Axiom 1 (lid): e * x = x
Axiom 2 (rid): x * e = x
Axiom 3 (b1): x \ (x * y) = y
Axiom 4 (b2): x * (x \ y) = y
Axiom 5 (s1): (x * y) / y = x
Axiom 6 (s2): (x / y) * y = x
Definition 1: a(x,y,z) is defined to be (x * (y * z)) \ ((x * y) * z)
Definition 2: K(x,y) is defined to be (y * x) \ (x * y)
Definition 3: T(u,x) is defined to be x \ (u * x)
Definition 4: L(u,x,y) is defined to be (y * x) \ (y * (x * u))
Definition 5: R(u,x,y) is defined to be ((u * x) * y) / (x * y)
Axiom 7 (TT): T(T(u,x),y) = T(T(u,y),x)
Axiom 8 (TL): T(L(u,x,y),z) = L(T(u,z),x,y)
Axiom 9 (TR): T(R(u,x,y),z) = R(T(u,z),x,y)
Axiom 10 (LR): L(R(u,x,y),z,w) = R(L(u,z,w),x,y)
Axiom 11 (LL): L(L(u,x,y),z,w) = L(L(u,z,w),x,y)
Axiom 12 (RR): R(R(u,x,y),z,w) = R(R(u,z,w),x,y)
Proposition 1 (pos): x * y = z → z / y = x
Proposition 2 (pob): x * y = z → x \ z = y
Proposition 3 (pso): z / y = x → x * y = z
Proposition 4 (psb): z / y = x → x \ z = y
Proposition 5 (pbo): x \ z = y → x * y = z
Proposition 6 (pbs): x \ z = y → z / y = x
Proposition 7 (co1): x * y = u ∧ x * z = u → y = z
Proposition 8 (co2): y * x = u ∧ z * x = u → y = z
Proposition 9 (co3): x * y = x * z → y = z
Proposition 10 (co4): y * x = z * x → y = z
Proposition 11 (cs1): x / y = u ∧ x / z = u → y = z
Proposition 12 (cs2): y / x = u ∧ z / x = u → y = z
Proposition 13 (cs3): x / y = x / z → y = z
Proposition 14 (cs4): y / x = z / x → y = z
Proposition 15 (cb1): x \ y = u ∧ x \ z = u → y = z
Proposition 16 (cb2): y \ x = u ∧ z \ x = u → y = z
Proposition 17 (cb3): x \ y = x \ z → y = z
Proposition 18 (cb4): y \ x = z \ x → y = z
Proposition 19 (comp1): a(x,y,z) = e → L(z,y,x) = z
Proposition 20 (comp2): L(z,y,x) = z → a(x,y,z) = e
Proposition 21 (comp3): T(x,y) = x → T(y,x) = y
Proposition 22 (comp4): T(x,y) = x → K(x,y) = e
Proposition 23 (comp5): K(x,y) = e → T(x,y) = x
Proposition 24 (id1): y / (x \ y) = x
Proposition 25 (id2): (y / x) \ y = x
Proposition 26 (id3): e \ x = x
Proposition 27 (id4): x / e = x
Proposition 28 (id5): x \ x = e
Proposition 29 (id6): x / x = e
Proposition 30 (bid): x \ y = e → x = y
Proposition 31 (sid): x / y = e → x = y
Proposition 32 (assoc_ae): x * (y * z) = (x * y) * z → a(x,y,z) = e
Proposition 33 (ae_assoc): a(x,y,z) = e → x * (y * z) = (x * y) * z
Proposition 34 (com_Ke): x * y = y * x → K(x,y) = e
Proposition 35 (Ke_com): K(x,y) = e → x * y = y * x
Definition 6: F(u,x) is defined to be (x * u) / x
Proposition 36 (Tinj): T(z,x) = T(u,x) → z = u
Proposition 37 (Linj): L(z,x,y) = L(u,x,y) → z = u
Proposition 38 (Rinj): R(z,x,y) = R(u,x,y) → z = u
Proposition 39 (Finj): F(z,x) = F(u,x) → z = u
Proposition 40 (TFid): T(F(u,x),x) = u
Proposition 41 (FTid): F(T(u,x),x) = u
Proposition 42 (FF): F(F(u,x),y) = F(F(u,y),x)
Proposition 43 (FL): F(L(u,x,y),z) = L(F(u,z),x,y)
Proposition 44 (FR): F(R(u,x,y),z) = R(F(u,z),x,y)
Proposition 45 (FT): F(T(u,x),y) = T(F(u,y),x)
Proposition 46 (prop_034cf5c5): x * T(y,x) = y * x
Proposition 47 (prop_66f8dd43): T(x / y,y) = y \ x
Proposition 48 (prop_81894ca4): (x * T(y,x)) / x = y
