fof(t79_xboole_1,theorem,(
    ! [A,B] : r1_xboole_0(k4_xboole_0(A,B),B) ),
    inference(mizar_proof,[status(thm)],[d3_xboole_0,d4_xboole_0,d1_xboole_0,d7_xboole_0]),
    [file(xboole_1,t79_xboole_1),interesting(1.00)]).

fof(t60_xboole_1,theorem,(
    ! [A,B] : ~ ( r1_tarski(A,B)
      & r2_xboole_0(B,A) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,d10_xboole_0]),
    [file(xboole_1,t60_xboole_1),interesting(0.93)]).

fof(t17_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_xboole_0]),
    [file(xboole_1,t17_xboole_1),interesting(0.92)]).

fof(t36_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0]),
    [file(xboole_1,t36_xboole_1),interesting(0.92)]).

fof(t7_xboole_1,theorem,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0]),
    [file(xboole_1,t7_xboole_1),interesting(0.92)]).

fof(t65_xboole_1,theorem,(
    ! [A] : r1_xboole_0(A,k1_xboole_0) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0]),
    [file(xboole_1,t65_xboole_1),interesting(0.91)]).

fof(t91_xboole_1,theorem,(
    ! [A,B,C] : k5_xboole_0(k5_xboole_0(A,B),C) = k5_xboole_0(A,k5_xboole_0(B,C)) ),
    inference(mizar_proof,[status(thm)],[t42_xboole_1,t41_xboole_1,t41_xboole_1,t55_xboole_1,t52_xboole_1,t4_xboole_1,t4_xboole_1,t4_xboole_1,t16_xboole_1,t52_xboole_1,t55_xboole_1,t41_xboole_1,t41_xboole_1,t42_xboole_1]),
    [file(xboole_1,t91_xboole_1),interesting(0.91)]).

fof(t95_xboole_1,theorem,(
    ! [A,B] : k3_xboole_0(A,B) = k5_xboole_0(k5_xboole_0(A,B),k2_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[l97_xboole_1,t93_xboole_1,t88_xboole_1,l98_xboole_1,t36_xboole_1,l32_xboole_1]),
    [file(xboole_1,t95_xboole_1),interesting(0.91)]).

fof(t63_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_xboole_0(B,C) )
     => r1_xboole_0(A,C) ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,t26_xboole_1,t3_xboole_1,d7_xboole_0]),
    [file(xboole_1,t63_xboole_1),interesting(0.90)]).

fof(t74_xboole_1,theorem,(
    ! [A,B,C] : ~ ( ~ r1_xboole_0(A,k3_xboole_0(B,C))
      & r1_xboole_0(A,B) ) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0,d3_xboole_0,t3_xboole_0]),
    [file(xboole_1,t74_xboole_1),interesting(0.88)]).

fof(t83_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
    <=> k4_xboole_0(A,B) = A ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,d4_xboole_0,d3_tarski,d10_xboole_0,d3_tarski,d3_xboole_0,d4_xboole_0,d3_xboole_0,d4_xboole_0,t4_xboole_0]),
    [file(xboole_1,t83_xboole_1),interesting(0.88)]).

fof(l97_xboole_1,theorem,(
    ! [A,B] : r1_xboole_0(k3_xboole_0(A,B),k5_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[t89_xboole_1,t70_xboole_1]),
    [file(xboole_1,l97_xboole_1),interesting(0.87)]).

fof(t2_xboole_1,theorem,(
    ! [A] : r1_tarski(k1_xboole_0,A) ),
    inference(mizar_proof,[status(thm)],[d3_tarski]),
    [file(xboole_1,t2_xboole_1),interesting(0.87)]).

fof(t1_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_tarski,d3_tarski]),
    [file(xboole_1,t1_xboole_1),interesting(0.86)]).

fof(l58_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r2_xboole_0(B,C) )
     => r2_xboole_0(A,C) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,t1_xboole_1,d10_xboole_0,d8_xboole_0]),
    [file(xboole_1,l58_xboole_1),interesting(0.86)]).

fof(t89_xboole_1,theorem,(
    ! [A,B] : r1_xboole_0(k3_xboole_0(A,B),k4_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[d3_xboole_0,d4_xboole_0,t3_xboole_0]),
    [file(xboole_1,t89_xboole_1),interesting(0.85)]).

fof(t109_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k4_xboole_0(A,C),B) ) ),
    inference(mizar_proof,[status(thm)],[t36_xboole_1,t1_xboole_1]),
    [file(xboole_1,t109_xboole_1),interesting(0.85)]).

fof(t62_xboole_1,theorem,(
    ! [A] : ~ r2_xboole_0(A,k1_xboole_0) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,t3_xboole_1]),
    [file(xboole_1,t62_xboole_1),interesting(0.85)]).

