fof(t1_birkhoff,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v2_relat_1(C) & m1_pboole(C,u1_struct_0(A)) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),C,u4_msualg_1(A,B)) => r2_pboole(u1_struct_0(A),k2_mssubfam(u1_struct_0(A),D),k2_mssubfam(u1_struct_0(A),k1_birkhoff(A,C,B,D))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d5_pboole,d3_tarski,t13_mssubfam,d5_funct_1,t14_extens_1,t13_extens_1,d3_msualg_3,d5_funct_1,d1_funct_2,d1_funct_2,d23_pboole,d5_pboole,t23_funct_1,t2_msualg_3,d1_birkhoff,d1_msafree,t72_funct_1,d5_funct_1,t13_mssubfam]), [file(birkhoff,t1_birkhoff),interesting(1.00)]). fof(s12_birkhoff,theorem, ( ( ! [A] : ( ( v5_msualg_1(A,f1_s12_birkhoff) & l3_msualg_1(A,f1_s12_birkhoff) ) => ! [B] : ( ( v5_msualg_1(B,f1_s12_birkhoff) & l3_msualg_1(B,f1_s12_birkhoff) ) => ( ( r6_msualg_3(f1_s12_birkhoff,A,B) & p1_s12_birkhoff(A) ) => p1_s12_birkhoff(B) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s12_birkhoff) & l3_msualg_1(A,f1_s12_birkhoff) ) => ! [B] : ( ( v4_msualg_1(B,f1_s12_birkhoff) & v5_msualg_1(B,f1_s12_birkhoff) & m1_msualg_2(B,f1_s12_birkhoff,A) ) => ( p1_s12_birkhoff(A) => p1_s12_birkhoff(B) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s12_birkhoff) & l3_msualg_1(A,f1_s12_birkhoff) ) => ! [B] : ( ( v3_msualg_4(B,f1_s12_birkhoff,A) & v4_msualg_4(B,f1_s12_birkhoff,A) & m1_msualg_4(B,u1_struct_0(f1_s12_birkhoff),u4_msualg_1(f1_s12_birkhoff,A),u4_msualg_1(f1_s12_birkhoff,A)) ) => ( p1_s12_birkhoff(A) => p1_s12_birkhoff(k14_msualg_4(f1_s12_birkhoff,A,B)) ) ) ) & ! [A,B] : ( m2_pralg_2(B,A,f1_s12_birkhoff) => ( ! [C] : ~ ( r2_hidden(C,A) & ! [D] : ( l3_msualg_1(D,f1_s12_birkhoff) => ~ ( D = k1_funct_1(B,C) & p1_s12_birkhoff(D) ) ) ) => p1_s12_birkhoff(k15_pralg_2(A,f1_s12_birkhoff,B)) ) ) ) => ? [A] : ( m4_pboole(A,u1_struct_0(f1_s12_birkhoff),k4_equation(f1_s12_birkhoff)) & ! [B] : ( ( v5_msualg_1(B,f1_s12_birkhoff) & l3_msualg_1(B,f1_s12_birkhoff) ) => ( p1_s12_birkhoff(B) <=> r2_equation(f1_s12_birkhoff,B,A) ) ) ) ), inference(mizar_proof,[status(thm)],[s3_pboole,d5_pboole,d23_pboole,d3_tarski,d6_equation,d5_equation,d5_equation,t36_equation,t34_equation,t40_equation,s2_birkhoff,d6_equation,d6_equation,s11_birkhoff,t34_equation,s7_birkhoff,s9_birkhoff]), [file(birkhoff,s12_birkhoff),interesting(1.00)]). fof(s8_birkhoff,theorem, ( ( ? [A] : ( m3_pboole(A,u1_struct_0(f1_s8_birkhoff),u4_msualg_1(f1_s8_birkhoff,f2_s8_birkhoff),u4_msualg_1(f1_s8_birkhoff,f3_s8_birkhoff)) & r2_msualg_3(f1_s8_birkhoff,f2_s8_birkhoff,f3_s8_birkhoff,A) ) & p1_s8_birkhoff(f2_s8_birkhoff) & ! [A] : ( ( v5_msualg_1(A,f1_s8_birkhoff) & l3_msualg_1(A,f1_s8_birkhoff) ) => ! [B] : ( ( v5_msualg_1(B,f1_s8_birkhoff) & l3_msualg_1(B,f1_s8_birkhoff) ) => ( ( r6_msualg_3(f1_s8_birkhoff,A,B) & p1_s8_birkhoff(A) ) => p1_s8_birkhoff(B) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s8_birkhoff) & l3_msualg_1(A,f1_s8_birkhoff) ) => ! [B] : ( ( v3_msualg_4(B,f1_s8_birkhoff,A) & v4_msualg_4(B,f1_s8_birkhoff,A) & m1_msualg_4(B,u1_struct_0(f1_s8_birkhoff),u4_msualg_1(f1_s8_birkhoff,A),u4_msualg_1(f1_s8_birkhoff,A)) ) => ( p1_s8_birkhoff(A) => p1_s8_birkhoff(k14_msualg_4(f1_s8_birkhoff,A,B)) ) ) ) ) => p1_s8_birkhoff(f3_s8_birkhoff) ), inference(mizar_proof,[status(thm)],[t6_msualg_4]), [file(birkhoff,s8_birkhoff),interesting(0.59)]). fof(s2_birkhoff,theorem, ( ( ! [A] : ( ( v5_msualg_1(A,f1_s2_birkhoff) & l3_msualg_1(A,f1_s2_birkhoff) ) => ! [B] : ( ( v5_msualg_1(B,f1_s2_birkhoff) & l3_msualg_1(B,f1_s2_birkhoff) ) => ( ( r6_msualg_3(f1_s2_birkhoff,A,B) & p1_s2_birkhoff(A) ) => p1_s2_birkhoff(B) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s2_birkhoff) & l3_msualg_1(A,f1_s2_birkhoff) ) => ! [B] : ( ( v4_msualg_1(B,f1_s2_birkhoff) & v5_msualg_1(B,f1_s2_birkhoff) & m1_msualg_2(B,f1_s2_birkhoff,A) ) => ( p1_s2_birkhoff(A) => p1_s2_birkhoff(B) ) ) ) & ! [A,B] : ( m2_pralg_2(B,A,f1_s2_birkhoff) => ( ! [C] : ~ ( r2_hidden(C,A) & ! [D] : ( l3_msualg_1(D,f1_s2_birkhoff) => ~ ( D = k1_funct_1(B,C) & p1_s2_birkhoff(D) ) ) ) => p1_s2_birkhoff(k15_pralg_2(A,f1_s2_birkhoff,B)) ) ) ) => ? [A] : ( v4_msualg_1(A,f1_s2_birkhoff) & v5_msualg_1(A,f1_s2_birkhoff) & l3_msualg_1(A,f1_s2_birkhoff) & ? [B] : ( m3_pboole(B,u1_struct_0(f1_s2_birkhoff),f2_s2_birkhoff,u4_msualg_1(f1_s2_birkhoff,A)) & p1_s2_birkhoff(A) & ! [C] : ( ( v5_msualg_1(C,f1_s2_birkhoff) & l3_msualg_1(C,f1_s2_birkhoff) ) => ! [D] : ( m3_pboole(D,u1_struct_0(f1_s2_birkhoff),f2_s2_birkhoff,u4_msualg_1(f1_s2_birkhoff,C)) => ~ ( p1_s2_birkhoff(C) & ! [E] : ( m3_pboole(E,u1_struct_0(f1_s2_birkhoff),u4_msualg_1(f1_s2_birkhoff,A),u4_msualg_1(f1_s2_birkhoff,C)) => ~ ( r1_msualg_3(f1_s2_birkhoff,A,C,E) & r6_pboole(u1_struct_0(f1_s2_birkhoff),k3_msualg_3(u1_struct_0(f1_s2_birkhoff),f2_s2_birkhoff,u4_msualg_1(f1_s2_birkhoff,A),u4_msualg_1(f1_s2_birkhoff,C),B,E),D) & ! [F] : ( m3_pboole(F,u1_struct_0(f1_s2_birkhoff),u4_msualg_1(f1_s2_birkhoff,A),u4_msualg_1(f1_s2_birkhoff,C)) => ( ( r1_msualg_3(f1_s2_birkhoff,A,C,F) & r6_pboole(u1_struct_0(f1_s2_birkhoff),k3_msualg_3(u1_struct_0(f1_s2_birkhoff),f2_s2_birkhoff,u4_msualg_1(f1_s2_birkhoff,A),u4_msualg_1(f1_s2_birkhoff,C),B,F),D) ) => r6_pboole(u1_struct_0(f1_s2_birkhoff),E,F) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[s1_birkhoff,d5_msafree,t14_extens_1,t13_extens_1,t12_msualg_9,t13_autalg_1,t5_msualg_3,t3_msualg_3,t8_extens_1,t13_autalg_1,t16_extens_1,d10_msualg_3,d10_msualg_3,t10_msualg_3,t13_autalg_1,t13_autalg_1,t13_autalg_1,t13_autalg_1,t5_msualg_3,t5_msualg_3,t3_msualg_3,t3_msualg_3,t8_extens_1,t8_extens_1,t18_extens_1,t16_extens_1]), [file(birkhoff,s2_birkhoff),interesting(0.47)]). fof(s6_birkhoff,theorem, ( ( ! [A] : ( ( v5_msualg_1(A,f1_s6_birkhoff) & l3_msualg_1(A,f1_s6_birkhoff) ) => ! [B] : ( m3_pboole(B,u1_struct_0(f1_s6_birkhoff),k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers),u4_msualg_1(f1_s6_birkhoff,A)) => ~ ( p1_s6_birkhoff(A) & ! [C] : ( m3_pboole(C,u1_struct_0(f1_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,f2_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,A)) => ~ ( r1_msualg_3(f1_s6_birkhoff,f2_s6_birkhoff,A,C) & r6_pboole(u1_struct_0(f1_s6_birkhoff),k3_msualg_3(u1_struct_0(f1_s6_birkhoff),k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers),u4_msualg_1(f1_s6_birkhoff,f2_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,A),f3_s6_birkhoff,C),B) & ! [D] : ( m3_pboole(D,u1_struct_0(f1_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,f2_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,A)) => ( ( r1_msualg_3(f1_s6_birkhoff,f2_s6_birkhoff,A,D) & r6_pboole(u1_struct_0(f1_s6_birkhoff),k3_msualg_3(u1_struct_0(f1_s6_birkhoff),k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers),u4_msualg_1(f1_s6_birkhoff,f2_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,A),f3_s6_birkhoff,D),B) ) => r6_pboole(u1_struct_0(f1_s6_birkhoff),C,D) ) ) ) ) ) ) ) & p1_s6_birkhoff(f2_s6_birkhoff) & ! [A] : ( ( v5_msualg_1(A,f1_s6_birkhoff) & l3_msualg_1(A,f1_s6_birkhoff) ) => ! [B] : ( ( v4_msualg_1(B,f1_s6_birkhoff) & v5_msualg_1(B,f1_s6_birkhoff) & m1_msualg_2(B,f1_s6_birkhoff,A) ) => ( p1_s6_birkhoff(A) => p1_s6_birkhoff(B) ) ) ) ) => r2_msualg_3(f1_s6_birkhoff,k11_msafree(f1_s6_birkhoff,k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers)),f2_s6_birkhoff,k1_birkhoff(f1_s6_birkhoff,k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers),f2_s6_birkhoff,f3_s6_birkhoff)) ), inference(mizar_proof,[status(thm)],[t9_msafree2,s5_birkhoff,d3_pzfmisc1,d16_pboole,t17_mssubfam,t14_equation,t1_birkhoff,t13_equation,t15_pboole,d1_birkhoff,t25_equation]), [file(birkhoff,s6_birkhoff),interesting(0.47)]). fof(s7_birkhoff,theorem, ( ( p2_s7_birkhoff(f2_s7_birkhoff) & p1_s7_birkhoff(f3_s7_birkhoff) & ! [A] : ( ( v5_msualg_1(A,f1_s7_birkhoff) & l3_msualg_1(A,f1_s7_birkhoff) ) => ! [B] : ( m3_pboole(B,u1_struct_0(f1_s7_birkhoff),k2_pre_circ(u1_struct_0(f1_s7_birkhoff),k5_numbers),u4_msualg_1(f1_s7_birkhoff,A)) => ~ ( p2_s7_birkhoff(A) & ! [C] : ( m3_pboole(C,u1_struct_0(f1_s7_birkhoff),u4_msualg_1(f1_s7_birkhoff,f3_s7_birkhoff),u4_msualg_1(f1_s7_birkhoff,A)) => ~ ( r1_msualg_3(f1_s7_birkhoff,f3_s7_birkhoff,A,C) & r6_pboole(u1_struct_0(f1_s7_birkhoff),B,k3_msualg_3(u1_struct_0(f1_s7_birkhoff),k2_pre_circ(u1_struct_0(f1_s7_birkhoff),k5_numbers),u4_msualg_1(f1_s7_birkhoff,f3_s7_birkhoff),u4_msualg_1(f1_s7_birkhoff,A),f4_s7_birkhoff,C)) ) ) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s7_birkhoff) & l3_msualg_1(A,f1_s7_birkhoff) ) => ! [B] : ( ( v5_msualg_1(B,f1_s7_birkhoff) & l3_msualg_1(B,f1_s7_birkhoff) ) => ( ( r6_msualg_3(f1_s7_birkhoff,A,B) & p1_s7_birkhoff(A) ) => p1_s7_birkhoff(B) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s7_birkhoff) & l3_msualg_1(A,f1_s7_birkhoff) ) => ! [B] : ( ( v3_msualg_4(B,f1_s7_birkhoff,A) & v4_msualg_4(B,f1_s7_birkhoff,A) & m1_msualg_4(B,u1_struct_0(f1_s7_birkhoff),u4_msualg_1(f1_s7_birkhoff,A),u4_msualg_1(f1_s7_birkhoff,A)) ) => ( p1_s7_birkhoff(A) => p1_s7_birkhoff(k14_msualg_4(f1_s7_birkhoff,A,B)) ) ) ) ) => p1_s7_birkhoff(f2_s7_birkhoff) ), inference(mizar_proof,[status(thm)],[d10_msafree2,t8_msualg_9,t13_msualg_9,d3_pzfmisc1,d16_pboole,t9_extens_1,d10_msualg_2,d23_pboole,d5_pboole,d23_pboole,d3_msualg_3,t13_funcop_1,d3_tarski,d5_funct_1,d1_funct_2,d1_funct_2,d5_funct_1,d1_funct_2,d1_funct_2,t27_funct_1,t2_msualg_3,t21_funct_2,d12_funct_1,d25_pboole,d14_msualg_3,t21_extens_1,t22_extens_1,t22_msualg_2,t21_msscyc_1,t3_msafree,t8_msualg_2,t19_msualg_3,t6_msualg_4]), [file(birkhoff,s7_birkhoff),interesting(0.46)]). fof(s9_birkhoff,theorem, ( ( ! [A] : ( ( v4_msualg_1(A,f1_s9_birkhoff) & v5_msualg_1(A,f1_s9_birkhoff) & v3_msafree2(A,f1_s9_birkhoff) & m1_msualg_2(A,f1_s9_birkhoff,f2_s9_birkhoff) ) => p1_s9_birkhoff(A) ) & ! [A] : ( ( v5_msualg_1(A,f1_s9_birkhoff) & l3_msualg_1(A,f1_s9_birkhoff) ) => ! [B] : ( ( v5_msualg_1(B,f1_s9_birkhoff) & l3_msualg_1(B,f1_s9_birkhoff) ) => ( ( r6_msualg_3(f1_s9_birkhoff,A,B) & p1_s9_birkhoff(A) ) => p1_s9_birkhoff(B) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s9_birkhoff) & l3_msualg_1(A,f1_s9_birkhoff) ) => ! [B] : ( ( v4_msualg_1(B,f1_s9_birkhoff) & v5_msualg_1(B,f1_s9_birkhoff) & m1_msualg_2(B,f1_s9_birkhoff,A) ) => ( p1_s9_birkhoff(A) => p1_s9_birkhoff(B) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s9_birkhoff) & l3_msualg_1(A,f1_s9_birkhoff) ) => ! [B] : ( ( v3_msualg_4(B,f1_s9_birkhoff,A) & v4_msualg_4(B,f1_s9_birkhoff,A) & m1_msualg_4(B,u1_struct_0(f1_s9_birkhoff),u4_msualg_1(f1_s9_birkhoff,A),u4_msualg_1(f1_s9_birkhoff,A)) ) => ( p1_s9_birkhoff(A) => p1_s9_birkhoff(k14_msualg_4(f1_s9_birkhoff,A,B)) ) ) ) & ! [A,B] : ( m2_pralg_2(B,A,f1_s9_birkhoff) => ( ! [C] : ~ ( r2_hidden(C,A) & ! [D] : ( l3_msualg_1(D,f1_s9_birkhoff) => ~ ( D = k1_funct_1(B,C) & p1_s9_birkhoff(D) ) ) ) => p1_s9_birkhoff(k15_pralg_2(A,f1_s9_birkhoff,B)) ) ) ) => p1_s9_birkhoff(f2_s9_birkhoff) ), inference(mizar_proof,[status(thm)],[d20_msualg_2,s3_pboole,d12_pralg_2,t29_equation,s8_birkhoff]), [file(birkhoff,s9_birkhoff),interesting(0.46)]). fof(s5_birkhoff,theorem, ( ( ! [A] : ( ( v5_msualg_1(A,f1_s5_birkhoff) & l3_msualg_1(A,f1_s5_birkhoff) ) => ! [B] : ( m3_pboole(B,u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,u4_msualg_1(f1_s5_birkhoff,A)) => ~ ( p1_s5_birkhoff(A) & ! [C] : ( m3_pboole(C,u1_struct_0(f1_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,f3_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,A)) => ~ ( r1_msualg_3(f1_s5_birkhoff,f3_s5_birkhoff,A,C) & r6_pboole(u1_struct_0(f1_s5_birkhoff),k3_msualg_3(u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,u4_msualg_1(f1_s5_birkhoff,f3_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,A),f4_s5_birkhoff,C),B) & ! [D] : ( m3_pboole(D,u1_struct_0(f1_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,f3_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,A)) => ( ( r1_msualg_3(f1_s5_birkhoff,f3_s5_birkhoff,A,D) & r6_pboole(u1_struct_0(f1_s5_birkhoff),k3_msualg_3(u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,u4_msualg_1(f1_s5_birkhoff,f3_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,A),f4_s5_birkhoff,D),B) ) => r6_pboole(u1_struct_0(f1_s5_birkhoff),C,D) ) ) ) ) ) ) ) & p1_s5_birkhoff(f3_s5_birkhoff) & ! [A] : ( ( v5_msualg_1(A,f1_s5_birkhoff) & l3_msualg_1(A,f1_s5_birkhoff) ) => ! [B] : ( ( v4_msualg_1(B,f1_s5_birkhoff) & v5_msualg_1(B,f1_s5_birkhoff) & m1_msualg_2(B,f1_s5_birkhoff,A) ) => ( p1_s5_birkhoff(A) => p1_s5_birkhoff(B) ) ) ) ) => ( v2_relat_1(k14_pboole(u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,f4_s5_birkhoff)) & m1_msafree(k14_pboole(u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,f4_s5_birkhoff),f1_s5_birkhoff,f3_s5_birkhoff) ) ), inference(mizar_proof,[status(thm)],[d16_pboole,d16_pboole,d18_pboole,d16_pboole,d1_funct_2,d25_pboole,t151_relat_1,d5_pboole,d23_pboole,d18_pboole,d25_pboole,d10_msualg_2,t9_extens_1,d3_pzfmisc1,d16_pboole,d3_pzfmisc1,d16_pboole,t17_mssubfam,t14_equation,d18_msualg_2,d23_pboole,t5_equation,t13_autalg_1,t22_msualg_3,d11_msualg_3,t10_msualg_3,t9_msualg_3,t4_msualg_3,t4_msualg_3,t24_autalg_1,t2_equation,t6_msualg_2,t16_equation,t10_equation,t10_msualg_2,t3_msafree]), [file(birkhoff,s5_birkhoff),interesting(0.45)]). fof(s3_birkhoff,theorem, ( ( p1_s3_birkhoff(f3_s3_birkhoff) & ! [A] : ( ( v5_msualg_1(A,f1_s3_birkhoff) & l3_msualg_1(A,f1_s3_birkhoff) ) => ! [B] : ( m3_pboole(B,u1_struct_0(f1_s3_birkhoff),k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers),u4_msualg_1(f1_s3_birkhoff,A)) => ~ ( p1_s3_birkhoff(A) & ! [C] : ( m3_pboole(C,u1_struct_0(f1_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,f2_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,A)) => ~ ( r1_msualg_3(f1_s3_birkhoff,f2_s3_birkhoff,A,C) & r6_pboole(u1_struct_0(f1_s3_birkhoff),B,k3_msualg_3(u1_struct_0(f1_s3_birkhoff),k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers),u4_msualg_1(f1_s3_birkhoff,f2_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,A),f4_s3_birkhoff,C)) ) ) ) ) ) ) => ? [A] : ( m3_pboole(A,u1_struct_0(f1_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,f2_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,f3_s3_birkhoff)) & r1_msualg_3(f1_s3_birkhoff,f2_s3_birkhoff,f3_s3_birkhoff,A) & r6_pboole(u1_struct_0(f1_s3_birkhoff),k1_birkhoff(f1_s3_birkhoff,k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers),f3_s3_birkhoff,f5_s3_birkhoff),k3_msualg_3(u1_struct_0(f1_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,k11_msafree(f1_s3_birkhoff,k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers))),u4_msualg_1(f1_s3_birkhoff,f2_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,f3_s3_birkhoff),k1_birkhoff(f1_s3_birkhoff,k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers),f2_s3_birkhoff,f4_s3_birkhoff),A)) ) ), inference(mizar_proof,[status(thm)],[d1_birkhoff,d1_birkhoff,t10_msualg_3,d1_birkhoff,t13_autalg_1,d1_birkhoff,t8_extens_1,t18_extens_1]), [file(birkhoff,s3_birkhoff),interesting(0.43)]). fof(s11_birkhoff,theorem, ( ( ! [A] : ( ( v5_msualg_1(A,f1_s11_birkhoff) & l3_msualg_1(A,f1_s11_birkhoff) ) => ( p2_s11_birkhoff(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(f1_s11_birkhoff)) => ! [C] : ( m1_subset_1(C,k1_funct_1(k4_equation(f1_s11_birkhoff),B)) => ( ! [D] : ( ( v5_msualg_1(D,f1_s11_birkhoff) & l3_msualg_1(D,f1_s11_birkhoff) ) => ( p1_s11_birkhoff(D) => r1_equation(f1_s11_birkhoff,D,B,C) ) ) => r1_equation(f1_s11_birkhoff,A,B,C) ) ) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s11_birkhoff) & l3_msualg_1(A,f1_s11_birkhoff) ) => ! [B] : ( m3_pboole(B,u1_struct_0(f1_s11_birkhoff),k2_pre_circ(u1_struct_0(f1_s11_birkhoff),k5_numbers),u4_msualg_1(f1_s11_birkhoff,A)) => ~ ( p2_s11_birkhoff(A) & ! [C] : ( m3_pboole(C,u1_struct_0(f1_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,f2_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,A)) => ~ ( r1_msualg_3(f1_s11_birkhoff,f2_s11_birkhoff,A,C) & r6_pboole(u1_struct_0(f1_s11_birkhoff),k3_msualg_3(u1_struct_0(f1_s11_birkhoff),k2_pre_circ(u1_struct_0(f1_s11_birkhoff),k5_numbers),u4_msualg_1(f1_s11_birkhoff,f2_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,A),f3_s11_birkhoff,C),B) & ! [D] : ( m3_pboole(D,u1_struct_0(f1_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,f2_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,A)) => ( ( r1_msualg_3(f1_s11_birkhoff,f2_s11_birkhoff,A,D) & r6_pboole(u1_struct_0(f1_s11_birkhoff),k3_msualg_3(u1_struct_0(f1_s11_birkhoff),k2_pre_circ(u1_struct_0(f1_s11_birkhoff),k5_numbers),u4_msualg_1(f1_s11_birkhoff,f2_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,A),f3_s11_birkhoff,D),B) ) => r6_pboole(u1_struct_0(f1_s11_birkhoff),C,D) ) ) ) ) ) ) ) & p2_s11_birkhoff(f2_s11_birkhoff) & ! [A] : ( ( v5_msualg_1(A,f1_s11_birkhoff) & l3_msualg_1(A,f1_s11_birkhoff) ) => ! [B] : ( ( v5_msualg_1(B,f1_s11_birkhoff) & l3_msualg_1(B,f1_s11_birkhoff) ) => ( ( r6_msualg_3(f1_s11_birkhoff,A,B) & p1_s11_birkhoff(A) ) => p1_s11_birkhoff(B) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s11_birkhoff) & l3_msualg_1(A,f1_s11_birkhoff) ) => ! [B] : ( ( v4_msualg_1(B,f1_s11_birkhoff) & v5_msualg_1(B,f1_s11_birkhoff) & m1_msualg_2(B,f1_s11_birkhoff,A) ) => ( p1_s11_birkhoff(A) => p1_s11_birkhoff(B) ) ) ) & ! [A,B] : ( m2_pralg_2(B,A,f1_s11_birkhoff) => ( ! [C] : ~ ( r2_hidden(C,A) & ! [D] : ( l3_msualg_1(D,f1_s11_birkhoff) => ~ ( D = k1_funct_1(B,C) & p1_s11_birkhoff(D) ) ) ) => p1_s11_birkhoff(k15_pralg_2(A,f1_s11_birkhoff,B)) ) ) ) => p1_s11_birkhoff(f2_s11_birkhoff) ), inference(mizar_proof,[status(thm)],[s2_birkhoff,s10_birkhoff,d1_birkhoff,s3_birkhoff,d11_msualg_3,d8_funct_1,t33_equation,s6_birkhoff,d10_msualg_3,d1_funct_2,d3_msualg_3,d5_funct_1,d5_funct_1,t2_msualg_3,t21_funct_2,t21_funct_2,t2_msualg_3,s4_birkhoff,t7_mcart_1,d5_equation,t7_mcart_1,t1_msualg_3,t17_msualg_9,t17_msualg_3]), [file(birkhoff,s11_birkhoff),interesting(0.41)]). fof(s4_birkhoff,theorem, ( ( k1_funct_1(k1_msualg_3(u1_struct_0(f1_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,k11_msafree(f1_s4_birkhoff,k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers))),u4_msualg_1(f1_s4_birkhoff,f2_s4_birkhoff),k1_birkhoff(f1_s4_birkhoff,k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers),f2_s4_birkhoff,f3_s4_birkhoff),f4_s4_birkhoff),f5_s4_birkhoff) = k1_funct_1(k1_msualg_3(u1_struct_0(f1_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,k11_msafree(f1_s4_birkhoff,k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers))),u4_msualg_1(f1_s4_birkhoff,f2_s4_birkhoff),k1_birkhoff(f1_s4_birkhoff,k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers),f2_s4_birkhoff,f3_s4_birkhoff),f4_s4_birkhoff),f6_s4_birkhoff) & ! [A] : ( ( v5_msualg_1(A,f1_s4_birkhoff) & l3_msualg_1(A,f1_s4_birkhoff) ) => ! [B] : ( m3_pboole(B,u1_struct_0(f1_s4_birkhoff),k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers),u4_msualg_1(f1_s4_birkhoff,A)) => ~ ( p1_s4_birkhoff(A) & ! [C] : ( m3_pboole(C,u1_struct_0(f1_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,f2_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,A)) => ~ ( r1_msualg_3(f1_s4_birkhoff,f2_s4_birkhoff,A,C) & r6_pboole(u1_struct_0(f1_s4_birkhoff),B,k3_msualg_3(u1_struct_0(f1_s4_birkhoff),k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers),u4_msualg_1(f1_s4_birkhoff,f2_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,A),f3_s4_birkhoff,C)) ) ) ) ) ) ) => ! [A] : ( ( v5_msualg_1(A,f1_s4_birkhoff) & l3_msualg_1(A,f1_s4_birkhoff) ) => ( p1_s4_birkhoff(A) => r1_equation(f1_s4_birkhoff,A,f4_s4_birkhoff,k5_equation(f1_s4_birkhoff,f4_s4_birkhoff,f5_s4_birkhoff,f6_s4_birkhoff)) ) ) ), inference(mizar_proof,[status(thm)],[d5_equation,s3_birkhoff,d1_birkhoff,t14_extens_1,t13_extens_1,t12_msualg_9,d1_birkhoff,t13_autalg_1,t5_msualg_3,t3_msualg_3,t18_extens_1,t2_msualg_3,t21_funct_2,t21_funct_2,t2_msualg_3,t7_mcart_1,t7_mcart_1]), [file(birkhoff,s4_birkhoff),interesting(0.28)]). fof(s3_pboole,theorem, ( ! [A] : ~ ( r2_hidden(A,f1_s3_pboole) & ! [B] : ~ p1_s3_pboole(A,B) ) => ? [A] : ( m1_pboole(A,f1_s3_pboole) & ! [B] : ( r2_hidden(B,f1_s3_pboole) => p1_s3_pboole(B,k1_funct_1(A,B)) ) ) ), file(pboole,s3_pboole), [interesting(0.00)]). fof(d5_pboole,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ( r2_pboole(A,B,C) <=> ! [D] : ( r2_hidden(D,A) => r1_tarski(k1_funct_1(B,D),k1_funct_1(C,D)) ) ) ) ) ), file(pboole,d5_pboole), [interesting(0.00)]). fof(d23_pboole,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ( m4_pboole(C,A,B) <=> r2_pboole(A,C,B) ) ) ) ), file(pboole,d23_pboole), [interesting(0.00)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.00)]). fof(d6_equation,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( m4_pboole(C,u1_struct_0(A),k4_equation(A)) => ( r2_equation(A,B,C) <=> ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,k1_funct_1(k4_equation(A),D)) => ( r2_hidden(E,k1_funct_1(C,D)) => r1_equation(A,B,D,E) ) ) ) ) ) ) ) ), file(equation,d6_equation), [interesting(0.00)]). fof(d5_equation,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,k1_funct_1(k4_equation(A),C)) => ( r1_equation(A,B,C,D) <=> ! [E] : ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,B)) => ( r1_msualg_3(A,k3_equation(A),B,E) => k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,B),E,C),k1_mcart_1(D)) = k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,B),E,C),k2_mcart_1(D)) ) ) ) ) ) ) ) ), file(equation,d5_equation), [interesting(0.00)]). fof(t36_equation,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m4_pboole(D,u1_struct_0(A),k4_equation(A)) => ( ( r6_msualg_3(A,B,C) & r2_equation(A,B,D) ) => r2_equation(A,C,D) ) ) ) ) ) ), file(equation,t36_equation), [interesting(0.00)]). fof(t34_equation,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( m4_pboole(C,u1_struct_0(A),k4_equation(A)) => ! [D] : ( ( v4_msualg_1(D,A) & v5_msualg_1(D,A) & m1_msualg_2(D,A,B) ) => ( r2_equation(A,B,C) => r2_equation(A,D,C) ) ) ) ) ) ), file(equation,t34_equation), [interesting(0.00)]). fof(t40_equation,theorem,( ! [A,B] : ( ( ~ v3_struct_0(B) & ~ v2_msualg_1(B) & l1_msualg_1(B) ) => ! [C] : ( m4_pboole(C,u1_struct_0(B),k4_equation(B)) => ! [D] : ( m2_pralg_2(D,A,B) => ( ! [E] : ~ ( r2_hidden(E,A) & ! [F] : ( l3_msualg_1(F,B) => ~ ( F = k1_funct_1(D,E) & r2_equation(B,F,C) ) ) ) => r2_equation(B,k15_pralg_2(A,B,D),C) ) ) ) ) ), file(equation,t40_equation), [interesting(0.00)]). fof(s7_domain_1,theorem,( m1_subset_1(a_0_0_domain_1,k1_zfmisc_1(f1_s7_domain_1)) ), file(domain_1,s7_domain_1), [interesting(0.00)]). fof(d6_msualg_5,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v3_lattices(C) & m2_nat_lat(C,k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B))) ) => ( C = k6_msualg_5(A,B) <=> ! [D] : ( r2_hidden(D,u1_struct_0(C)) <=> ( v3_msualg_4(D,A,B) & v4_msualg_4(D,A,B) & m1_msualg_4(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) ) ) ) ) ) ), file(msualg_5,d6_msualg_5), [interesting(0.00)]). fof(t6_msualg_9,theorem,( ! [A,B] : ? [C] : ( m1_pboole(C,A) & r6_pboole(A,k1_pzfmisc1(A,C),k2_pre_circ(A,k1_tarski(B))) ) ), file(msualg_9,t6_msualg_9), [interesting(0.00)]). fof(d17_pralg_2,definition,( ! [A,B] : ( ( ~ v3_struct_0(B) & l1_msualg_1(B) ) => ! [C] : ( m2_pralg_2(C,A,B) => ! [D] : ( m1_pboole(D,u1_struct_0(B)) => ( D = k11_pralg_2(A,B,C) <=> ! [E] : ( m1_subset_1(E,u1_struct_0(B)) => k1_funct_1(D,E) = k4_card_3(k10_pralg_2(A,B,E,C)) ) ) ) ) ) ), file(pralg_2,d17_pralg_2), [interesting(0.00)]). fof(t19_card_3,theorem,( k4_card_3(k1_xboole_0) = k1_tarski(k1_xboole_0) ), file(card_3,t19_card_3), [interesting(0.00)]). fof(d16_pralg_2,definition,( ! [A,B] : ( ( ~ v3_struct_0(B) & l1_msualg_1(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m2_pralg_2(D,A,B) => ! [E] : ( m1_pboole(E,A) => ( ( A != k1_xboole_0 => ( E = k10_pralg_2(A,B,C,D) <=> ! [F] : ~ ( r2_hidden(F,A) & ! [G] : ( l3_msualg_1(G,B) => ~ ( G = k1_funct_1(D,F) & k1_funct_1(E,F) = k1_funct_1(u4_msualg_1(B,G),C) ) ) ) ) ) & ( A = k1_xboole_0 => ( E = k10_pralg_2(A,B,C,D) <=> E = k1_xboole_0 ) ) ) ) ) ) ) ), file(pralg_2,d16_pralg_2), [interesting(0.00)]). fof(t13_funcop_1,theorem,( ! [A,B,C] : ( r2_hidden(B,A) => k1_funct_1(k2_funcop_1(A,C),B) = C ) ), file(funcop_1,t13_funcop_1), [interesting(0.00)]). fof(t3_pboole,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ( ! [D] : ( r2_hidden(D,A) => k1_funct_1(B,D) = k1_funct_1(C,D) ) => B = C ) ) ) ), file(pboole,t3_pboole), [interesting(0.00)]). fof(t27_msualg_9,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ( ? [C] : ( m1_pboole(C,u1_struct_0(A)) & r6_pboole(u1_struct_0(A),u4_msualg_1(A,B),k1_pzfmisc1(u1_struct_0(A),C)) ) => r6_msualg_3(A,B,k8_msafree2(A)) ) ) ) ), file(msualg_9,t27_msualg_9), [interesting(0.00)]). fof(t20_msualg_5,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ( v3_msualg_4(k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)),A,B) & v4_msualg_4(k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)),A,B) & m1_msualg_4(k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)),u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) ) ) ), file(msualg_5,t20_msualg_5), [interesting(0.00)]). fof(t30_msualg_9,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v3_msualg_4(C,A,B) & v4_msualg_4(C,A,B) & m1_msualg_4(C,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) => ( r6_pboole(u1_struct_0(A),C,k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B))) => r6_pboole(u1_struct_0(A),u4_msualg_1(A,k14_msualg_4(A,B,C)),k1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,B))) ) ) ) ) ), file(msualg_9,t30_msualg_9), [interesting(0.00)]). fof(t17_msualg_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ( r6_msualg_3(A,B,C) => r6_msualg_3(A,C,B) ) ) ) ) ), file(msualg_3,t17_msualg_3), [interesting(0.00)]). fof(d5_msualg_5,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_pboole(B,A) => ! [C] : ( ( ~ v3_struct_0(C) & v3_lattices(C) & v10_lattices(C) & l3_lattices(C) ) => ( C = k5_msualg_5(A,B) <=> ( ! [D] : ( r2_hidden(D,u1_struct_0(C)) <=> ( v2_msualg_4(D,A,B) & m1_msualg_4(D,A,B,B) ) ) & ! [D] : ( ( v2_msualg_4(D,A,B) & m1_msualg_4(D,A,B,B) ) => ! [E] : ( ( v2_msualg_4(E,A,B) & m1_msualg_4(E,A,B,B) ) => ( k1_binop_1(u1_lattices(C),D,E) = k3_pboole(A,D,E) & k1_binop_1(u2_lattices(C),D,E) = k4_msualg_5(A,B,D,E) ) ) ) ) ) ) ) ) ), file(msualg_5,d5_msualg_5), [interesting(0.00)]). fof(t6_msualg_7,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_pboole(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B)))) => m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B)))) ) ) ) ), file(msualg_7,t6_msualg_7), [interesting(0.00)]). fof(t9_msualg_7,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_pboole(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B)))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B)))) => ( D = C => ( v1_xboole_0(C) | ( v2_msualg_4(k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)),A,B) & m1_msualg_4(k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)),A,B,B) ) ) ) ) ) ) ) ), file(msualg_7,t9_msualg_7), [interesting(0.00)]). fof(d3_msualg_4,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_msualg_4(C,A,B,B) => ( v2_msualg_4(C,A,B) <=> ! [D,E] : ( m2_relset_1(E,k1_funct_1(B,D),k1_funct_1(B,D)) => ( ( r2_hidden(D,A) & k1_funct_1(C,D) = E ) => ( v3_relat_2(E) & v8_relat_2(E) & v1_partfun1(E,k1_funct_1(B,D),k1_funct_1(B,D)) & m2_relset_1(E,k1_funct_1(B,D),k1_funct_1(B,D)) ) ) ) ) ) ) ), file(msualg_4,d3_msualg_4), [interesting(0.00)]). fof(d5_msualg_4,definition,( ! [A] : ( ( ~ v3_struct_0(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( m1_msualg_4(C,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) => ( v3_msualg_4(C,A,B) <=> v2_msualg_4(C,u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ) ), file(msualg_4,d5_msualg_4), [interesting(0.00)]). fof(t29_msualg_9,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_msualg_5(A,B)))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(u1_struct_0(A),k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B))))) => ( C = D => ( v3_msualg_4(k6_mssubfam(u1_struct_0(A),k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)),k5_closure2(u1_struct_0(A),k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)),D)),A,B) & v4_msualg_4(k6_mssubfam(u1_struct_0(A),k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)),k5_closure2(u1_struct_0(A),k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)),D)),A,B) & m1_msualg_4(k6_mssubfam(u1_struct_0(A),k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)),k5_closure2(u1_struct_0(A),k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)),D)),u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) ) ) ) ) ) ), file(msualg_9,t29_msualg_9), [interesting(0.00)]). fof(d12_pralg_2,definition,( ! [A,B] : ( ( ~ v3_struct_0(B) & l1_msualg_1(B) ) => ! [C] : ( m1_pboole(C,A) => ( m2_pralg_2(C,A,B) <=> ! [D] : ( r2_hidden(D,A) => ( v5_msualg_1(k1_funct_1(C,D),B) & l3_msualg_1(k1_funct_1(C,D),B) ) ) ) ) ) ), file(pralg_2,d12_pralg_2), [interesting(0.00)]). fof(t3_msualg_4,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v3_msualg_4(C,A,B) & v4_msualg_4(C,A,B) & m1_msualg_4(C,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) => r2_msualg_3(A,B,k14_msualg_4(A,B,C),k16_msualg_4(A,B,C)) ) ) ) ), file(msualg_4,t3_msualg_4), [interesting(0.00)]). fof(d10_msualg_3,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( l3_msualg_1(C,A) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( r2_msualg_3(A,B,C,D) <=> ( r1_msualg_3(A,B,C,D) & v2_msualg_3(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) ) ) ) ) ) ) ), file(msualg_3,d10_msualg_3), [interesting(0.00)]). fof(s1_msualg_9,theorem, ( ! [A] : ( m1_subset_1(A,f1_s1_msualg_9) => ? [B] : p1_s1_msualg_9(A,B) ) => ? [A] : ( m1_pboole(A,f1_s1_msualg_9) & ! [B] : ( m1_subset_1(B,f1_s1_msualg_9) => p1_s1_msualg_9(B,k1_funct_1(A,B)) ) ) ), file(msualg_9,s1_msualg_9), [interesting(0.00)]). fof(d6_funcop_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_funcop_1(A) <=> ! [B] : ( r2_hidden(B,k1_relat_1(A)) => ( v1_relat_1(k1_funct_1(A,B)) & v1_funct_1(k1_funct_1(A,B)) ) ) ) ) ), file(funcop_1,d6_funcop_1), [interesting(0.00)]). fof(d3_pboole,definition,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( m1_pboole(B,A) <=> k1_relat_1(B) = A ) ) ), file(pboole,d3_pboole), [interesting(0.00)]). fof(t30_pralg_3,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_msualg_1(B) & l1_msualg_1(B) ) => ! [C] : ( m2_pralg_2(C,A,B) => ! [D] : ( ( v5_msualg_1(D,B) & l3_msualg_1(D,B) ) => ! [E] : ( ( v1_funcop_1(E) & m1_pboole(E,A) ) => ~ ( ! [F] : ( m1_subset_1(F,A) => ? [G] : ( m3_pboole(G,u1_struct_0(B),u4_msualg_1(B,D),u4_msualg_1(B,k6_pralg_2(A,B,C,F))) & G = k1_funct_1(E,F) & r1_msualg_3(B,D,k6_pralg_2(A,B,C,F),G) ) ) & ! [F] : ( m3_pboole(F,u1_struct_0(B),u4_msualg_1(B,D),u4_msualg_1(B,k15_pralg_2(A,B,C))) => ~ ( r1_msualg_3(B,D,k15_pralg_2(A,B,C),F) & ! [G] : ( m1_subset_1(G,A) => k3_msualg_3(u1_struct_0(B),u4_msualg_1(B,D),u4_msualg_1(B,k15_pralg_2(A,B,C)),u4_msualg_1(B,k6_pralg_2(A,B,C,G)),F,k4_pralg_3(A,B,C,G)) = k1_funct_1(E,G) ) ) ) ) ) ) ) ) ) ), file(pralg_3,t30_pralg_3), [interesting(0.00)]). fof(t4_msualg_4,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( r1_msualg_3(A,B,C,D) => r3_msualg_3(A,k14_msualg_4(A,B,k18_msualg_4(A,B,C,D)),C,k20_msualg_4(A,B,C,D)) ) ) ) ) ) ), file(msualg_4,t4_msualg_4), [interesting(0.00)]). fof(t17_msualg_9,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( r3_msualg_3(A,B,C,D) => r6_msualg_3(A,B,k7_msualg_3(A,B,C,D)) ) ) ) ) ) ), file(msualg_9,t17_msualg_9), [interesting(0.00)]). fof(d2_mssubfam,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m4_pboole(C,A,k1_mboolean(A,B)) => ! [D] : ( m1_pboole(D,A) => ( D = k5_mssubfam(A,B,C) <=> ! [E] : ~ ( r2_hidden(E,A) & ! [F] : ( m1_subset_1(F,k1_zfmisc_1(k1_zfmisc_1(k1_funct_1(B,E)))) => ~ ( F = k1_funct_1(C,E) & k1_funct_1(D,E) = k8_setfam_1(k1_funct_1(B,E),F) ) ) ) ) ) ) ) ), file(mssubfam,d2_mssubfam), [interesting(0.00)]). fof(t15_closure2,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) => ( ~ v1_xboole_0(C) => ! [D] : ( r2_hidden(D,A) => k1_funct_1(k4_closure2(A,B,C),D) = a_4_0_closure2(A,B,C,D) ) ) ) ) ), file(closure2,t15_closure2), [interesting(0.00)]). fof(d2_relat_1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( A = B <=> ! [C,D] : ( r2_hidden(k4_tarski(C,D),A) <=> r2_hidden(k4_tarski(C,D),B) ) ) ) ) ), file(relat_1,d2_relat_1), [interesting(0.00)]). fof(t106_zfmisc_1,theorem,( ! [A,B,C,D] : ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D)) <=> ( r2_hidden(A,C) & r2_hidden(B,D) ) ) ), file(zfmisc_1,t106_zfmisc_1), [interesting(0.00)]). fof(d21_pboole,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m1_pboole(D,A) => ( D = k11_pboole(A,B,C) <=> ! [E] : ( r2_hidden(E,A) => k1_funct_1(D,E) = k2_zfmisc_1(k1_funct_1(B,E),k1_funct_1(C,E)) ) ) ) ) ) ), file(pboole,d21_pboole), [interesting(0.00)]). fof(d20_msualg_4,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( r1_msualg_3(A,B,C,D) => ! [E] : ( ( v3_msualg_4(E,A,B) & v4_msualg_4(E,A,B) & m1_msualg_4(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) => ( E = k18_msualg_4(A,B,C,D) <=> ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => k2_msualg_4(A,B,E,F) = k17_msualg_4(A,B,C,D,F) ) ) ) ) ) ) ) ) ), file(msualg_4,d20_msualg_4), [interesting(0.00)]). fof(t2_msualg_3,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m1_pboole(D,A) => ! [E] : ( m3_pboole(E,A,B,C) => ! [F] : ( m3_pboole(F,A,C,D) => ( k1_relat_1(k13_pboole(E,F)) = A & ! [G] : ( r2_hidden(G,A) => k1_funct_1(k13_pboole(E,F),G) = k5_relat_1(k1_funct_1(E,G),k1_funct_1(F,G)) ) ) ) ) ) ) ) ), file(msualg_3,t2_msualg_3), [interesting(0.00)]). fof(t21_funct_2,theorem,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ! [E] : ( ( v1_relat_1(E) & v1_funct_1(E) ) => ( r2_hidden(C,A) => ( B = k1_xboole_0 | k1_funct_1(k5_relat_1(D,E),C) = k1_funct_1(E,k1_funct_1(D,C)) ) ) ) ) ), file(funct_2,t21_funct_2), [interesting(0.00)]). fof(d19_msualg_4,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( ( v3_relat_2(F) & v8_relat_2(F) & v1_partfun1(F,k1_funct_1(u4_msualg_1(A,B),E),k1_funct_1(u4_msualg_1(A,B),E)) & m2_relset_1(F,k1_funct_1(u4_msualg_1(A,B),E),k1_funct_1(u4_msualg_1(A,B),E)) ) => ( F = k17_msualg_4(A,B,C,D,E) <=> ! [G] : ( m1_subset_1(G,k1_funct_1(u4_msualg_1(A,B),E)) => ! [H] : ( m1_subset_1(H,k1_funct_1(u4_msualg_1(A,B),E)) => ( r2_hidden(k4_tarski(G,H),F) <=> k8_funct_2(k1_funct_1(u4_msualg_1(A,B),E),k1_funct_1(u4_msualg_1(A,C),E),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C),D,E),G) = k8_funct_2(k1_funct_1(u4_msualg_1(A,B),E),k1_funct_1(u4_msualg_1(A,C),E),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C),D,E),H) ) ) ) ) ) ) ) ) ) ) ), file(msualg_4,d19_msualg_4), [interesting(0.00)]). fof(t34_msualg_9,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v3_msualg_4(C,A,B) & v4_msualg_4(C,A,B) & m1_msualg_4(C,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,B),D)) => ! [F] : ( m1_subset_1(F,k1_funct_1(u4_msualg_1(A,B),D)) => ( k8_funct_2(k1_funct_1(u4_msualg_1(A,B),D),k1_funct_1(u4_msualg_1(A,k14_msualg_4(A,B,C)),D),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,k14_msualg_4(A,B,C)),k16_msualg_4(A,B,C),D),E) = k8_funct_2(k1_funct_1(u4_msualg_1(A,B),D),k1_funct_1(u4_msualg_1(A,k14_msualg_4(A,B,C)),D),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,k14_msualg_4(A,B,C)),k16_msualg_4(A,B,C),D),F) <=> r2_hidden(k4_tarski(E,F),k2_msualg_4(A,B,C,D)) ) ) ) ) ) ) ) ), file(msualg_9,t34_msualg_9), [interesting(0.00)]). fof(t58_setfam_1,theorem,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) => ( r2_hidden(B,A) => ( r2_hidden(B,k8_setfam_1(A,C)) <=> ! [D] : ( r2_hidden(D,C) => r2_hidden(B,D) ) ) ) ) ), file(setfam_1,t58_setfam_1), [interesting(0.00)]). fof(t7_funct_2,theorem,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( r2_hidden(C,A) => ( B = k1_xboole_0 | r2_hidden(k1_funct_1(D,C),B) ) ) ) ), file(funct_2,t7_funct_2), [interesting(0.00)]). fof(d1_closure2,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( C = k1_closure2(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> m4_pboole(D,A,B) ) ) ) ), file(closure2,d1_closure2), [interesting(0.00)]). fof(d3_pralg_3,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_msualg_1(B) & l1_msualg_1(B) ) => ! [C] : ( m2_pralg_2(C,A,B) => ! [D] : ( m1_subset_1(D,A) => ! [E] : ( m3_pboole(E,u1_struct_0(B),u4_msualg_1(B,k15_pralg_2(A,B,C)),u4_msualg_1(B,k6_pralg_2(A,B,C,D))) => ( E = k4_pralg_3(A,B,C,D) <=> ! [F] : ( m1_subset_1(F,u1_struct_0(B)) => k1_msualg_3(u1_struct_0(B),u4_msualg_1(B,k15_pralg_2(A,B,C)),u4_msualg_1(B,k6_pralg_2(A,B,C,D)),E,F) = k3_pralg_3(k10_pralg_2(A,B,F,C),D) ) ) ) ) ) ) ) ), file(pralg_3,d3_pralg_3), [interesting(0.00)]). fof(t15_msualg_9,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m2_pralg_2(D,B,A) => ! [E] : ( m1_pralg_2(E,k4_card_3(k10_pralg_2(B,A,C,D))) => ! [F] : ( m1_pralg_2(F,k4_card_3(k10_pralg_2(B,A,C,D))) => ( ! [G] : ( m1_subset_1(G,B) => k1_funct_1(k3_pralg_3(k10_pralg_2(B,A,C,D),G),E) = k1_funct_1(k3_pralg_3(k10_pralg_2(B,A,C,D),G),F) ) => r6_pboole(k1_pralg_2(k4_card_3(k10_pralg_2(B,A,C,D))),E,F) ) ) ) ) ) ) ) ), file(msualg_9,t15_msualg_9), [interesting(0.00)]). fof(t22_closure2,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) => ( r2_hidden(C,D) => r1_pboole(A,C,k5_closure2(A,B,D)) ) ) ) ) ), file(closure2,t22_closure2), [interesting(0.00)]). fof(t43_mssubfam,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m4_pboole(D,A,k1_mboolean(A,B)) => ( r1_pboole(A,C,D) => r2_pboole(A,k6_mssubfam(A,B,D),C) ) ) ) ) ), file(mssubfam,t43_mssubfam), [interesting(0.00)]). fof(t37_msualg_9,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( r1_msualg_3(A,B,C,D) => ! [E] : ( ( v3_msualg_4(E,A,B) & v4_msualg_4(E,A,B) & m1_msualg_4(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) => ~ ( r2_pboole(u1_struct_0(A),E,k18_msualg_4(A,B,C,D)) & ! [F] : ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,k14_msualg_4(A,B,E)),u4_msualg_1(A,C)) => ~ ( r1_msualg_3(A,k14_msualg_4(A,B,E),C,F) & r6_pboole(u1_struct_0(A),D,k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,k14_msualg_4(A,B,E)),u4_msualg_1(A,C),k16_msualg_4(A,B,E),F)) ) ) ) ) ) ) ) ) ) ), file(msualg_9,t37_msualg_9), [interesting(0.00)]). fof(t16_extens_1,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( ( v2_relat_1(C) & m1_pboole(C,A) ) => ! [D] : ( m3_pboole(D,A,B,C) => ( v2_msualg_3(D,A,B,C) <=> ! [E] : ( ( v2_relat_1(E) & m1_pboole(E,A) ) => ! [F] : ( m3_pboole(F,A,C,E) => ! [G] : ( m3_pboole(G,A,C,E) => ( r6_pboole(A,k3_msualg_3(A,B,C,E,D,F),k3_msualg_3(A,B,C,E,D,G)) => r6_pboole(A,F,G) ) ) ) ) ) ) ) ) ), file(extens_1,t16_extens_1), [interesting(0.00)]). fof(s1_birkhoff,theorem, ( ( ! [A] : ( ( v5_msualg_1(A,f1_s1_birkhoff) & l3_msualg_1(A,f1_s1_birkhoff) ) => ! [B] : ( ( v5_msualg_1(B,f1_s1_birkhoff) & l3_msualg_1(B,f1_s1_birkhoff) ) => ( ( r6_msualg_3(f1_s1_birkhoff,A,B) & p1_s1_birkhoff(A) ) => p1_s1_birkhoff(B) ) ) ) & ! [A] : ( ( v5_msualg_1(A,f1_s1_birkhoff) & l3_msualg_1(A,f1_s1_birkhoff) ) => ! [B] : ( ( v4_msualg_1(B,f1_s1_birkhoff) & v5_msualg_1(B,f1_s1_birkhoff) & m1_msualg_2(B,f1_s1_birkhoff,A) ) => ( p1_s1_birkhoff(A) => p1_s1_birkhoff(B) ) ) ) & ! [A,B] : ( m2_pralg_2(B,A,f1_s1_birkhoff) => ( ! [C] : ~ ( r2_hidden(C,A) & ! [D] : ( l3_msualg_1(D,f1_s1_birkhoff) => ~ ( D = k1_funct_1(B,C) & p1_s1_birkhoff(D) ) ) ) => p1_s1_birkhoff(k15_pralg_2(A,f1_s1_birkhoff,B)) ) ) ) => ? [A] : ( v4_msualg_1(A,f1_s1_birkhoff) & v5_msualg_1(A,f1_s1_birkhoff) & l3_msualg_1(A,f1_s1_birkhoff) & ? [B] : ( m3_pboole(B,u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,f2_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A)) & p1_s1_birkhoff(A) & r2_msualg_3(f1_s1_birkhoff,f2_s1_birkhoff,A,B) & ! [C] : ( ( v5_msualg_1(C,f1_s1_birkhoff) & l3_msualg_1(C,f1_s1_birkhoff) ) => ! [D] : ( m3_pboole(D,u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,f2_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,C)) => ~ ( r1_msualg_3(f1_s1_birkhoff,f2_s1_birkhoff,C,D) & p1_s1_birkhoff(C) & ! [E] : ( m3_pboole(E,u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A),u4_msualg_1(f1_s1_birkhoff,C)) => ~ ( r1_msualg_3(f1_s1_birkhoff,A,C,E) & r6_pboole(u1_struct_0(f1_s1_birkhoff),k3_msualg_3(u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,f2_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A),u4_msualg_1(f1_s1_birkhoff,C),B,E),D) & ! [F] : ( m3_pboole(F,u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A),u4_msualg_1(f1_s1_birkhoff,C)) => ( r6_pboole(u1_struct_0(f1_s1_birkhoff),k3_msualg_3(u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,f2_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A),u4_msualg_1(f1_s1_birkhoff,C),B,F),D) => r6_pboole(u1_struct_0(f1_s1_birkhoff),E,F) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[s7_domain_1,d6_msualg_5,t6_msualg_9,d17_pralg_2,t19_card_3,d16_pralg_2,t13_funcop_1,t3_pboole,t27_msualg_9,t20_msualg_5,d6_msualg_5,t20_msualg_5,t30_msualg_9,t27_msualg_9,t17_msualg_3,d3_tarski,d5_msualg_5,t6_msualg_7,t9_msualg_7,t9_msualg_7,d3_msualg_4,d5_msualg_4,d3_msualg_4,t29_msualg_9,s3_pboole,d12_pralg_2,t3_msualg_4,d10_msualg_3,s1_msualg_9,d6_funcop_1,d3_pboole,t30_pralg_3,t4_msualg_4,t17_msualg_9,d2_mssubfam,t15_closure2,d2_relat_1,t106_zfmisc_1,d21_pboole,d20_msualg_4,t2_msualg_3,t21_funct_2,d19_msualg_4,t21_funct_2,t2_msualg_3,t34_msualg_9,t58_setfam_1,t106_zfmisc_1,t7_funct_2,d17_pralg_2,t7_funct_2,d17_pralg_2,d1_closure2,t58_setfam_1,d3_pralg_3,t21_funct_2,t2_msualg_3,t34_msualg_9,t2_msualg_3,t21_funct_2,d3_pralg_3,t15_msualg_9,d19_msualg_4,d20_msualg_4,t3_pboole,t17_msualg_3,t3_msualg_4,t4_msualg_4,t17_msualg_9,t17_msualg_3,d6_msualg_5,t22_closure2,t43_mssubfam,t37_msualg_9,d10_msualg_3,t16_extens_1]), [file(birkhoff,s1_birkhoff),interesting(0.00)]). fof(d5_msafree,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( m1_msafree(C,A,B) => ( v1_msafree(C,A,B) <=> ! [D] : ( ( v5_msualg_1(D,A) & l3_msualg_1(D,A) ) => ! [E] : ( m3_pboole(E,u1_struct_0(A),C,u4_msualg_1(A,D)) => ? [F] : ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,D)) & r1_msualg_3(A,B,D,F) & r6_pboole(u1_struct_0(A),k1_msafree(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,D),C,F),E) ) ) ) ) ) ) ) ), file(msafree,d5_msafree), [interesting(0.00)]). fof(t14_extens_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v2_relat_1(B) & m1_pboole(B,u1_struct_0(A)) ) => r6_pboole(u1_struct_0(A),k2_extens_1(u1_struct_0(A),k15_msafree(A,B)),B) ) ) ), file(extens_1,t14_extens_1), [interesting(0.00)]). fof(t13_extens_1,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m3_pboole(D,A,B,C) => ( v2_msualg_3(D,A,B,C) <=> r6_pboole(A,k2_extens_1(A,D),C) ) ) ) ) ), file(extens_1,t13_extens_1), [interesting(0.00)]). fof(t12_msualg_9,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v2_relat_1(B) & m1_pboole(B,u1_struct_0(A)) ) => v1_msualg_3(k15_msafree(A,B)) ) ) ), file(msualg_9,t12_msualg_9), [interesting(0.00)]). fof(t13_autalg_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_funcop_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_funcop_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_funcop_1(C) ) => k13_pboole(A,k13_pboole(B,C)) = k13_pboole(k13_pboole(A,B),C) ) ) ) ), file(autalg_1,t13_autalg_1), [interesting(0.00)]). fof(t5_msualg_3,theorem,( ! [A,B] : ( ( v2_relat_1(B) & m1_pboole(B,A) ) => ! [C] : ( ( v2_relat_1(C) & m1_pboole(C,A) ) => ! [D] : ( m3_pboole(D,A,B,C) => ! [E] : ( m3_pboole(E,A,C,B) => ( ( v1_msualg_3(D) & v2_msualg_3(D,A,B,C) & r6_pboole(A,E,k4_msualg_3(A,B,C,D)) ) => ( r6_pboole(A,k3_msualg_3(A,C,B,C,E,D),k2_msualg_3(A,C)) & r6_pboole(A,k3_msualg_3(A,B,C,B,D,E),k2_msualg_3(A,B)) ) ) ) ) ) ) ), file(msualg_3,t5_msualg_3), [interesting(0.00)]). fof(t3_msualg_3,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m3_pboole(D,A,B,C) => k13_pboole(k2_msualg_3(A,B),D) = D ) ) ) ), file(msualg_3,t3_msualg_3), [interesting(0.00)]). fof(t8_extens_1,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( ( v2_relat_1(C) & m1_pboole(C,A) ) => ! [D] : ( ( v2_relat_1(D) & m1_pboole(D,A) ) => ! [E] : ( m3_pboole(E,A,B,C) => ! [F] : ( m3_pboole(F,A,C,D) => ! [G] : ( m4_pboole(G,A,B) => r6_pboole(A,k1_msafree(A,B,D,G,k3_msualg_3(A,B,C,D,E,F)),k3_msualg_3(A,G,C,D,k1_msafree(A,B,C,G,E),F)) ) ) ) ) ) ) ), file(extens_1,t8_extens_1), [interesting(0.00)]). fof(t10_msualg_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( ( v5_msualg_1(D,A) & l3_msualg_1(D,A) ) => ! [E] : ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ! [F] : ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D)) => ( ( r1_msualg_3(A,B,C,E) & r1_msualg_3(A,C,D,F) ) => r1_msualg_3(A,B,D,k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C),u4_msualg_1(A,D),E,F)) ) ) ) ) ) ) ) ), file(msualg_3,t10_msualg_3), [interesting(0.00)]). fof(t18_extens_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v2_relat_1(C) & m1_pboole(C,u1_struct_0(A)) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B)) => ! [E] : ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B)) => ( ( r1_msualg_3(A,k11_msafree(A,C),B,D) & r1_msualg_3(A,k11_msafree(A,C),B,E) & r6_pboole(u1_struct_0(A),k1_msafree(u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B),k13_msafree(A,C),D),k1_msafree(u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B),k13_msafree(A,C),E)) ) => r6_pboole(u1_struct_0(A),D,E) ) ) ) ) ) ) ), file(extens_1,t18_extens_1), [interesting(0.00)]). fof(s10_birkhoff,theorem, ( ( ! [A] : ( ( v5_msualg_1(A,f1_s10_birkhoff) & l3_msualg_1(A,f1_s10_birkhoff) ) => ( p2_s10_birkhoff(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(f1_s10_birkhoff)) => ! [C] : ( m1_subset_1(C,k1_funct_1(k4_equation(f1_s10_birkhoff),B)) => ( ! [D] : ( ( v5_msualg_1(D,f1_s10_birkhoff) & l3_msualg_1(D,f1_s10_birkhoff) ) => ( p1_s10_birkhoff(D) => r1_equation(f1_s10_birkhoff,D,B,C) ) ) => r1_equation(f1_s10_birkhoff,A,B,C) ) ) ) ) ) & p1_s10_birkhoff(f2_s10_birkhoff) ) => p2_s10_birkhoff(f2_s10_birkhoff) ), file(birkhoff,s10_birkhoff), [interesting(0.00)]). fof(d1_birkhoff,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v2_relat_1(B) & m1_pboole(B,u1_struct_0(A)) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),B,u4_msualg_1(A,C)) => ! [E] : ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,C)) => ( E = k1_birkhoff(A,B,C,D) <=> ( r1_msualg_3(A,k11_msafree(A,B),C,E) & r6_pboole(u1_struct_0(A),k1_msafree(u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,C),k13_msafree(A,B),E),k3_msualg_3(u1_struct_0(A),k13_msafree(A,B),B,u4_msualg_1(A,C),k15_msafree(A,B),D)) ) ) ) ) ) ) ) ), file(birkhoff,d1_birkhoff), [interesting(0.00)]). fof(d11_msualg_3,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( l3_msualg_1(C,A) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( r3_msualg_3(A,B,C,D) <=> ( r1_msualg_3(A,B,C,D) & v1_msualg_3(D) ) ) ) ) ) ) ), file(msualg_3,d11_msualg_3), [interesting(0.00)]). fof(d8_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) <=> ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) & r2_hidden(C,k1_relat_1(A)) & k1_funct_1(A,B) = k1_funct_1(A,C) ) => B = C ) ) ) ), file(funct_1,d8_funct_1), [interesting(0.00)]). fof(t33_equation,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,k1_funct_1(k4_equation(A),C)) => ! [E] : ( ( v4_msualg_1(E,A) & v5_msualg_1(E,A) & m1_msualg_2(E,A,B) ) => ( r1_equation(A,B,C,D) => r1_equation(A,E,C,D) ) ) ) ) ) ) ), file(equation,t33_equation), [interesting(0.00)]). fof(t9_msafree2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => m1_msafree(u4_msualg_1(A,B),A,B) ) ) ), file(msafree2,t9_msafree2), [interesting(0.00)]). fof(d16_pboole,definition,( ! [A,B] : ( m1_pboole(B,A) => ( v2_relat_1(B) <=> ! [C] : ~ ( r2_hidden(C,A) & v1_xboole_0(k1_funct_1(B,C)) ) ) ) ), file(pboole,d16_pboole), [interesting(0.00)]). fof(d18_pboole,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m1_pboole(D,A) => ( m3_pboole(D,A,B,C) <=> ! [E] : ( r2_hidden(E,A) => ( v1_funct_1(k1_funct_1(D,E)) & v1_funct_2(k1_funct_1(D,E),k1_funct_1(B,E),k1_funct_1(C,E)) & m2_relset_1(k1_funct_1(D,E),k1_funct_1(B,E),k1_funct_1(C,E)) ) ) ) ) ) ) ), file(pboole,d18_pboole), [interesting(0.00)]). fof(d1_funct_2,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), [interesting(0.00)]). fof(d25_pboole,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( ( v1_funcop_1(C) & m1_pboole(C,A) ) => ! [D] : ( m1_pboole(D,A) => ( D = k14_pboole(A,B,C) <=> ! [E] : ( r2_hidden(E,A) => k1_funct_1(D,E) = k9_relat_1(k1_funct_1(C,E),k1_funct_1(B,E)) ) ) ) ) ) ), file(pboole,d25_pboole), [interesting(0.00)]). fof(t151_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => ( k9_relat_1(B,A) = k1_xboole_0 <=> r1_xboole_0(k1_relat_1(B),A) ) ) ), file(relat_1,t151_relat_1), [interesting(0.00)]). fof(d10_msualg_2,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( l3_msualg_1(C,A) => ( m1_msualg_2(C,A,B) <=> ( m4_pboole(u4_msualg_1(A,C),u1_struct_0(A),u4_msualg_1(A,B)) & ! [D] : ( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B)) => ( r6_pboole(u1_struct_0(A),D,u4_msualg_1(A,C)) => ( v3_msualg_2(D,A,B) & r6_pboole(u1_msualg_1(A),u5_msualg_1(A,C),k4_msualg_2(A,B,D)) ) ) ) ) ) ) ) ) ), file(msualg_2,d10_msualg_2), [interesting(0.00)]). fof(t9_extens_1,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ( r1_pzfmisc1(A,B,C) => ! [D] : ( m3_pboole(D,A,B,C) => ! [E] : ( m1_pboole(E,A) => ( m4_pboole(C,A,E) => m3_pboole(D,A,B,E) ) ) ) ) ) ) ), file(extens_1,t9_extens_1), [interesting(0.00)]). fof(d3_pzfmisc1,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ( r1_pzfmisc1(A,B,C) <=> ! [D] : ( ( r2_hidden(D,A) & k1_funct_1(C,D) = k1_xboole_0 ) => k1_funct_1(B,D) = k1_xboole_0 ) ) ) ) ), file(pzfmisc1,d3_pzfmisc1), [interesting(0.00)]). fof(t17_mssubfam,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( ( v1_funcop_1(D) & m1_pboole(D,A) ) => ( ( r1_pzfmisc1(A,B,C) & m3_pboole(D,A,B,C) ) => ( r6_pboole(A,k1_mssubfam(A,D),B) & r2_pboole(A,k2_mssubfam(A,D),C) ) ) ) ) ) ), file(mssubfam,t17_mssubfam), [interesting(0.00)]). fof(t14_equation,theorem,( ! [A,B] : ( ( v1_funcop_1(B) & m1_pboole(B,A) ) => r6_pboole(A,k14_pboole(A,k1_extens_1(A,B),B),k2_extens_1(A,B)) ) ), file(equation,t14_equation), [interesting(0.00)]). fof(d18_msualg_2,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) => ! [D] : ( ( v4_msualg_1(D,A) & m1_msualg_2(D,A,B) ) => ( D = k12_msualg_2(A,B,C) <=> ( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,D)) & ! [E] : ( m1_msualg_2(E,A,B) => ( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,E)) => m1_msualg_2(D,A,E) ) ) ) ) ) ) ) ) ), file(msualg_2,d18_msualg_2), [interesting(0.00)]). fof(t5_equation,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ( r1_pzfmisc1(A,B,C) => ! [D] : ( ( v1_funcop_1(D) & m1_pboole(D,A) ) => ( ( r6_pboole(A,k1_extens_1(A,D),B) & r2_pboole(A,k2_extens_1(A,D),C) ) => m3_pboole(D,A,B,C) ) ) ) ) ) ), file(equation,t5_equation), [interesting(0.00)]). fof(t22_msualg_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & m1_msualg_2(C,A,B) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B)) => ( r6_pboole(u1_struct_0(A),D,k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,C))) => r3_msualg_3(A,C,B,D) ) ) ) ) ) ), file(msualg_3,t22_msualg_3), [interesting(0.00)]). fof(t9_msualg_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => r1_msualg_3(A,B,B,k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B))) ) ) ), file(msualg_3,t9_msualg_3), [interesting(0.00)]). fof(t4_msualg_3,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m3_pboole(D,A,B,C) => k13_pboole(D,k2_msualg_3(A,C)) = D ) ) ) ), file(msualg_3,t4_msualg_3), [interesting(0.00)]). fof(t24_autalg_1,theorem,( ! [A,B] : ( m1_pboole(B,A) => v2_msualg_3(k2_msualg_3(A,B),A,B,B) ) ), file(autalg_1,t24_autalg_1), [interesting(0.00)]). fof(t2_equation,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( ( v2_relat_1(C) & m1_pboole(C,A) ) => ! [D] : ( ( v2_relat_1(D) & m1_pboole(D,A) ) => ! [E] : ( m3_pboole(E,A,B,C) => ! [F] : ( m3_pboole(F,A,C,D) => ( v2_msualg_3(k3_msualg_3(A,B,C,D,E,F),A,B,D) => v2_msualg_3(F,A,C,D) ) ) ) ) ) ) ), file(equation,t2_equation), [interesting(0.00)]). fof(t6_msualg_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => m1_msualg_2(B,A,B) ) ) ), file(msualg_2,t6_msualg_2), [interesting(0.00)]). fof(t16_equation,theorem,( ! [A,B] : ( m1_pboole(B,A) => r6_pboole(A,k14_pboole(A,B,k2_msualg_3(A,B)),B) ) ), file(equation,t16_equation), [interesting(0.00)]). fof(t10_equation,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m3_pboole(D,A,B,C) => ( v2_msualg_3(D,A,B,C) => r6_pboole(A,k14_pboole(A,B,D),C) ) ) ) ) ), file(equation,t10_equation), [interesting(0.00)]). fof(t10_msualg_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( ( v4_msualg_1(C,A) & m1_msualg_2(C,A,B) ) => ! [D] : ( ( v4_msualg_1(D,A) & m1_msualg_2(D,A,B) ) => ( r6_pboole(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D)) => C = D ) ) ) ) ) ), file(msualg_2,t10_msualg_2), [interesting(0.00)]). fof(t3_msafree,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v4_msualg_1(B,A) & v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) => ( m1_msafree(C,A,B) <=> k12_msualg_2(A,B,C) = B ) ) ) ) ), file(msafree,t3_msafree), [interesting(0.00)]). fof(t13_mssubfam,theorem,( ! [A,B,C] : ( ( v1_funcop_1(C) & m1_pboole(C,B) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) ) => ( ( r2_hidden(A,B) & D = k1_funct_1(C,A) ) => k1_funct_1(k2_mssubfam(B,C),A) = k2_relat_1(D) ) ) ) ), file(mssubfam,t13_mssubfam), [interesting(0.00)]). fof(d5_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), [interesting(0.00)]). fof(d3_msualg_3,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m3_pboole(D,A,B,C) => ( v2_msualg_3(D,A,B,C) <=> ! [E] : ( r2_hidden(E,A) => k2_relat_1(k1_funct_1(D,E)) = k1_funct_1(C,E) ) ) ) ) ) ), file(msualg_3,d3_msualg_3), [interesting(0.00)]). fof(t23_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ), file(funct_1,t23_funct_1), [interesting(0.00)]). fof(d1_msafree,definition,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m4_pboole(D,A,B) => ! [E] : ( m3_pboole(E,A,B,C) => ! [F] : ( m3_pboole(F,A,D,C) => ( F = k1_msafree(A,B,C,D,E) <=> ! [G] : ( r2_hidden(G,A) => ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,k1_funct_1(B,G),k1_funct_1(C,G)) & m2_relset_1(H,k1_funct_1(B,G),k1_funct_1(C,G)) ) => ( H = k1_funct_1(E,G) => k1_funct_1(F,G) = k8_relset_1(k1_funct_1(B,G),k1_funct_1(C,G),H,k1_funct_1(D,G)) ) ) ) ) ) ) ) ) ) ), file(msafree,d1_msafree), [interesting(0.00)]). fof(t72_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(B,A) => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ), file(funct_1,t72_funct_1), [interesting(0.00)]). fof(t13_equation,theorem,( ! [A,B] : ( ( v1_funcop_1(B) & m1_pboole(B,A) ) => ! [C] : ( m1_pboole(C,A) => r2_pboole(A,k14_pboole(A,C,B),k2_extens_1(A,B)) ) ) ), file(equation,t13_equation), [interesting(0.00)]). fof(t15_pboole,theorem,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m1_pboole(D,A) => ( ( r2_pboole(A,B,C) & r2_pboole(A,C,D) ) => r2_pboole(A,B,D) ) ) ) ) ), file(pboole,t15_pboole), [interesting(0.00)]). fof(t25_equation,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v4_msualg_1(C,A) & v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m1_msafree(D,A,B) => ! [E] : ( ( v2_relat_1(E) & m1_msafree(E,A,C) ) => ! [F] : ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( ( r2_pboole(u1_struct_0(A),E,k14_pboole(u1_struct_0(A),D,F)) & r1_msualg_3(A,B,C,F) ) => r2_msualg_3(A,B,C,F) ) ) ) ) ) ) ) ), file(equation,t25_equation), [interesting(0.00)]). fof(t7_mcart_1,theorem,( ! [A,B] : ( k1_mcart_1(k4_tarski(A,B)) = A & k2_mcart_1(k4_tarski(A,B)) = B ) ), file(mcart_1,t7_mcart_1), [interesting(0.00)]). fof(t1_msualg_3,theorem,( ! [A,B] : ( ( v1_funcop_1(B) & m1_pboole(B,A) ) => ( v1_msualg_3(B) <=> ! [C] : ( r2_hidden(C,A) => v2_funct_1(k1_funct_1(B,C)) ) ) ) ), file(msualg_3,t1_msualg_3), [interesting(0.00)]). fof(d10_msafree2,definition,( ! [A] : ( ( ~ v3_struct_0(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ( ( ~ v2_msualg_1(A) => ( v3_msafree2(B,A) <=> ! [C] : ( ( ~ v3_struct_0(C) & ~ v2_msualg_1(C) & l1_msualg_1(C) ) => ( C = A => ! [D] : ( l3_msualg_1(D,C) => ~ ( D = B & ! [E] : ( m1_msafree(E,C,D) => ~ v1_pre_circ(E,u1_struct_0(C)) ) ) ) ) ) ) ) & ( v2_msualg_1(A) => ( v3_msafree2(B,A) <=> v1_pre_circ(u4_msualg_1(A,B),u1_struct_0(A)) ) ) ) ) ) ), file(msafree2,d10_msafree2), [interesting(0.00)]). fof(t8_msualg_9,theorem,( ! [A,B] : ( ( v2_relat_1(B) & m1_pboole(B,A) ) => ! [C] : ( ( v1_pre_circ(C,A) & m4_pboole(C,A,B) ) => ? [D] : ( v2_relat_1(D) & v1_pre_circ(D,A) & m4_pboole(D,A,B) & r2_pboole(A,C,D) ) ) ) ), file(msualg_9,t8_msualg_9), [interesting(0.00)]). fof(t13_msualg_9,theorem,( ! [A,B] : ( ( v2_relat_1(B) & v1_pre_circ(B,A) & m1_pboole(B,A) ) => ? [C] : ( m3_pboole(C,A,k2_pre_circ(A,k5_numbers),B) & v2_msualg_3(C,A,k2_pre_circ(A,k5_numbers),B) ) ) ), file(msualg_9,t13_msualg_9), [interesting(0.00)]). fof(t27_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( k1_relat_1(k5_relat_1(B,A)) = k1_relat_1(B) => r1_tarski(k2_relat_1(B),k1_relat_1(A)) ) ) ) ), file(funct_1,t27_funct_1), [interesting(0.00)]). fof(d12_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k9_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E] : ( r2_hidden(E,k1_relat_1(A)) & r2_hidden(E,B) & D = k1_funct_1(A,E) ) ) ) ) ), file(funct_1,d12_funct_1), [interesting(0.00)]). fof(d14_msualg_3,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( r1_msualg_3(A,B,C,D) => ! [E] : ( ( v4_msualg_1(E,A) & v5_msualg_1(E,A) & m1_msualg_2(E,A,C) ) => ( E = k7_msualg_3(A,B,C,D) <=> r6_pboole(u1_struct_0(A),u4_msualg_1(A,E),k14_pboole(u1_struct_0(A),u4_msualg_1(A,B),D)) ) ) ) ) ) ) ) ), file(msualg_3,d14_msualg_3), [interesting(0.00)]). fof(t21_extens_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) => ! [D] : ( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B)) => ( m4_pboole(C,u1_struct_0(A),D) => m1_msualg_2(k12_msualg_2(A,B,C),A,k12_msualg_2(A,B,D)) ) ) ) ) ) ), file(extens_1,t21_extens_1), [interesting(0.00)]). fof(t22_extens_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( m1_msualg_2(C,A,B) => ! [D] : ( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B)) => ! [E] : ( m4_pboole(E,u1_struct_0(A),u4_msualg_1(A,C)) => ( r6_pboole(u1_struct_0(A),D,E) => k12_msualg_2(A,B,D) = k12_msualg_2(A,C,E) ) ) ) ) ) ) ), file(extens_1,t22_extens_1), [interesting(0.00)]). fof(t22_msualg_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v4_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) => ( r6_pboole(u1_struct_0(A),C,u4_msualg_1(A,B)) => k12_msualg_2(A,B,C) = B ) ) ) ) ), file(msualg_2,t22_msualg_2), [interesting(0.00)]). fof(t21_msscyc_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( m1_msafree(C,A,B) => ! [D] : ( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B)) => ( r2_pboole(u1_struct_0(A),C,D) => m1_msafree(D,A,B) ) ) ) ) ) ), file(msscyc_1,t21_msscyc_1), [interesting(0.00)]). fof(t8_msualg_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( l3_msualg_1(C,A) => ( ( v4_msualg_1(B,A) & m1_msualg_2(B,A,C) & v4_msualg_1(C,A) & m1_msualg_2(C,A,B) ) => B = C ) ) ) ) ), file(msualg_2,t8_msualg_2), [interesting(0.00)]). fof(t19_msualg_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v4_msualg_1(C,A) & v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( r1_msualg_3(A,B,C,D) => ( r2_msualg_3(A,B,C,D) <=> k7_msualg_3(A,B,C,D) = C ) ) ) ) ) ) ), file(msualg_3,t19_msualg_3), [interesting(0.00)]). fof(t6_msualg_4,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( ( v5_msualg_1(C,A) & l3_msualg_1(C,A) ) => ! [D] : ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)) => ( r2_msualg_3(A,B,C,D) => r6_msualg_3(A,k14_msualg_4(A,B,k18_msualg_4(A,B,C,D)),C) ) ) ) ) ) ), file(msualg_4,t6_msualg_4), [interesting(0.00)]). fof(d20_msualg_2,definition,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( l3_msualg_1(B,A) => ! [C] : ( C = k14_msualg_2(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( v4_msualg_1(D,A) & m1_msualg_2(D,A,B) ) ) ) ) ) ), file(msualg_2,d20_msualg_2), [interesting(0.00)]). fof(t29_equation,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v2_msualg_1(A) & l1_msualg_1(A) ) => ! [B] : ( ( v5_msualg_1(B,A) & l3_msualg_1(B,A) ) => ! [C] : ( C = a_2_0_equation(A,B) => ! [D] : ( m2_pralg_2(D,C,A) => ~ ( ! [E] : ( r2_hidden(E,C) => E = k1_funct_1(D,E) ) & ! [E] : ( ( v4_msualg_1(E,A) & v5_msualg_1(E,A) & m1_msualg_2(E,A,k15_pralg_2(C,A,D)) ) => ! [F] : ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,E),u4_msualg_1(A,B)) => ~ r2_msualg_3(A,E,B,F) ) ) ) ) ) ) ) ), file(equation,t29_equation), [interesting(0.00)]).