Proposition 49 (prop_da4738e2): T(x,x \ y) = (x \ y) \ y
Proposition 50 (prop_4a90a23f): x * T(T(y,x),z) = T(y,z) * x
Proposition 51 (prop_73fa4877): T(T(x / y,z),y) = T(y \ x,z)
Proposition 52 (prop_d10a3b1a): (x * y) * L(z,y,x) = x * (y * z)
Proposition 53 (prop_293cbc16): L(x \ y,x,z) = (z * x) \ (z * y)
Proposition 54 (prop_55464ec9): R(x,y,z) * (y * z) = (x * y) * z
Proposition 55 (prop_61fb8127): R(x / y,y,z) = (x * z) / (y * z)
Proposition 56 (prop_ddd1c86f): x * ((x \ e) * y) = L(y,x \ e,x)
Proposition 57 (prop_0d7e7151): (x \ e) * y = x \ L(y,x \ e,x)
Proposition 58 (prop_1aae4a83): x * L(T(y,x),z,w) = L(y,z,w) * x
Proposition 59 (prop_1db0183a): x * R(T(y,x),z,w) = R(y,z,w) * x
Proposition 60 (prop_a4abb1e0): T(R(x / y,z,w),y) = R(y \ x,z,w)
Proposition 61 (prop_55842885): T(x / y,z) * y = y * T(y \ x,z)
Proposition 62 (prop_1a725917): (x * y) \ (x * (z * y)) = L(T(z,y),y,x)
Proposition 63 (prop_c626af2d): T(x,x * y) = L(T(x,y),y,x)
Proposition 64 (prop_526a359c): T((x * y) / (z * y),z) = R(z \ x,z,y)
Proposition 65 (prop_deeac89a): T(R(x,y,z),w) * (y * z) = (T(x,w) * y) * z
Proposition 66 (prop_575d1ed9): R(x,x \ e,y) = y / ((x \ e) * y)
Proposition 67 (prop_f338e359): (e / x) * (x * y) = L(y,x,e / x)
Proposition 68 (prop_5a914e30): R(x,y,y \ e) = (x * y) * (y \ e)
Proposition 69 (prop_cc8a9ae6): R(x,e / y,y) = (x * (e / y)) * y
Proposition 70 (prop_c2aa8580): L(x \ (y \ z),x,y) = (y * x) \ z
Proposition 71 (prop_b3890d2c): R(x / y,y,y \ e) = x * (y \ e)
Proposition 72 (prop_f14899ed): T(L(x / y,z,w),y) = L(y \ x,z,w)
Proposition 73 (prop_f2b7d0ab): L(x \ T(y,z),x,z) = (z * x) \ (y * z)
Proposition 74 (prop_be6cad0a): R((x / y) / z,z,y) = x / (z * y)
Proposition 75 (prop_1e558562): L((x \ y) / z,z,x) = (z * ((x * z) \ y)) / z
Proposition 76 (prop_e9eef609): K(x \ e,x) = (x \ e) * x
Proposition 77 (prop_9ee87fb5): K(x,e / x) = x * (e / x)
Proposition 78 (prop_da958b3f): L(x \ e,x,y) = (y * x) \ y
Proposition 79 (prop_c3fa51e8): R(e / x,x,y) = y / (x * y)
Proposition 80 (prop_ba6418d1): R(e / x,x,y) \ y = x * y
Proposition 81 (prop_d7dd57dd): (x * y) \ ((z * x) * y) = R(T(z,x * y),x,y)
Proposition 82 (prop_e1aa92db): (x * y) * K(y,x) = y * x
Proposition 83 (prop_19fcac9b2): R(x / y,z,w) * y = y * R(y \ x,z,w)
Proposition 84 (prop_3d75df700): (x * y) * R((x * y) \ y,z,w) = (x * R(x \ e,z,w)) * y
Proposition 85 (prop_acafcc6f0): L(R(x \ (y \ z),w,u),x,y) = R((y * x) \ z,w,u)
Proposition 86 (prop_203fc9151): x * (y * R(y \ (x \ y),z,w)) = (x * R(x \ e,z,w)) * y
Proposition 87 (prop_2e844a2a9): (x * y) * R((x * y) \ z,w,u) = x * (y * R(y \ (x \ z),w,u))
Proposition 88 (prop_d9f457e09): (x \ y) * R((x \ y) \ y,z,w) = R(x,z,w) * (x \ y)
Proposition 89 (prop_ce2987245): x \ R(x,y,z) = R(x,y,z) * (x \ e)
Proposition 90 (prop_b7fe5fbfb): K(x \ e,x) = (e / x) \ (x \ e)
Theorem 1 (prov9_9c01197768): a(x,y / z,z) = (x * y) \ ((x * (y / z)) * z)
Proof. We have a(x,y / z,z) = (x * ((y / z) * z)) \ ((x * (y / z)) * z) by Definition 1. Hence we are done by Axiom 6.
