SET007 Axioms: SET007+616.ax
%------------------------------------------------------------------------------
% File : SET007+616 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Irrationality of e
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : irrat_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 55 ( 4 unt; 0 def)
% Number of atoms : 279 ( 52 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 252 ( 28 ~; 6 |; 90 &)
% ( 5 <=>; 123 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 34 ( 34 usr; 9 con; 0-4 aty)
% Number of variables : 104 ( 98 !; 6 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( v1_int_2(A)
& v1_rat_1(k9_square_1(A)) ) ) ).
fof(t2_irrat_1,axiom,
? [A] :
( v1_xreal_0(A)
& ? [B] :
( v1_xreal_0(B)
& ~ v1_rat_1(A)
& ~ v1_rat_1(B)
& v1_rat_1(k3_power(A,B)) ) ) ).
fof(d1_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k1_irrat_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(k6_xcmplx_0(C,A),C) ) ) ) ) ).
fof(d2_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k2_irrat_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k3_xcmplx_0(k8_newton(A,C),k3_power(C,k4_xcmplx_0(A))) ) ) ) ) ).
fof(d3_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k3_irrat_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k3_xcmplx_0(k8_newton(C,A),k3_power(A,k4_xcmplx_0(C))) ) ) ) ) ).
fof(t3_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k3_irrat_1(A),B) = k2_seq_1(k5_numbers,k1_numbers,k2_irrat_1(B),A) ) ) ).
fof(d4_irrat_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( A = k4_irrat_1
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k3_power(k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,B)),B) ) ) ) ).
fof(d5_irrat_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( A = k5_irrat_1
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k7_xcmplx_0(np__1,k11_newton(B)) ) ) ) ).
fof(t4_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k3_power(A,k4_xcmplx_0(k1_nat_1(B,np__1))) = k7_xcmplx_0(k3_power(A,k4_xcmplx_0(B)),A) ) ) ) ).
fof(t5_irrat_1,axiom,
$true ).
fof(t6_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k8_newton(k1_nat_1(B,np__1),A) = k3_xcmplx_0(k7_xcmplx_0(k6_xcmplx_0(A,B),k1_nat_1(B,np__1)),k8_newton(B,A)) ) ) ).
fof(t7_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,k2_irrat_1(k1_nat_1(B,np__1)),A) = k3_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(np__1,k1_nat_1(B,np__1)),k2_seq_1(k5_numbers,k1_numbers,k2_irrat_1(B),A)),k2_seq_1(k5_numbers,k1_numbers,k1_irrat_1(B),A)) ) ) ) ).
fof(t8_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,k1_irrat_1(B),A) = k6_xcmplx_0(np__1,k7_xcmplx_0(B,A)) ) ) ) ).
fof(t9_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( v4_seq_2(k1_irrat_1(A))
& k2_seq_2(k1_irrat_1(A)) = np__1 ) ) ).
fof(t10_irrat_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = A )
=> ( v4_seq_2(B)
& k2_seq_2(B) = A ) ) ) ) ).
fof(t11_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k2_irrat_1(np__0),A) = np__1 ) ).
fof(t12_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_xcmplx_0(k7_xcmplx_0(np__1,k1_nat_1(A,np__1)),k7_xcmplx_0(np__1,k11_newton(A))) = k7_xcmplx_0(np__1,k11_newton(k1_nat_1(A,np__1))) ) ).
fof(t13_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( v4_seq_2(k2_irrat_1(A))
& k2_seq_2(k2_irrat_1(A)) = k7_xcmplx_0(np__1,k11_newton(A))
& k2_seq_2(k2_irrat_1(A)) = k2_seq_1(k5_numbers,k1_numbers,k5_irrat_1,A) ) ) ).
fof(t14_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,A)
=> ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_irrat_1(A),B),np__0)
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_irrat_1(A),B),np__1) ) ) ) ) ).