fof(t22_xboole_1,theorem,(
    ! [A,B] : k2_xboole_0(A,k3_xboole_0(A,B)) = A ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d3_xboole_0,d10_xboole_0,d3_tarski,d2_xboole_0]),
    [file(xboole_1,t22_xboole_1),interesting(0.84)]).

fof(t84_xboole_1,theorem,(
    ! [A,B,C] : ~ ( ~ r1_xboole_0(A,B)
      & r1_xboole_0(A,C)
      & r1_xboole_0(A,k4_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[t50_xboole_1,d7_xboole_0,d7_xboole_0,d7_xboole_0]),
    [file(xboole_1,t84_xboole_1),interesting(0.83)]).

fof(t28_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k3_xboole_0(A,B) = A ) ),
    inference(mizar_proof,[status(thm)],[t17_xboole_1,d10_xboole_0,d3_tarski,d3_tarski,d3_xboole_0]),
    [file(xboole_1,t28_xboole_1),interesting(0.82)]).

fof(t12_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k2_xboole_0(A,B) = B ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d3_tarski,d10_xboole_0,d3_tarski,d2_xboole_0]),
    [file(xboole_1,t12_xboole_1),interesting(0.80)]).

fof(t56_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r2_xboole_0(A,B)
        & r2_xboole_0(B,C) )
     => r2_xboole_0(A,C) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,l58_xboole_1]),
    [file(xboole_1,t56_xboole_1),interesting(0.79)]).

fof(t112_xboole_1,theorem,(
    ! [A,B,C] : k5_xboole_0(k3_xboole_0(A,B),k3_xboole_0(C,B)) = k3_xboole_0(k5_xboole_0(A,C),B) ),
    inference(mizar_proof,[status(thm)],[t111_xboole_1,t111_xboole_1,t23_xboole_1]),
    [file(xboole_1,t112_xboole_1),interesting(0.78)]).

fof(t64_xboole_1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D)
        & r1_xboole_0(B,D) )
     => r1_xboole_0(A,C) ) ),
    inference(mizar_proof,[status(thm)],[t63_xboole_1,t63_xboole_1]),
    [file(xboole_1,t64_xboole_1),interesting(0.78)]).

fof(t103_xboole_1,theorem,(
    ! [A,B] : r1_xboole_0(k3_xboole_0(A,B),k5_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[l97_xboole_1]),
    [file(xboole_1,t103_xboole_1),interesting(0.75)]).

fof(t90_xboole_1,theorem,(
    ! [A,B] : r1_xboole_0(k4_xboole_0(A,k3_xboole_0(A,B)),B) ),
    inference(mizar_proof,[status(thm)],[t47_xboole_1,t79_xboole_1]),
    [file(xboole_1,t90_xboole_1),interesting(0.75)]).

fof(t88_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
     => k4_xboole_0(k2_xboole_0(A,B),B) = A ) ),
    inference(mizar_proof,[status(thm)],[t42_xboole_1,l32_xboole_1,t83_xboole_1]),
    [file(xboole_1,t88_xboole_1),interesting(0.75)]).

fof(t9_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d3_tarski,d2_xboole_0]),
    [file(xboole_1,t9_xboole_1),interesting(0.74)]).

fof(t8_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k2_xboole_0(A,C),B) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d3_tarski]),
    [file(xboole_1,t8_xboole_1),interesting(0.74)]).

fof(t34_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k4_xboole_0(C,B),k4_xboole_0(C,A)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0,d3_tarski,d4_xboole_0]),
    [file(xboole_1,t34_xboole_1),interesting(0.74)]).

fof(t111_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(k3_xboole_0(A,B),k3_xboole_0(C,B)) = k3_xboole_0(k4_xboole_0(A,C),B) ),
    inference(mizar_proof,[status(thm)],[l36_xboole_1,t49_xboole_1,l32_xboole_1,t49_xboole_1]),
    [file(xboole_1,t111_xboole_1),interesting(0.74)]).

fof(t19_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(A,C) )
     => r1_tarski(A,k3_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_tarski,d3_xboole_0]),
    [file(xboole_1,t19_xboole_1),interesting(0.74)]).

fof(t33_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k4_xboole_0(A,C),k4_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0,d3_tarski,d4_xboole_0]),
    [file(xboole_1,t33_xboole_1),interesting(0.74)]).

fof(t26_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_xboole_0,d3_tarski,d3_xboole_0]),
    [file(xboole_1,t26_xboole_1),interesting(0.74)]).

fof(t50_xboole_1,theorem,(
    ! [A,B,C] : k3_xboole_0(A,k4_xboole_0(B,C)) = k4_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) ),
    inference(mizar_proof,[status(thm)],[t17_xboole_1,l36_xboole_1,l32_xboole_1,t49_xboole_1]),
    [file(xboole_1,t50_xboole_1),interesting(0.74)]).