Theorem 2 (prov9_845295062d): K(y \ x,y) = x \ ((y \ x) * y)
Proof. We have K(y \ x,y) = (y * (y \ x)) \ ((y \ x) * y) by Definition 2. Hence we are done by Axiom 4.
Theorem 3 (prov9_d7b75a47a4): K(y,x / y) = x \ (y * (x / y))
Proof. We have K(y,x / y) = ((x / y) * y) \ (y * (x / y)) by Definition 2. Hence we are done by Axiom 6.
Theorem 4 (prov9_8f0630f296): L(z,y,x / y) = x \ ((x / y) * (y * z))
Proof. We have L(z,y,x / y) = ((x / y) * y) \ ((x / y) * (y * z)) by Definition 4. Hence we are done by Axiom 6.
Theorem 5 (prov9_56306af1a4): (x * y) / T(x,y) = y
Proof. We have (x * y) / (y \ (x * y)) = y by Proposition 24. Hence we are done by Definition 3.
Theorem 6 (prov9_dc338e99f5): R(x,y,z) \ ((x * y) * z) = y * z
Proof. We have (((x * y) * z) / (y * z)) \ ((x * y) * z) = y * z by Proposition 25. Hence we are done by Definition 5.
Theorem 7 (prov9_3a38636154): T((x * y) / x,x) = y
Proof. We have
(L1) x * T((x * y) / x,x) = ((x * y) / x) * x
by Proposition 46. We have x * y = ((x * y) / x) * x by Axiom 6. Hence we are done by (L1) and Proposition 7.
Theorem 8 (prov9_111601b442): (x \ y) * T(x,x \ y) = y
Proof. We have x * (x \ y) = y by Axiom 4. Hence we are done by Proposition 46.
Theorem 9 (prov9_b2c1eb82ce): z * (x * (x \ T(y,z))) = y * z
Proof. We have z * T(y,z) = y * z by Proposition 46. Hence we are done by Axiom 4.
Theorem 10 (prov9_bf75bab344): x * T(y / x,x) = y
Proof. We have (y / x) * x = y by Axiom 6. Hence we are done by Proposition 46.
Theorem 11 (prov9_74d78eb007): x * y = y * z → T(x,y) = z
Proof. Assume
(H1) x * y = y * z.
We have y * T(x,y) = x * y by Proposition 46. Then y * T(x,y) = y * z by (H1). Hence we are done by Proposition 9.
Theorem 12 (prov9_64380845d2): K(T(x,y),y) = (x * y) \ (T(x,y) * y)
Proof. We have K(T(x,y),y) = (y * T(x,y)) \ (T(x,y) * y) by Definition 2. Hence we are done by Proposition 46.
Theorem 13 (prov9_ec40c6abb5): T(y,T(x,y)) = T(x,y) \ (x * y)
Proof. We have T(y,T(x,y)) = T(x,y) \ (y * T(x,y)) by Definition 3. Hence we are done by Proposition 46.
Theorem 14 (prov9_e121dd2cc9): T(x / (x / y),x / y) = y
Proof. We have (x / y) \ x = y by Proposition 25. Hence we are done by Proposition 47.
Theorem 15 (prov9_9c36d4f6ef): y / T(x,x \ y) = x \ y
Proof. We have (x * (x \ y)) / T(x,x \ y) = x \ y by Theorem 5. Hence we are done by Axiom 4.
Theorem 16 (prov9_e48169321d): (x * T(T(y,x),z)) / x = T(y,z)
Proof. We have (x * T(T(y,z),x)) / x = T(y,z) by Proposition 48. Hence we are done by Axiom 7.
Theorem 17 (prov9_4c82c6ab39): x * K(y \ x,y) = (y \ x) * y
Proof. We have (y * (y \ x)) * K(y \ x,y) = (y \ x) * y by Proposition 82. Hence we are done by Axiom 4.
Theorem 18 (prov9_f18fbc71dc): T((y \ z) \ z,x) = T(T(y,x),y \ z)
Proof. We have T(T(y,y \ z),x) = T(T(y,x),y \ z) by Axiom 7. Hence we are done by Proposition 49.
Theorem 19 (prov9_00cd9e5d9b): (x / z) \ x = y * (y \ z)
Proof. We have y * (y \ z) = z by Axiom 4. Then
(L2) ((y * (y \ z)) / e) * e = z
by Axiom 6. We have ((y * (y \ z)) / e) * e = y * (y \ z) by Axiom 6. Then (x / (((y * (y \ z)) / e) * e)) \ x = y * (y \ z) by Proposition 25. Hence we are done by (L2).
Theorem 20 (prov9_51c11aebbd): y = (x \ y) * z → T(x,x \ y) = z
Proof. Assume
(H1) y = (x \ y) * z.
