% Mizar problem: l32_arytm_1,arytm_1,447,5 
fof(l32_arytm_1, conjecture,  (! [A] :  (m1_subset_1(A, k2_arytm_2) =>  (! [B] :  (m1_subset_1(B, k2_arytm_2) =>  (! [C] :  (m1_subset_1(C, k2_arytm_2) => k8_arytm_2(A, k1_arytm_1(B, C))=k1_arytm_1(k8_arytm_2(A, B), k8_arytm_2(A, C))) ) ) ) ) ) ).
fof(commutativity_k7_arytm_2, axiom,  (! [A, B] :  ( (m1_subset_1(A, k2_arytm_2) & m1_subset_1(B, k2_arytm_2))  => k7_arytm_2(A, B)=k7_arytm_2(B, A)) ) ).
fof(commutativity_k8_arytm_2, axiom,  (! [A, B] :  ( (m1_subset_1(A, k2_arytm_2) & m1_subset_1(B, k2_arytm_2))  => k8_arytm_2(A, B)=k8_arytm_2(B, A)) ) ).
fof(connectedness_r1_arytm_2, axiom,  (! [A, B] :  ( (m1_subset_1(A, k2_arytm_2) & m1_subset_1(B, k2_arytm_2))  =>  (r1_arytm_2(A, B) | r1_arytm_2(B, A)) ) ) ).
fof(d1_arytm_1, axiom,  (! [A] :  (m1_subset_1(A, k2_arytm_2) =>  (! [B] :  (m1_subset_1(B, k2_arytm_2) =>  (! [C] :  (m1_subset_1(C, k2_arytm_2) =>  ( (r1_arytm_2(B, A) =>  (C=k1_arytm_1(A, B) <=> k7_arytm_2(C, B)=A) )  &  ( ~ (r1_arytm_2(B, A))  =>  (C=k1_arytm_1(A, B) <=> C=k11_arytm_3) ) ) ) ) ) ) ) ) ).
fof(dt_k11_arytm_3, axiom, m1_subset_1(k11_arytm_3, k5_arytm_3)).
fof(dt_k1_arytm_1, axiom,  (! [A, B] :  ( (m1_subset_1(A, k2_arytm_2) & m1_subset_1(B, k2_arytm_2))  => m1_subset_1(k1_arytm_1(A, B), k2_arytm_2)) ) ).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k2_arytm_2, axiom, $true).
fof(dt_k5_arytm_3, axiom, $true).
fof(dt_k7_arytm_2, axiom,  (! [A, B] :  ( (m1_subset_1(A, k2_arytm_2) & m1_subset_1(B, k2_arytm_2))  => m1_subset_1(k7_arytm_2(A, B), k2_arytm_2)) ) ).
fof(dt_k8_arytm_2, axiom,  (! [A, B] :  ( (m1_subset_1(A, k2_arytm_2) & m1_subset_1(B, k2_arytm_2))  => m1_subset_1(k8_arytm_2(A, B), k2_arytm_2)) ) ).
fof(dt_m1_subset_1, axiom, $true).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(l10_arytm_1, axiom,  (! [A] :  (m1_subset_1(A, k2_arytm_2) =>  (! [B] :  (m1_subset_1(B, k2_arytm_2) =>  (! [C] :  (m1_subset_1(C, k2_arytm_2) =>  (r1_arytm_2(k8_arytm_2(A, B), k8_arytm_2(A, C)) =>  (A=k11_arytm_3 | r1_arytm_2(B, C)) ) ) ) ) ) ) ) ).
fof(l16_arytm_1, axiom,  (! [A] :  (m1_subset_1(A, k2_arytm_2) =>  (! [B] :  (m1_subset_1(B, k2_arytm_2) =>  (A=k11_arytm_3 => k1_arytm_1(B, A)=B) ) ) ) ) ).
fof(redefinition_k11_arytm_3, axiom, k11_arytm_3=k1_xboole_0).
fof(t11_arytm_2, axiom,  (! [A] :  (m1_subset_1(A, k2_arytm_2) =>  (! [B] :  (m1_subset_1(B, k2_arytm_2) =>  (! [C] :  (m1_subset_1(C, k2_arytm_2) =>  (k7_arytm_2(A, B)=k7_arytm_2(A, C) => B=C) ) ) ) ) ) ) ).
fof(t13_arytm_2, axiom,  (! [A] :  (m1_subset_1(A, k2_arytm_2) =>  (! [B] :  (m1_subset_1(B, k2_arytm_2) =>  (! [C] :  (m1_subset_1(C, k2_arytm_2) => k8_arytm_2(A, k7_arytm_2(B, C))=k7_arytm_2(k8_arytm_2(A, B), k8_arytm_2(A, C))) ) ) ) ) ) ).
fof(t4_arytm_2, axiom,  (! [A] :  (m1_subset_1(A, k2_arytm_2) =>  (! [B] :  (m1_subset_1(B, k2_arytm_2) =>  (A=k11_arytm_3 => k8_arytm_2(A, B)=k11_arytm_3) ) ) ) ) ).
fof(t8_arytm_1, axiom,  (! [A] :  (m1_subset_1(A, k2_arytm_2) =>  (! [B] :  (m1_subset_1(B, k2_arytm_2) =>  (! [C] :  (m1_subset_1(C, k2_arytm_2) =>  (r1_arytm_2(A, B) => r1_arytm_2(k8_arytm_2(A, C), k8_arytm_2(B, C))) ) ) ) ) ) ) ).