fof(t47_xboole_1,theorem,(
    ! [A,B] : k4_xboole_0(A,k3_xboole_0(A,B)) = k4_xboole_0(A,B) ),
    inference(mizar_proof,[status(thm)],[d3_xboole_0,d4_xboole_0,t2_tarski]),
    [file(xboole_1,t47_xboole_1),interesting(0.73)]).

fof(t59_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r2_xboole_0(B,C) )
     => r2_xboole_0(A,C) ) ),
    inference(mizar_proof,[status(thm)],[l58_xboole_1]),
    [file(xboole_1,t59_xboole_1),interesting(0.73)]).

fof(t75_xboole_1,theorem,(
    ! [A,B] : ~ ( ~ r1_xboole_0(A,B)
      & r1_xboole_0(k3_xboole_0(A,B),B) ) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0,d3_xboole_0,t3_xboole_0]),
    [file(xboole_1,t75_xboole_1),interesting(0.73)]).

fof(t16_xboole_1,theorem,(
    ! [A,B,C] : k3_xboole_0(k3_xboole_0(A,B),C) = k3_xboole_0(A,k3_xboole_0(B,C)) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_xboole_0,d3_xboole_0,d3_xboole_0,d3_xboole_0,d10_xboole_0,d3_tarski,d3_xboole_0,d3_xboole_0,d3_xboole_0,d3_xboole_0]),
    [file(xboole_1,t16_xboole_1),interesting(0.72)]).

fof(t106_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,k4_xboole_0(B,C))
     => ( r1_tarski(A,B)
        & r1_xboole_0(A,C) ) ) ),
    inference(mizar_proof,[status(thm)],[t36_xboole_1,t1_xboole_1,d3_tarski,d4_xboole_0,t3_xboole_0]),
    [file(xboole_1,t106_xboole_1),interesting(0.72)]).

fof(t10_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(A,k2_xboole_0(C,B)) ) ),
    inference(mizar_proof,[status(thm)],[t9_xboole_1,t7_xboole_1,t1_xboole_1]),
    [file(xboole_1,t10_xboole_1),interesting(0.72)]).

fof(t4_xboole_1,theorem,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d2_xboole_0,d2_xboole_0,d2_xboole_0,d10_xboole_0,d3_tarski,d2_xboole_0,d2_xboole_0,d2_xboole_0,d2_xboole_0]),
    [file(xboole_1,t4_xboole_1),interesting(0.72)]).

fof(t39_xboole_1,theorem,(
    ! [A,B] : k2_xboole_0(A,k4_xboole_0(B,A)) = k2_xboole_0(A,B) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d4_xboole_0,d2_xboole_0,d10_xboole_0,d3_tarski,d2_xboole_0,d4_xboole_0,d2_xboole_0]),
    [file(xboole_1,t39_xboole_1),interesting(0.72)]).

fof(t80_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_xboole_0(A,B)
     => r1_xboole_0(A,k4_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0,d4_xboole_0,t3_xboole_0]),
    [file(xboole_1,t80_xboole_1),interesting(0.71)]).

fof(t82_xboole_1,theorem,(
    ! [A,B] : r1_xboole_0(k4_xboole_0(A,B),k4_xboole_0(B,A)) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0,d4_xboole_0]),
    [file(xboole_1,t82_xboole_1),interesting(0.71)]).

fof(t108_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k3_xboole_0(A,C),B) ) ),
    inference(mizar_proof,[status(thm)],[t17_xboole_1,t1_xboole_1]),
    [file(xboole_1,t108_xboole_1),interesting(0.70)]).

fof(t110_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k5_xboole_0(A,C),B) ) ),
    inference(mizar_proof,[status(thm)],[t109_xboole_1,t8_xboole_1]),
    [file(xboole_1,t110_xboole_1),interesting(0.70)]).

fof(t11_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(k2_xboole_0(A,B),C)
     => r1_tarski(A,C) ) ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1,t1_xboole_1]),
    [file(xboole_1,t11_xboole_1),interesting(0.70)]).

fof(t18_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,k3_xboole_0(B,C))
     => r1_tarski(A,B) ) ),
    inference(mizar_proof,[status(thm)],[t17_xboole_1,t1_xboole_1]),
    [file(xboole_1,t18_xboole_1),interesting(0.70)]).

fof(t58_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r2_xboole_0(A,B)
        & r1_tarski(B,C) )
     => r2_xboole_0(A,C) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,t1_xboole_1,d10_xboole_0,d8_xboole_0]),
    [file(xboole_1,t58_xboole_1),interesting(0.70)]).

fof(t85_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_xboole_0(A,k4_xboole_0(C,B)) ) ),
    inference(mizar_proof,[status(thm)],[t49_xboole_1,t49_xboole_1,l32_xboole_1,d7_xboole_0]),
    [file(xboole_1,t85_xboole_1),interesting(0.69)]).