fof(t15_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k2_irrat_1(B),A))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k2_irrat_1(B),A),k7_xcmplx_0(np__1,k11_newton(B)))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k2_irrat_1(B),A),k2_seq_1(k5_numbers,k1_numbers,k5_irrat_1,B))
& r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k3_irrat_1(A),B))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k3_irrat_1(A),B),k7_xcmplx_0(np__1,k11_newton(B)))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k3_irrat_1(A),B),k2_seq_1(k5_numbers,k1_numbers,k5_irrat_1,B)) ) ) ) ) ).
fof(t16_irrat_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_series_1(k1_seqm_3(A,np__1))
=> ( v1_series_1(A)
& k2_series_1(A) = k2_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,np__0),k2_series_1(k1_seqm_3(A,np__1))) ) ) ) ).
fof(t17_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(k3_finseq_1(C),A)
| k1_funct_1(k1_rfinseq(B,C,np__1),A) = k1_funct_1(C,k1_nat_1(A,np__1)) ) ) ) ) ) ).
fof(t18_irrat_1,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> k15_rvsum_1(A) = k2_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,np__1),k15_rvsum_1(k1_rfinseq(k1_numbers,A,np__1))) ) ) ).
fof(t19_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ( ( k3_finseq_1(C) = A
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,D)
=> k2_seq_1(k5_numbers,k1_numbers,B,D) = k2_seq_1(k5_numbers,k1_numbers,C,k1_nat_1(D,np__1)) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,D)
=> k2_seq_1(k5_numbers,k1_numbers,B,D) = np__0 ) ) )
=> ( v1_series_1(B)
& k2_series_1(B) = k15_rvsum_1(C) ) ) ) ) ) ).
fof(t20_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_xreal_0(A,B)
=> ( C = np__0
| D = np__0
| k2_seq_1(k5_numbers,k1_numbers,k9_newton(C,D,B),k1_nat_1(A,np__1)) = k3_xcmplx_0(k3_xcmplx_0(k8_newton(A,B),k3_power(C,k6_xcmplx_0(B,A))),k3_power(D,A)) ) ) ) ) ) ) ).
fof(t21_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,A)
=> ( r1_xreal_0(A,np__0)
| k2_seq_1(k5_numbers,k1_numbers,k3_irrat_1(A),B) = k2_seq_1(k5_numbers,k1_numbers,k9_newton(np__1,k7_xcmplx_0(np__1,A),A),k1_nat_1(B,np__1)) ) ) ) ) ).
fof(t22_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ( v1_series_1(k3_irrat_1(A))
& k2_series_1(k3_irrat_1(A)) = k3_power(k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,A)),A)
& k2_series_1(k3_irrat_1(A)) = k2_seq_1(k5_numbers,k1_numbers,k4_irrat_1,A) ) ) ) ).
fof(t23_irrat_1,axiom,
( v4_seq_2(k4_irrat_1)
& k2_seq_2(k4_irrat_1) = k8_power ) ).
fof(t24_irrat_1,axiom,
( v1_series_1(k5_irrat_1)
& k2_series_1(k5_irrat_1) = k28_sin_cos(np__1) ) ).
fof(t25_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k3_irrat_1(C)),A) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k5_irrat_1),A) ) ) ) ) ).
fof(t26_irrat_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(B)
& k2_seq_2(B) = A )
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,E),k6_xcmplx_0(A,C)) ) ) ) ) ) ) ) ).
fof(t27_irrat_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,E),k6_xcmplx_0(A,C)) ) ) ) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(C,D)
& ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),A) ) )
| ( v4_seq_2(B)
& k2_seq_2(B) = A ) ) ) ) ) ).
fof(t28_irrat_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_series_1(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),C),k6_xcmplx_0(k2_series_1(A),B)) ) ) ) ) ) ).
fof(t29_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k4_irrat_1,A),k2_series_1(k5_irrat_1)) ) ) ).