We have (x \ y) * T(x,x \ y) = y by Theorem 8. Then (x \ y) * T(x,x \ y) = (x \ y) * z by (H1). Hence we are done by Proposition 9.
Theorem 21 (prov9_e602b43d9b): (x \ e) \ K(x \ e,x) = x
Proof. We have (x \ e) * x = K(x \ e,x) by Proposition 76. Hence we are done by Proposition 2.
Theorem 22 (prov9_0f1891cc96): x \ K(x,e / x) = e / x
Proof. We have x * (e / x) = K(x,e / x) by Proposition 77. Hence we are done by Proposition 2.
Theorem 23 (prov9_105f0176b3): R(x,y / z,z) * y = (x * (y / z)) * z
Proof. We have e * y = y by Axiom 1. Then
(L2) ((x * (y / z)) * z) * (((x * (y / z)) * z) \ (e * y)) = y
by Axiom 4. We have ((x * (y / z)) * z) * (((x * (y / z)) * z) \ (e * y)) = e * y by Axiom 4. Then ((((x * (y / z)) * z) * (((x * (y / z)) * z) \ (e * y))) / z) * z = e * y by Axiom 6. Then
(L5) (y / z) * z = e * y
by (L2). We have
(L6) e * y = (((x * (y / z)) * z) / (e * y)) \ ((x * (y / z)) * z)
by Proposition 25. We have e * ((((x * (y / z)) * z) / (e * y)) \ ((x * (y / z)) * z)) = (((x * (y / z)) * z) / (e * y)) \ ((x * (y / z)) * z) by Axiom 1. Then
(L8) y = (((x * (y / z)) * z) / (e * y)) \ ((x * (y / z)) * z)
by (L6) and Proposition 7. We have (((x * (y / z)) * z) / (e * y)) * ((((x * (y / z)) * z) / (e * y)) \ ((x * (y / z)) * z)) = (x * (y / z)) * z by Axiom 4. Then (((x * (y / z)) * z) / (e * y)) * y = (x * (y / z)) * z by (L8). Then (((x * (y / z)) * z) / ((y / z) * z)) * y = (x * (y / z)) * z by (L5). Hence we are done by Definition 5.
Theorem 24 (prov9_6cef0cef0a): x / (z \ ((x / y) * y)) = z
Proof. We have ((x / (z \ ((x / y) * y))) * (z \ ((x / y) * y))) / (z \ ((x / y) * y)) = x / (z \ ((x / y) * y)) by Axiom 5. Then ((((x / (z \ ((x / y) * y))) * (z \ ((x / y) * y))) / y) * y) / (z \ ((x / y) * y)) = x / (z \ ((x / y) * y)) by Axiom 6. Then
(L3) ((x / y) * y) / (z \ ((x / y) * y)) = x / (z \ ((x / y) * y))
by Axiom 6. We have ((x / y) * y) / (z \ ((x / y) * y)) = z by Proposition 24. Hence we are done by (L3).
Theorem 25 (prov9_7e82de43cd): R(x,y,y \ z) * z = (x * y) * (y \ z)
Proof. We have (((x * y) * (y \ z)) / (y * (y \ z))) * ((((x * y) * (y \ z)) / (y * (y \ z))) \ ((((x * y) * (y \ z)) / (y * (y \ z))) * z)) = (((x * y) * (y \ z)) / (y * (y \ z))) * z by Axiom 4. Then (((x * y) * (y \ z)) / (y * (y \ z))) * (y * (y \ ((((x * y) * (y \ z)) / (y * (y \ z))) \ ((((x * y) * (y \ z)) / (y * (y \ z))) * z)))) = (((x * y) * (y \ z)) / (y * (y \ z))) * z by Axiom 4. Then
(L3) (((x * y) * (y \ z)) / (y * (y \ z))) * (y * (y \ z)) = (((x * y) * (y \ z)) / (y * (y \ z))) * z
by Axiom 3. We have (((x * y) * (y \ z)) / (y * (y \ z))) * (y * (y \ z)) = (x * y) * (y \ z) by Axiom 6. Then (((x * y) * (y \ z)) / (y * (y \ z))) * z = (x * y) * (y \ z) by (L3). Hence we are done by Definition 5.
Theorem 26 (prov9_6e97ce8b3a): ((x / y) \ x) * z = y * z
Proof. We have ((y * z) / z) * z = y * z by Axiom 6. Then ((((y * z) / z) / (e \ y)) * (e \ y)) * z = y * z by Axiom 6. Then ((y / (e \ y)) * (e \ y)) * z = y * z by Axiom 5. Then (e * (e \ y)) * z = y * z by Proposition 24. Hence we are done by Theorem 19.