fof(t21_xboole_1,theorem,(
    ! [A,B] : k3_xboole_0(A,k2_xboole_0(A,B)) = A ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_xboole_0,d10_xboole_0,d3_tarski,d2_xboole_0,d3_xboole_0]),
    [file(xboole_1,t21_xboole_1),interesting(0.68)]).

fof(t76_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_xboole_0(A,B)
     => r1_xboole_0(k3_xboole_0(C,A),k3_xboole_0(C,B)) ) ),
    inference(mizar_proof,[status(thm)],[t74_xboole_1,t74_xboole_1]),
    [file(xboole_1,t76_xboole_1),interesting(0.68)]).

fof(t94_xboole_1,theorem,(
    ! [A,B] : k2_xboole_0(A,B) = k5_xboole_0(k5_xboole_0(A,B),k3_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[l97_xboole_1,t83_xboole_1,t93_xboole_1]),
    [file(xboole_1,t94_xboole_1),interesting(0.67)]).

fof(t29_xboole_1,theorem,(
    ! [A,B,C] : r1_tarski(k3_xboole_0(A,B),k2_xboole_0(A,C)) ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1,t17_xboole_1,t1_xboole_1]),
    [file(xboole_1,t29_xboole_1),interesting(0.66)]).

fof(t3_xboole_1,theorem,(
    ! [A] : 
      ( r1_tarski(A,k1_xboole_0)
     => A = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[t2_xboole_1,d10_xboole_0]),
    [file(xboole_1,t3_xboole_1),interesting(0.66)]).

fof(t81_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_xboole_0(A,k4_xboole_0(B,C))
     => r1_xboole_0(B,k4_xboole_0(A,C)) ) ),
    inference(mizar_proof,[status(thm)],[t49_xboole_1,t49_xboole_1,d7_xboole_0]),
    [file(xboole_1,t81_xboole_1),interesting(0.66)]).

fof(t86_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_xboole_0(A,C) )
     => r1_tarski(A,k4_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,d3_tarski,d3_tarski,d3_xboole_0,d4_xboole_0]),
    [file(xboole_1,t86_xboole_1),interesting(0.65)]).

fof(t92_xboole_1,theorem,(
    ! [A] : k5_xboole_0(A,A) = k1_xboole_0 ),
    inference(mizar_proof,[status(thm)],[l32_xboole_1]),
    [file(xboole_1,t92_xboole_1),interesting(0.65)]).

fof(t93_xboole_1,theorem,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(k5_xboole_0(A,B),k3_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[t51_xboole_1,t4_xboole_1,t22_xboole_1,t51_xboole_1,t4_xboole_1]),
    [file(xboole_1,t93_xboole_1),interesting(0.65)]).

fof(t49_xboole_1,theorem,(
    ! [A,B,C] : k3_xboole_0(A,k4_xboole_0(B,C)) = k4_xboole_0(k3_xboole_0(A,B),C) ),
    inference(mizar_proof,[status(thm)],[d3_xboole_0,d4_xboole_0,d3_xboole_0,d4_xboole_0,t2_tarski]),
    [file(xboole_1,t49_xboole_1),interesting(0.64)]).

fof(t6_xboole_1,theorem,(
    ! [A,B] : k2_xboole_0(A,k2_xboole_0(A,B)) = k2_xboole_0(A,B) ),
    inference(mizar_proof,[status(thm)],[t4_xboole_1]),
    [file(xboole_1,t6_xboole_1),interesting(0.64)]).

fof(t41_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(k4_xboole_0(A,B),C) = k4_xboole_0(A,k2_xboole_0(B,C)) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,d4_xboole_0,d2_xboole_0,d4_xboole_0,d3_tarski,d10_xboole_0,d4_xboole_0,d2_xboole_0,d4_xboole_0,d4_xboole_0,d3_tarski]),
    [file(xboole_1,t41_xboole_1),interesting(0.63)]).

fof(t51_xboole_1,theorem,(
    ! [A,B] : k2_xboole_0(k3_xboole_0(A,B),k4_xboole_0(A,B)) = A ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d3_xboole_0,d4_xboole_0,d10_xboole_0,d3_tarski,d4_xboole_0,d3_xboole_0,d2_xboole_0]),
    [file(xboole_1,t51_xboole_1),interesting(0.62)]).

fof(t32_xboole_1,theorem,(
    ! [A,B] : 
      ( k4_xboole_0(A,B) = k4_xboole_0(B,A)
     => A = B ) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,d4_xboole_0,t2_tarski]),
    [file(xboole_1,t32_xboole_1),interesting(0.61)]).

fof(t69_xboole_1,theorem,(
    ! [A,B] : 
      ( ~ v1_xboole_0(B)
     => ~ ( r1_tarski(B,A)
          & r1_xboole_0(B,A) ) ) ),
    inference(mizar_proof,[status(thm)],[t28_xboole_1,d7_xboole_0]),
    [file(xboole_1,t69_xboole_1),interesting(0.59)]).

