% Mizar problem: t25_sysrel,sysrel,325,55 
fof(t25_sysrel,conjecture,(
    ! [A,B,C,D,E] : 
      ( v1_relat_1(E)
     => ( ( r1_tarski(E,k2_zfmisc_1(A,B))
          & A = k2_xboole_0(C,D) )
       => E = k2_xboole_0(k7_relat_1(E,C),k7_relat_1(E,D)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[commutativity_k2_xboole_0,commutativity_k3_xboole_0,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k3_xboole_0,dt_k7_relat_1,dt_k8_relat_1,dt_m1_subset_1,existence_m1_subset_1,fc1_relat_1,fc1_sysrel,fc2_relat_1,idempotence_k2_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,t120_zfmisc_1,t23_xboole_1,t24_sysrel,t28_xboole_1,t3_subset,t7_xboole_1]),
    [file(sysrel,t25_sysrel)]).

fof(commutativity_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    []).

fof(commutativity_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ),
    file(xboole_0,k3_xboole_0),
    []).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    []).

fof(dt_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_xboole_0),
    []).

fof(dt_k7_relat_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(A)
     => v1_relat_1(k7_relat_1(A,B)) ) ),
    file(relat_1,k7_relat_1),
    []).

fof(dt_k8_relat_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => v1_relat_1(k8_relat_1(A,B)) ) ),
    file(relat_1,k8_relat_1),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(fc1_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k3_xboole_0(A,B)) ) ),
    file(relat_1,fc1_relat_1),
    []).

fof(fc1_sysrel,axiom,(
    ! [A,B] : v1_relat_1(k2_zfmisc_1(A,B)) ),
    file(sysrel,fc1_sysrel),
    []).

fof(fc2_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k2_xboole_0(A,B)) ) ),
    file(relat_1,fc2_relat_1),
    []).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    []).

fof(idempotence_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,A) = A ),
    file(xboole_0,k3_xboole_0),
    []).

fof(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

fof(t120_zfmisc_1,axiom,(
    ! [A,B,C] : 
      ( k2_zfmisc_1(k2_xboole_0(A,B),C) = k2_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C))
      & k2_zfmisc_1(C,k2_xboole_0(A,B)) = k2_xboole_0(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ),
    file(zfmisc_1,t120_zfmisc_1),
    []).

fof(t23_xboole_1,axiom,(
    ! [A,B,C] : k3_xboole_0(A,k2_xboole_0(B,C)) = k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) ),
    file(xboole_1,t23_xboole_1),
    []).

fof(t24_sysrel,axiom,(
    ! [A,B,C,D] : 
      ( v1_relat_1(D)
     => ( ( ( r1_tarski(D,k2_zfmisc_1(A,B))
            & r1_tarski(C,A) )
         => k7_relat_1(D,C) = k3_xboole_0(D,k2_zfmisc_1(C,B)) )
        & ( ( r1_tarski(D,k2_zfmisc_1(A,B))
            & r1_tarski(C,B) )
         => k8_relat_1(C,D) = k3_xboole_0(D,k2_zfmisc_1(A,C)) ) ) ) ),
    file(sysrel,t24_sysrel),
    []).

fof(t28_xboole_1,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k3_xboole_0(A,B) = A ) ),
    file(xboole_1,t28_xboole_1),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t7_xboole_1,axiom,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    file(xboole_1,t7_xboole_1),
    []).