Theorem 27 (prov9_7f77a2fd10): x * (((y / x) \ y) \ z) = z
Proof. We have ((y / x) \ y) * (((y / x) \ y) \ z) = z by Axiom 4. Hence we are done by Theorem 26.
Theorem 28 (prov9_b3a84cb14a): L((y \ x) \ e,y \ x,y) = x \ y
Proof. We have L((y \ x) \ e,y \ x,y) = (y * (y \ x)) \ y by Proposition 78. Hence we are done by Axiom 4.
Theorem 29 (prov9_ccb493e02d): L(y \ e,y,x / y) = x \ (x / y)
Proof. We have L(y \ e,y,x / y) = ((x / y) * y) \ (x / y) by Proposition 78. Hence we are done by Axiom 6.
Theorem 30 (prov9_1ffb5e2572): L(T(x \ e,z),x,y) = T((y * x) \ y,z)
Proof. We have L(T(x \ e,z),x,y) = T(L(x \ e,x,y),z) by Axiom 8. Hence we are done by Proposition 78.
Theorem 31 (prov9_48bd3a10d9): R((w * z) \ w,x,y) = L(R(z \ e,x,y),z,w)
Proof. We have R(L(z \ e,z,w),x,y) = L(R(z \ e,x,y),z,w) by Axiom 10. Hence we are done by Proposition 78.
Theorem 32 (prov9_888e80a1bd): L((w * z) \ w,x,y) = L(L(z \ e,x,y),z,w)
Proof. We have L(L(z \ e,z,w),x,y) = L(L(z \ e,x,y),z,w) by Axiom 11. Hence we are done by Proposition 78.
Theorem 33 (prov9_9e92ee5a25): x / L(y \ e,y,x) = x * y
Proof. We have x / ((x * y) \ x) = x * y by Proposition 24. Hence we are done by Proposition 78.
Theorem 34 (prov9_2d217e97e1): L(x,e / x,y) = (y * (e / x)) \ y
Proof. We have L((e / x) \ e,e / x,y) = (y * (e / x)) \ y by Proposition 78. Hence we are done by Proposition 25.
Theorem 35 (prov9_bb158ae903): L(x \ e,x,e / x) = e / x
Proof. We have e \ (e / x) = e / x by Proposition 26. Then ((e / x) * x) \ (e / x) = e / x by Axiom 6. Hence we are done by Proposition 78.
Theorem 36 (prov9_cc402e4a6c): L(x \ e,x,x \ e) = K(x \ e,x) \ (x \ e)
Proof. We have L(x \ e,x,x \ e) = ((x \ e) * x) \ (x \ e) by Proposition 78. Hence we are done by Proposition 76.
Theorem 37 (prov9_881108467c): R(e / x,x,x \ y) = (x \ y) / y
Proof. We have R(e / x,x,x \ y) = (x \ y) / (x * (x \ y)) by Proposition 79. Hence we are done by Axiom 4.
Theorem 38 (prov9_b509691b11): R(e / (y / x),y / x,x) = x / y
Proof. We have R(e / (y / x),y / x,x) = x / ((y / x) * x) by Proposition 79. Hence we are done by Axiom 6.
Theorem 39 (prov9_7c96e347d4): R(T(e / x,z),x,y) = T(y / (x * y),z)
Proof. We have R(T(e / x,z),x,y) = T(R(e / x,x,y),z) by Axiom 9. Hence we are done by Proposition 79.
Theorem 40 (prov9_a1ea40d446): R(w / (z * w),x,y) = R(R(e / z,x,y),z,w)
Proof. We have R(R(e / z,z,w),x,y) = R(R(e / z,x,y),z,w) by Axiom 12. Hence we are done by Proposition 79.
Theorem 41 (prov9_86a831c50b): R(e / x,x,x \ e) = x \ e
Proof. We have (x \ e) / e = x \ e by Proposition 27. Then (x \ e) / (x * (x \ e)) = x \ e by Axiom 4. Hence we are done by Proposition 79.
Theorem 42 (prov9_16d9106c18): ((y * x) \ y) * z = L(x \ e,x,y) * z
Proof. We have ((y / L(x \ e,x,y)) \ y) * z = L(x \ e,x,y) * z by Theorem 26. Hence we are done by Theorem 33.
Theorem 43 (prov9_bd60bde900): L(x \ e,x,y) * (((y * x) \ y) \ z) = z
Proof. We have L(x \ e,x,y) * (((y / L(x \ e,x,y)) \ y) \ z) = z by Theorem 27. Hence we are done by Theorem 33.
Theorem 44 (prov9_3cb093d8d3): L(T(x \ e,y),x,e / x) = T(e / x,y)
Proof. We have L(T(x \ e,y),x,e / x) = T(L(x \ e,x,e / x),y) by Axiom 8. Hence we are done by Theorem 35.