fof(t73_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,k2_xboole_0(B,C))
        & r1_xboole_0(A,C) )
     => r1_tarski(A,B) ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,t28_xboole_1,t23_xboole_1,t17_xboole_1]),
    [file(xboole_1,t73_xboole_1),interesting(0.58)]).

fof(t98_xboole_1,theorem,(
    ! [A,B] : k2_xboole_0(A,B) = k5_xboole_0(A,k4_xboole_0(B,A)) ),
    inference(mizar_proof,[status(thm)],[t41_xboole_1,t36_xboole_1,t52_xboole_1,t12_xboole_1,t39_xboole_1]),
    [file(xboole_1,t98_xboole_1),interesting(0.57)]).

fof(l32_xboole_1,theorem,(
    ! [A,B] : 
      ( k4_xboole_0(A,B) = k1_xboole_0
    <=> r1_tarski(A,B) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0,d3_tarski,d4_xboole_0,t2_tarski]),
    [file(xboole_1,l32_xboole_1),interesting(0.56)]).

fof(t42_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k4_xboole_0(A,C),k4_xboole_0(B,C)) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0,d2_xboole_0,d4_xboole_0,d2_xboole_0,d10_xboole_0,d3_tarski,d2_xboole_0,d4_xboole_0,d2_xboole_0,d4_xboole_0]),
    [file(xboole_1,t42_xboole_1),interesting(0.56)]).

fof(t23_xboole_1,theorem,(
    ! [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)) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_xboole_0,d2_xboole_0,d3_xboole_0,d2_xboole_0,d10_xboole_0,d3_tarski,d2_xboole_0,d3_xboole_0,d2_xboole_0,d3_xboole_0]),
    [file(xboole_1,t23_xboole_1),interesting(0.56)]).

fof(t24_xboole_1,theorem,(
    ! [A,B,C] : k2_xboole_0(A,k3_xboole_0(B,C)) = k3_xboole_0(k2_xboole_0(A,B),k2_xboole_0(A,C)) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d3_xboole_0,d2_xboole_0,d3_xboole_0,d10_xboole_0,d3_tarski,d3_xboole_0,d2_xboole_0,d3_xboole_0,d2_xboole_0]),
    [file(xboole_1,t24_xboole_1),interesting(0.56)]).

fof(t35_xboole_1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k4_xboole_0(A,D),k4_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[t33_xboole_1,t34_xboole_1,t1_xboole_1]),
    [file(xboole_1,t35_xboole_1),interesting(0.56)]).

fof(t77_xboole_1,theorem,(
    ! [A,B,C] : ~ ( ~ r1_xboole_0(A,B)
      & r1_tarski(A,C)
      & r1_xboole_0(A,k3_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[t28_xboole_1,t16_xboole_1,d7_xboole_0,d7_xboole_0]),
    [file(xboole_1,t77_xboole_1),interesting(0.56)]).

fof(t96_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),k5_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1]),
    [file(xboole_1,t96_xboole_1),interesting(0.56)]).

fof(t70_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ~ ( ~ r1_xboole_0(A,k2_xboole_0(B,C))
          & r1_xboole_0(A,B)
          & r1_xboole_0(A,C) )
      & ~ ( ~ ( r1_xboole_0(A,B)
              & r1_xboole_0(A,C) )
          & r1_xboole_0(A,k2_xboole_0(B,C)) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0,d2_xboole_0,t3_xboole_0,t3_xboole_0,d2_xboole_0,t3_xboole_0,t3_xboole_0,d2_xboole_0,t3_xboole_0]),
    [file(xboole_1,t70_xboole_1),interesting(0.55)]).

fof(t40_xboole_1,theorem,(
    ! [A,B] : k4_xboole_0(k2_xboole_0(A,B),B) = k4_xboole_0(A,B) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,d2_xboole_0,d4_xboole_0,d3_tarski,d10_xboole_0,d4_xboole_0,d2_xboole_0,d4_xboole_0,d3_tarski]),
    [file(xboole_1,t40_xboole_1),interesting(0.54)]).

fof(t48_xboole_1,theorem,(
    ! [A,B] : k4_xboole_0(A,k4_xboole_0(A,B)) = k3_xboole_0(A,B) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,d4_xboole_0,d3_xboole_0,d3_tarski,d10_xboole_0,d3_xboole_0,d4_xboole_0,d4_xboole_0,d3_tarski]),
    [file(xboole_1,t48_xboole_1),interesting(0.54)]).

fof(t5_xboole_1,theorem,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k2_xboole_0(A,C),k2_xboole_0(B,C)) ),
    inference(mizar_proof,[status(thm)],[t4_xboole_1,t4_xboole_1,t4_xboole_1]),
    [file(xboole_1,t5_xboole_1),interesting(0.53)]).

