprobability_2Efinite__marginal__product__space__POW,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( 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))))) ) ) ) )))))).
fof(conj_thm_2Eprobability_2Efinite__marginal__product__space__POW2,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1s1,arr(A_27b,bool)) => ! [V2s2] : ( mem(V2s2,arr(A_27c,bool)) => ! [V3X] : ( mem(V3X,arr(A_27a,A_27b)) => ! [V4Y] : ( 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))))) ) ) ) ) ) )))))).
fof(conj_thm_2Eprobability_2Efinite__marginal__product__space__POW3,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) => ! [V0p] : ( 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] : ( mem(V1s1,arr(A_27b,bool)) => ! [V2s2] : ( mem(V2s2,arr(A_27c,bool)) => ! [V3s3] : ( mem(V3s3,arr(A_27d,bool)) => ! [V4X] : ( mem(V4X,arr(A_27a,A_27b)) => ! [V5Y] : ( mem(V5Y,arr(A_27a,A_27c)) => ! [V6Z] : ( 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))))) ) ) ) ) ) ) ) ))))))).
fof(conj_thm_2Eprobability_2Eprob__x__eq__1__imp__prob__y__eq__0,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( 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] : ( 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))) & ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) => ! [V2y] : ( 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 ) ) ) => ( ap(ap(c_2Eprobability_2Eprob(A_27a),V0p),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2y),c_2Epred__set_2EEMPTY(A_27a))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) )))).
fof(conj_thm_2Eprobability_2Edistribution__x__eq__1__imp__distribution__y__eq__0,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0X] : ( mem(V0X,arr(A_27a,A_27b)) => ! [V1p] : ( 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] : ( 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)))))) & ( 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))) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) => ! [V3y] : ( mem(V3y,A_27b) => ( ( ~~ ( V3y = V2x ) ) => ( 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))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ))))).
fof(conj_thm_2Eprobability_2Ejoint__distribution__sym,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3a] : ( mem(V3a,arr(A_27b,bool)) => ! [V4b] : ( mem(V4b,arr(A_27c,bool)) => ( p(ap(c_2Eprobability_2Eprob__space(A_27a),V0p)) => ( 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(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)) ) ) ) ) ) ) )))))).
fof(conj_thm_2Eprobability_2Ejoint__distribution__pos,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3a] : ( 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,c_2Enum_2E0)),ap(ap(ap(ap(c_2Eprobability_2Ejoint__distribution(A_27b,A_27c,A_27a),V0p),V1X),V2Y),V3a))) ) ) ) ) )))))).
fof(conj_thm_2Eprobability_2Ejoint__distribution__le__1,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3a] : ( 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,c_2Earithmetic_2EZERO))))) ) ) ) ) )))))).
fof(conj_thm_2Eprobability_2Ejoint__distribution__le,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3a] : ( mem(V3a,arr(A_27b,bool)) => ! [V4b] : ( 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))) ) ) ) ) ) )))))).
fof(conj_thm_2Eprobability_2Ejoint__distribution__le2,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3a] : ( mem(V3a,arr(A_27b,bool)) => ! [V4b] : ( 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))) ) ) ) ) ) )))))).
fof(conj_thm_2Eprobability_2Ejoint__conditional,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3a] : ( mem(V3a,arr(A_27b,bool)) => ! [V4b] : ( 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)) ) ) => ( 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(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)) ) ) ) ) ) ) )))))).
fof(conj_thm_2Eprobability_2Edistribution__pos,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2a] : ( 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,c_2Enum_2E0)),ap(ap(ap(c_2Eprobability_2Edistribution(A_27b,A_27a),V0p),V1X),V2a))) ) ) ) ))))).
fof(conj_thm_2Eprobability_2Econditional__distribution__pos,conjecture,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0p] : ( 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] : ( mem(V1X,arr(A_27a,A_27b)) => ! [V2Y] : ( mem(V2Y,arr(A_27a,A_27c)) => ! [V3a] : ( mem(V3a,arr(A_27b,bool)) => ! [V4b] : ( 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,c_2Enum_2E0)),ap(ap(ap(ap(ap(c_2Eprobability_2Econditional__distribution(A_27a,A_27b,A_27c),V0p),V1X),V2Y),V3a),V4b))) ) ) ) ) ) )))))).