Theorem 45 (prov9_3e047dc57d): R(x,x \ e,y) \ y = (x \ e) * y
Proof. We have R(e / (x \ e),x \ e,y) \ y = (x \ e) * y by Proposition 80. Hence we are done by Proposition 24.
Theorem 46 (prov9_353af6ed74): x \ (y * T(y \ x,y)) = K(y \ x,y)
Proof. We have x \ ((y \ x) * y) = K(y \ x,y) by Theorem 2. Hence we are done by Proposition 46.
Theorem 47 (prov9_2c6ff9f17a): z * T(x,y) = T((z * x) / z,y) * z
Proof. We have z * T(T((z * x) / z,z),y) = T((z * x) / z,y) * z by Proposition 50. Hence we are done by Theorem 7.
Theorem 48 (prov9_372d93a908): (y \ z) * T((y \ z) \ z,x) = T(y,x) * (y \ z)
Proof. We have (y \ z) * T(T(y,y \ z),x) = T(y,x) * (y \ z) by Proposition 50. Hence we are done by Proposition 49.
Theorem 49 (prov9_bff3f02308): (y * T(y \ x,z)) / y = T(x / y,z)
Proof. We have (y * T(T(x / y,z),y)) / y = T(x / y,z) by Proposition 48. Hence we are done by Proposition 51.
Theorem 50 (prov9_e81158a000): (y * T(x,z)) / y = T((y * x) / y,z)
Proof. We have (y * T(T((y * x) / y,y),z)) / y = T((y * x) / y,z) by Theorem 16. Hence we are done by Theorem 7.
Theorem 51 (prov9_42fd5c4565): ((x / y) * T(y,z)) / (x / y) = T(x / (x / y),z)
Proof. We have ((x / y) * T(T(x / (x / y),x / y),z)) / (x / y) = T(x / (x / y),z) by Theorem 16. Hence we are done by Theorem 14.
Theorem 52 (prov9_bd00283a61): y * L(z,x \ y,x) = x * ((x \ y) * z)
Proof. We have (x * (x \ y)) * L(z,x \ y,x) = x * ((x \ y) * z) by Proposition 52. Hence we are done by Axiom 4.
Theorem 53 (prov9_a6675ab488): x * L(z,y,x / y) = (x / y) * (y * z)
Proof. We have ((x / y) * y) * L(z,y,x / y) = (x / y) * (y * z) by Proposition 52. Hence we are done by Axiom 6.
Theorem 54 (prov9_07f4728f13): z * (y * x) = (z * y) * w → L(x,y,z) = w
Proof. Assume
(H1) z * (y * x) = (z * y) * w.
We have (z * y) * L(x,y,z) = z * (y * x) by Proposition 52. Then (z * y) * L(x,y,z) = (z * y) * w by (H1). Hence we are done by Proposition 9.
Theorem 55 (prov9_3aca1b8d7f): (x * y) * T(L(z,y,x),w) = x * (y * T(z,w))
Proof. We have (x * y) * L(T(z,w),y,x) = x * (y * T(z,w)) by Proposition 52. Hence we are done by Axiom 8.
Theorem 56 (prov9_50cfb81303): (x * y) * R(L(z,y,x),w,u) = x * (y * R(z,w,u))
Proof. We have (x * y) * L(R(z,w,u),y,x) = x * (y * R(z,w,u)) by Proposition 52. Hence we are done by Axiom 10.
Theorem 57 (prov9_9ed4d18c14): (x * y) * L(L(z,y,x),w,u) = x * (y * L(z,w,u))
Proof. We have (x * y) * L(L(z,w,u),y,x) = x * (y * L(z,w,u)) by Proposition 52. Hence we are done by Axiom 11.
Theorem 58 (prov9_dc13711b88): (x * y) * L(z,T(x,y),y) = y * (T(x,y) * z)
Proof. We have (y * T(x,y)) * L(z,T(x,y),y) = y * (T(x,y) * z) by Proposition 52. Hence we are done by Proposition 46.
Theorem 59 (prov9_cda7f0b937): K(x \ e,x) * L(y,x,x \ e) = (x \ e) * (x * y)
Proof. We have ((x \ e) * x) * L(y,x,x \ e) = (x \ e) * (x * y) by Proposition 52. Hence we are done by Proposition 76.
Theorem 60 (prov9_91aa1685a0): L((y \ x) \ z,y \ x,y) = x \ (y * z)
Proof. We have L((y \ x) \ z,y \ x,y) = (y * (y \ x)) \ (y * z) by Proposition 53. Hence we are done by Axiom 4.
Theorem 61 (prov9_b3c0beca0a): (y * x) / L(z \ x,z,y) = y * z
Proof. We have (y * x) / ((y * z) \ (y * x)) = y * z by Proposition 24. Hence we are done by Proposition 53.