fof(t114_xboole_1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_xboole_0(A,D)
        & r1_xboole_0(B,D)
        & r1_xboole_0(C,D) )
     => r1_xboole_0(k2_xboole_0(k2_xboole_0(A,B),C),D) ) ),
    inference(mizar_proof,[status(thm)],[t70_xboole_1,t70_xboole_1]),
    [file(xboole_1,t114_xboole_1),interesting(0.50)]).

fof(l36_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(A,k3_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(A,C)) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0,d3_xboole_0,d4_xboole_0,d2_xboole_0,d10_xboole_0,t17_xboole_1,t34_xboole_1,t8_xboole_1]),
    [file(xboole_1,l36_xboole_1),interesting(0.49)]).

fof(t13_xboole_1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,D)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d3_tarski,d2_xboole_0]),
    [file(xboole_1,t13_xboole_1),interesting(0.49)]).

fof(t27_xboole_1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,D)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_xboole_0,d3_tarski,d3_xboole_0]),
    [file(xboole_1,t27_xboole_1),interesting(0.49)]).

fof(t72_xboole_1,theorem,(
    ! [A,B,C,D] : 
      ( ( k2_xboole_0(A,B) = k2_xboole_0(C,D)
        & r1_xboole_0(C,A)
        & r1_xboole_0(D,B) )
     => C = B ) ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1,d7_xboole_0,t28_xboole_1,t23_xboole_1,t23_xboole_1,t28_xboole_1]),
    [file(xboole_1,t72_xboole_1),interesting(0.49)]).

fof(l98_xboole_1,theorem,(
    ! [A,B] : k5_xboole_0(A,B) = k4_xboole_0(k2_xboole_0(A,B),k3_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[t47_xboole_1,t47_xboole_1,t42_xboole_1]),
    [file(xboole_1,l98_xboole_1),interesting(0.48)]).

fof(t113_xboole_1,theorem,(
    ! [A,B,C,D] : k2_xboole_0(k2_xboole_0(k2_xboole_0(A,B),C),D) = k2_xboole_0(A,k2_xboole_0(k2_xboole_0(B,C),D)) ),
    inference(mizar_proof,[status(thm)],[t4_xboole_1,t4_xboole_1,t4_xboole_1]),
    [file(xboole_1,t113_xboole_1),interesting(0.47)]).

fof(t43_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,k2_xboole_0(B,C))
     => r1_tarski(k4_xboole_0(A,B),C) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0,d3_tarski,d2_xboole_0]),
    [file(xboole_1,t43_xboole_1),interesting(0.47)]).

fof(t44_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(k4_xboole_0(A,B),C)
     => r1_tarski(A,k2_xboole_0(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_tarski,d4_xboole_0,d2_xboole_0]),
    [file(xboole_1,t44_xboole_1),interesting(0.47)]).

fof(t105_xboole_1,theorem,(
    ! [A,B] : ~ ( r2_xboole_0(A,B)
      & k4_xboole_0(B,A) = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[l32_xboole_1,t60_xboole_1]),
    [file(xboole_1,t105_xboole_1),interesting(0.46)]).

fof(t100_xboole_1,theorem,(
    ! [A,B] : k4_xboole_0(A,B) = k5_xboole_0(A,k3_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[t17_xboole_1,l32_xboole_1,t47_xboole_1]),
    [file(xboole_1,t100_xboole_1),interesting(0.45)]).

fof(t46_xboole_1,theorem,(
    ! [A,B] : k4_xboole_0(A,k2_xboole_0(A,B)) = k1_xboole_0 ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1,l32_xboole_1]),
    [file(xboole_1,t46_xboole_1),interesting(0.44)]).

fof(t52_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(A,k4_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,B),k3_xboole_0(A,C)) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,d4_xboole_0,d3_xboole_0,d4_xboole_0,d2_xboole_0,d3_tarski,d10_xboole_0,d2_xboole_0,d3_xboole_0,d4_xboole_0,d4_xboole_0,d4_xboole_0,d3_tarski]),
    [file(xboole_1,t52_xboole_1),interesting(0.43)]).

fof(t68_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ~ v1_xboole_0(C)
     => ~ ( r1_tarski(C,A)
          & r1_tarski(C,B)
          & r1_xboole_0(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_xboole_0,d3_tarski,t3_xboole_0]),
    [file(xboole_1,t68_xboole_1),interesting(0.42)]).

fof(t61_xboole_1,theorem,(
    ! [A] : 
      ( A != k1_xboole_0
     => r2_xboole_0(k1_xboole_0,A) ) ),
    inference(mizar_proof,[status(thm)],[t2_xboole_1,d8_xboole_0]),
    [file(xboole_1,t61_xboole_1),interesting(0.41)]).

fof(t45_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => B = k2_xboole_0(A,k4_xboole_0(B,A)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0,d2_xboole_0,t2_tarski]),
    [file(xboole_1,t45_xboole_1),interesting(0.39)]).