fof(t30_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v1_series_1(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,C)) ) )
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A),k2_series_1(B)) ) ) ) ).
fof(t31_irrat_1,axiom,
( v4_seq_2(k4_irrat_1)
& k2_seq_2(k4_irrat_1) = k2_series_1(k5_irrat_1) ) ).
fof(d6_irrat_1,axiom,
k7_power = k2_series_1(k5_irrat_1) ).
fof(d7_irrat_1,axiom,
k7_power = k28_sin_cos(np__1) ).
fof(t32_irrat_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( v1_rat_1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,B)
& v1_int_1(k3_xcmplx_0(k11_newton(B),A)) ) ) ) ) ).
fof(t33_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k3_xcmplx_0(k11_newton(A),k2_seq_1(k5_numbers,k1_numbers,k5_irrat_1,B)) = k7_xcmplx_0(k11_newton(A),k11_newton(B)) ) ) ).
fof(t34_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k7_xcmplx_0(k11_newton(A),k11_newton(B)),np__0) ) ) ).
fof(t35_irrat_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( v1_series_1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),np__0) )
& r1_xreal_0(k2_series_1(A),np__0) ) ) ).
fof(t36_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k3_xcmplx_0(k11_newton(A),k2_series_1(k1_seqm_3(k5_irrat_1,k1_nat_1(A,np__1)))),np__0) ) ).
fof(t37_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> v1_int_1(k7_xcmplx_0(k11_newton(B),k11_newton(A))) ) ) ) ).
fof(t38_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v1_int_1(k3_xcmplx_0(k11_newton(A),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k5_irrat_1),A))) ) ).
fof(t39_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( v1_xreal_0(C)
=> ( C = k7_xcmplx_0(np__1,k1_nat_1(A,np__1))
=> r1_xreal_0(k7_xcmplx_0(k11_newton(A),k11_newton(k1_nat_1(k1_nat_1(A,B),np__1))),k3_power(C,k1_nat_1(B,np__1))) ) ) ) ) ).
fof(t40_irrat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ( B = k7_xcmplx_0(np__1,k1_nat_1(A,np__1))
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(k3_xcmplx_0(k11_newton(A),k2_series_1(k1_seqm_3(k5_irrat_1,k1_nat_1(A,np__1)))),k7_xcmplx_0(B,k6_xcmplx_0(np__1,B))) ) ) ) ) ).
fof(t41_irrat_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r1_xreal_0(np__2,B)
& A = k7_xcmplx_0(np__1,k2_xcmplx_0(B,np__1))
& r1_xreal_0(np__1,k7_xcmplx_0(A,k6_xcmplx_0(np__1,A))) ) ) ) ).
fof(t42_irrat_1,axiom,
~ v1_rat_1(k7_power) ).
fof(s1_irrat_1,axiom,
( ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = f1_s1_irrat_1(B) ) )
& ! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = f1_s1_irrat_1(C) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = f1_s1_irrat_1(C) ) )
=> A = B ) ) ) ) ).
fof(dt_k1_irrat_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k1_irrat_1(A))
& v1_funct_2(k1_irrat_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k1_irrat_1(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k2_irrat_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k2_irrat_1(A))
& v1_funct_2(k2_irrat_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k2_irrat_1(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k3_irrat_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k3_irrat_1(A))
& v1_funct_2(k3_irrat_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k3_irrat_1(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k4_irrat_1,axiom,
( v1_funct_1(k4_irrat_1)
& v1_funct_2(k4_irrat_1,k5_numbers,k1_numbers)
& m2_relset_1(k4_irrat_1,k5_numbers,k1_numbers) ) ).
fof(dt_k5_irrat_1,axiom,
( v1_funct_1(k5_irrat_1)
& v1_funct_2(k5_irrat_1,k5_numbers,k1_numbers)
& m2_relset_1(k5_irrat_1,k5_numbers,k1_numbers) ) ).
%------------------------------------------------------------------------------