Theorem 62 (prov9_65b83fe1bd): L((x \ e) \ y,x \ e,x) = x * y
Proof. We have e \ (x * y) = x * y by Proposition 26. Then (x * (x \ e)) \ (x * y) = x * y by Axiom 4. Hence we are done by Proposition 53.
Theorem 63 (prov9_5c4ce47872): L(x \ y,x,e / x) = (e / x) * y
Proof. We have e \ ((e / x) * y) = (e / x) * y by Proposition 26. Then ((e / x) * x) \ ((e / x) * y) = (e / x) * y by Axiom 6. Hence we are done by Proposition 53.
Theorem 64 (prov9_e31740b326): (x * y) * z = w * (y * z) → R(x,y,z) = w
Proof. Assume
(H1) (x * y) * z = w * (y * z).
We have R(x,y,z) * (y * z) = (x * y) * z by Proposition 54. Then R(x,y,z) * (y * z) = w * (y * z) by (H1). Hence we are done by Proposition 10.
Theorem 65 (prov9_b51d69c9ff): L(R(x,w,u),y,z) * (w * u) = (L(x,y,z) * w) * u
Proof. We have R(L(x,y,z),w,u) * (w * u) = (L(x,y,z) * w) * u by Proposition 54. Hence we are done by Axiom 10.
Theorem 66 (prov9_3e0b9638c4): R(x / y,y,y \ z) = (x * (y \ z)) / z
Proof. We have R(x / y,y,y \ z) = (x * (y \ z)) / (y * (y \ z)) by Proposition 55. Hence we are done by Axiom 4.
Theorem 67 (prov9_21f5c76235): R(x / (z / y),z / y,y) = (x * y) / z
Proof. We have R(x / (z / y),z / y,y) = (x * y) / ((z / y) * y) by Proposition 55. Hence we are done by Axiom 6.
Theorem 68 (prov9_9ac2e22b67): R(x / (e / y),e / y,y) = x * y
Proof. We have (x * y) / e = x * y by Proposition 27. Then (x * y) / ((e / y) * y) = x * y by Axiom 6. Hence we are done by Proposition 55.
Theorem 69 (prov9_e86c30861e): L(x,y \ e,y) / ((y \ e) * x) = y
Proof. We have y * ((y \ e) * x) = L(x,y \ e,y) by Proposition 56. Hence we are done by Proposition 1.
Theorem 70 (prov9_bc3902f258): x * K(x \ e,x) = L(x,x \ e,x)
Proof. We have x * ((x \ e) * x) = L(x,x \ e,x) by Proposition 56. Hence we are done by Proposition 76.
Theorem 71 (prov9_de3066f53e): (e / x) \ L(y,x,e / x) = x * y
Proof. We have (e / x) * (x * y) = L(y,x,e / x) by Proposition 67. Hence we are done by Proposition 2.
Theorem 72 (prov9_e17216d269): (e / y) * (x * y) = L(T(x,y),y,e / y)
Proof. We have (e / y) * (y * T(x,y)) = L(T(x,y),y,e / y) by Proposition 67. Hence we are done by Proposition 46.
Theorem 73 (prov9_5ac80d7449): R(x,y,y \ e) / (y \ e) = x * y
Proof. We have (x * y) * (y \ e) = R(x,y,y \ e) by Proposition 68. Hence we are done by Proposition 1.
Theorem 74 (prov9_9e549c298b): R(x,e / y,y) / y = x * (e / y)
Proof. We have (x * (e / y)) * y = R(x,e / y,y) by Proposition 69. Hence we are done by Proposition 1.
Theorem 75 (prov9_8a4e701251): L(T(y \ e,z),y,x / y) = T(x \ (x / y),z)
Proof. We have L(T(y \ e,z),y,x / y) = T(L(y \ e,y,x / y),z) by Axiom 8. Hence we are done by Theorem 29.
Theorem 76 (prov9_1fb145d496): L(x,e / x,x) = K(x,e / x) \ x
Proof. We have L(x,e / x,x) = (x * (e / x)) \ x by Theorem 34. Hence we are done by Proposition 77.
Theorem 77 (prov9_04f8f002cd): T(x,x * y) = T(L(x,y,x),y)
Proof. We have L(T(x,y),y,x) = T(L(x,y,x),y) by Axiom 8. Hence we are done by Proposition 63.
Theorem 78 (prov9_d633abd527): L(T(T(x,y),z),y,x) = T(T(x,x * y),z)
Proof. We have L(T(T(x,y),z),y,x) = T(L(T(x,y),y,x),z) by Axiom 8. Hence we are done by Proposition 63.
Theorem 79 (prov9_7f64a796c0): y * T(x,x * y) = L(x,y,x) * y
Proof. We have y * T(L(x,y,x),y) = L(x,y,x) * y by Proposition 46. Hence we are done by Theorem 77.