fof(t104_xboole_1,theorem,(
    ! [A,B] : 
      ( ~ ( ~ r2_xboole_0(A,B)
          & A != B
          & ~ r2_xboole_0(B,A) )
    <=> r3_xboole_0(A,B) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,d9_xboole_0,d9_xboole_0,d8_xboole_0]),
    [file(xboole_1,t104_xboole_1),interesting(0.38)]).

fof(t31_xboole_1,theorem,(
    ! [A,B,C] : r1_tarski(k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)),k2_xboole_0(B,C)) ),
    inference(mizar_proof,[status(thm)],[d2_xboole_0,d3_xboole_0,d2_xboole_0,d3_tarski]),
    [file(xboole_1,t31_xboole_1),interesting(0.38)]).

fof(t71_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( k2_xboole_0(A,B) = k2_xboole_0(C,B)
        & r1_xboole_0(A,B)
        & r1_xboole_0(C,B) )
     => A = C ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,d3_tarski,d3_xboole_0,t7_xboole_1,d3_tarski,d2_xboole_0,d10_xboole_0,d3_tarski,d3_xboole_0,t7_xboole_1,d3_tarski,d2_xboole_0]),
    [file(xboole_1,t71_xboole_1),interesting(0.37)]).

fof(t37_xboole_1,theorem,(
    ! [A,B] : 
      ( k4_xboole_0(A,B) = k1_xboole_0
    <=> r1_tarski(A,B) ) ),
    inference(mizar_proof,[status(thm)],[l32_xboole_1]),
    [file(xboole_1,t37_xboole_1),interesting(0.36)]).

fof(t38_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,k4_xboole_0(B,A))
     => A = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_tarski,d4_xboole_0,d10_xboole_0,t2_xboole_1]),
    [file(xboole_1,t38_xboole_1),interesting(0.36)]).

fof(t55_xboole_1,theorem,(
    ! [A,B] : k4_xboole_0(k2_xboole_0(A,B),k3_xboole_0(A,B)) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(B,A)) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,d2_xboole_0,d3_xboole_0,d4_xboole_0,d2_xboole_0,d2_xboole_0,d4_xboole_0,d2_xboole_0,d3_xboole_0,d4_xboole_0,t2_tarski]),
    [file(xboole_1,t55_xboole_1),interesting(0.36)]).

fof(t15_xboole_1,theorem,(
    ! [A,B] : 
      ( k2_xboole_0(A,B) = k1_xboole_0
     => A = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[d2_xboole_0,d1_xboole_0]),
    [file(xboole_1,t15_xboole_1),interesting(0.35)]).

fof(t14_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B)
        & ! [D] : 
            ( ( r1_tarski(A,D)
              & r1_tarski(C,D) )
           => r1_tarski(B,D) ) )
     => B = k2_xboole_0(A,C) ) ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1,d10_xboole_0,t8_xboole_1]),
    [file(xboole_1,t14_xboole_1),interesting(0.30)]).

fof(t20_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(A,C)
        & ! [D] : 
            ( ( r1_tarski(D,B)
              & r1_tarski(D,C) )
           => r1_tarski(D,A) ) )
     => A = k3_xboole_0(B,C) ) ),
    inference(mizar_proof,[status(thm)],[t19_xboole_1,d10_xboole_0,t17_xboole_1]),
    [file(xboole_1,t20_xboole_1),interesting(0.30)]).

fof(t53_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(A,k2_xboole_0(B,C)) = k3_xboole_0(k4_xboole_0(A,B),k4_xboole_0(A,C)) ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1,t34_xboole_1,t7_xboole_1,t34_xboole_1,t19_xboole_1,d10_xboole_0,d3_tarski,d3_xboole_0,d4_xboole_0,d2_xboole_0,d4_xboole_0]),
    [file(xboole_1,t53_xboole_1),interesting(0.30)]).

fof(t97_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(k4_xboole_0(A,B),C)
        & r1_tarski(k4_xboole_0(B,A),C) )
     => r1_tarski(k5_xboole_0(A,B),C) ) ),
    inference(mizar_proof,[status(thm)],[t8_xboole_1]),
    [file(xboole_1,t97_xboole_1),interesting(0.30)]).

fof(t66_xboole_1,theorem,(
    ! [A] : 
      ( ~ ( ~ r1_xboole_0(A,A)
          & A = k1_xboole_0 )
      & ~ ( A != k1_xboole_0
          & r1_xboole_0(A,A) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0,d7_xboole_0]),
    [file(xboole_1,t66_xboole_1),interesting(0.29)]).

fof(t54_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(A,k3_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(A,C)) ),
    inference(mizar_proof,[status(thm)],[l36_xboole_1]),
    [file(xboole_1,t54_xboole_1),interesting(0.28)]).

fof(t67_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(A,C)
        & r1_xboole_0(B,C) )
     => A = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,t19_xboole_1,t3_xboole_1]),
    [file(xboole_1,t67_xboole_1),interesting(0.28)]).

fof(t78_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_xboole_0(A,B)
     => k3_xboole_0(A,k2_xboole_0(B,C)) = k3_xboole_0(A,C) ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,t23_xboole_1]),
    [file(xboole_1,t78_xboole_1),interesting(0.28)]).

fof(t101_xboole_1,theorem,(
    ! [A,B] : k5_xboole_0(A,B) = k4_xboole_0(k2_xboole_0(A,B),k3_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[l98_xboole_1]),
    [file(xboole_1,t101_xboole_1),interesting(0.27)]).

fof(t87_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_xboole_0(A,B)
     => k2_xboole_0(k4_xboole_0(C,A),B) = k4_xboole_0(k2_xboole_0(C,B),A) ) ),
    inference(mizar_proof,[status(thm)],[t42_xboole_1,t83_xboole_1]),
    [file(xboole_1,t87_xboole_1),interesting(0.26)]).

fof(t25_xboole_1,theorem,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(B,C)),k3_xboole_0(C,A)) = k3_xboole_0(k3_xboole_0(k2_xboole_0(A,B),k2_xboole_0(B,C)),k2_xboole_0(C,A)) ),
    inference(mizar_proof,[status(thm)],[t24_xboole_1,t4_xboole_1,t22_xboole_1,t4_xboole_1,t22_xboole_1,t24_xboole_1,t24_xboole_1,t16_xboole_1,t16_xboole_1,t16_xboole_1]),
    [file(xboole_1,t25_xboole_1),interesting(0.25)]).

fof(t107_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,k5_xboole_0(B,C))
    <=> ( r1_tarski(A,k2_xboole_0(B,C))
        & r1_xboole_0(A,k3_xboole_0(B,C)) ) ) ),
    inference(mizar_proof,[status(thm)],[l98_xboole_1,t86_xboole_1,t106_xboole_1]),
    [file(xboole_1,t107_xboole_1),interesting(0.23)]).

fof(t30_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => k2_xboole_0(A,k3_xboole_0(C,B)) = k3_xboole_0(k2_xboole_0(A,C),B) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_xboole_0,d3_xboole_0,d3_tarski,d2_xboole_0,d3_xboole_0,d10_xboole_0,d3_tarski,d3_xboole_0,d2_xboole_0,d3_xboole_0,d2_xboole_0]),
    [file(xboole_1,t30_xboole_1),interesting(0.17)]).

fof(t99_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(k5_xboole_0(A,B),C) = k2_xboole_0(k4_xboole_0(A,k2_xboole_0(B,C)),k4_xboole_0(B,k2_xboole_0(A,C))) ),
    inference(mizar_proof,[status(thm)],[t42_xboole_1,t41_xboole_1,t41_xboole_1]),
    [file(xboole_1,t99_xboole_1),interesting(0.11)]).

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(d3_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k3_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d3_xboole_0),
    [interesting(0.00)]).

fof(d4_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    [interesting(0.00)]).

fof(t2_tarski,theorem,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    [interesting(0.00)]).

fof(d2_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    [interesting(0.00)]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.00)]).

fof(t102_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(A,k5_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,k2_xboole_0(B,C)),k3_xboole_0(k3_xboole_0(A,B),C)) ),
    inference(mizar_proof,[status(thm)],[l98_xboole_1,t52_xboole_1,t16_xboole_1]),
    [file(xboole_1,t102_xboole_1),interesting(0.00)]).

fof(t3_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ ( r2_hidden(C,A)
              & r2_hidden(C,B) ) )
      & ~ ( ? [C] : 
              ( r2_hidden(C,A)
              & r2_hidden(C,B) )
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t3_xboole_0),
    [interesting(0.00)]).

fof(d8_xboole_0,definition,(
    ! [A,B] : 
      ( r2_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        & A != B ) ) ),
    file(xboole_0,d8_xboole_0),
    [interesting(0.00)]).

fof(d9_xboole_0,definition,(
    ! [A,B] : 
      ( r3_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        | r1_tarski(B,A) ) ) ),
    file(xboole_0,d9_xboole_0),
    [interesting(0.00)]).

fof(d7_xboole_0,definition,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
    <=> k3_xboole_0(A,B) = k1_xboole_0 ) ),
    file(xboole_0,d7_xboole_0),
    [interesting(0.00)]).

fof(d1_xboole_0,definition,(
    ! [A] : 
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) ),
    file(xboole_0,d1_xboole_0),
    [interesting(0.00)]).

fof(t57_xboole_1,theorem,(
    ! [A,B] : ~ ( r2_xboole_0(A,B)
      & r2_xboole_0(B,A) ) ),
    file(xboole_1,t57_xboole_1),
    [interesting(0.00)]).

fof(t4_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ r2_hidden(C,k3_xboole_0(A,B)) )
      & ~ ( ? [C] : r2_hidden(C,k3_xboole_0(A,B))
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t4_xboole_0),
    [interesting(0.00)]).