Theorem 80 (prov9_fc5c32616b): (x * T(y,y * x)) / x = L(y,x,y)
Proof. We have (x * T(L(y,x,y),x)) / x = L(y,x,y) by Proposition 48. Hence we are done by Theorem 77.
Theorem 81 (prov9_9491eaca3d): R((e / w) * x,y,z) = L(R(w \ x,y,z),w,e / w)
Proof. We have R(L(w \ x,w,e / w),y,z) = L(R(w \ x,y,z),w,e / w) by Axiom 10. Hence we are done by Theorem 63.
Theorem 82 (prov9_3ce9b639ff): (e / x) \ L(L(y,x,e / x),z,w) = x * L(y,z,w)
Proof. We have (e / x) \ L(L(y,z,w),x,e / x) = x * L(y,z,w) by Theorem 71. Hence we are done by Axiom 11.
Theorem 83 (prov9_4dd4fc483d): L(R(x,w,w \ e),y,z) / (w \ e) = L(x,y,z) * w
Proof. We have R(L(x,y,z),w,w \ e) / (w \ e) = L(x,y,z) * w by Theorem 73. Hence we are done by Axiom 10.
Theorem 84 (prov9_cb38e45b35): L(R(x,e / w,w),y,z) / w = L(x,y,z) * (e / w)
Proof. We have R(L(x,y,z),e / w,w) / w = L(x,y,z) * (e / w) by Theorem 74. Hence we are done by Axiom 10.
Theorem 85 (prov9_ac5c0f43a6): w * L(w \ x,y,z) = L(x / w,y,z) * w
Proof. We have w * L(T(x / w,w),y,z) = L(x / w,y,z) * w by Proposition 58. Hence we are done by Proposition 47.
Theorem 86 (prov9_ede6c7fae6): w * L(x,y,z) = L((w * x) / w,y,z) * w
Proof. We have w * L(T((w * x) / w,w),y,z) = L((w * x) / w,y,z) * w by Proposition 58. Hence we are done by Theorem 7.
Theorem 87 (prov9_d00de280a7): w * R(x,y,z) = R((w * x) / w,y,z) * w
Proof. We have w * R(T((w * x) / w,w),y,z) = R((w * x) / w,y,z) * w by Proposition 59. Hence we are done by Theorem 7.
Theorem 88 (prov9_8b992aaa66): (y * R(y \ x,z,w)) / y = R(x / y,z,w)
Proof. We have (y * T(R(x / y,z,w),y)) / y = R(x / y,z,w) by Proposition 48. Hence we are done by Proposition 60.
Theorem 89 (prov9_e4fcc12aaa): T(y / (x * y),x) = R(x \ e,x,y)
Proof. We have T(R(e / x,x,y),x) = R(x \ e,x,y) by Proposition 60. Hence we are done by Proposition 79.
Theorem 90 (prov9_f5ce0eb528): y * R(y \ e,y,x) = (x / (y * x)) * y
Proof. We have y * T(x / (y * x),y) = (x / (y * x)) * y by Proposition 46. Hence we are done by Theorem 89.
Theorem 91 (prov9_b7f69918ab): T(y / x,(x \ y) * x) = L(x \ y,x,x \ y)
Proof. We have
(L1) T(y / x,(x \ y) * x) * x = x * T(x \ y,(x \ y) * x)
by Proposition 61. We have L(x \ y,x,x \ y) * x = x * T(x \ y,(x \ y) * x) by Theorem 79. Hence we are done by (L1) and Proposition 8.
Theorem 92 (prov9_5c56ab88f3): T((x * y) / x,y * x) = L(y,x,y)
Proof. We have
(L1) T((x * y) / x,y * x) * x = x * T(y,y * x)
by Theorem 47. We have L(y,x,y) * x = x * T(y,y * x) by Theorem 79. Hence we are done by (L1) and Proposition 8.
Theorem 93 (prov9_4f1ac4df82): z * ((y * T(x,z)) / y) = ((y * x) / y) * z
Proof. We have z * T((y * x) / y,z) = ((y * x) / y) * z by Proposition 46. Hence we are done by Theorem 50.
Theorem 94 (prov9_d38bdcbfda): (y * (z \ x)) / y = T((y * (x / z)) / y,z)
Proof. We have (y * T(x / z,z)) / y = T((y * (x / z)) / y,z) by Theorem 50. Hence we are done by Proposition 47.
Theorem 95 (prov9_92aac183a3): y \ ((y / z) * (x * z)) = L(T(x,z),z,y / z)
Proof. We have y \ ((y / z) * (z * T(x,z))) = L(T(x,z),z,y / z) by Theorem 4. Hence we are done by Proposition 46.
Theorem 96 (prov9_2b25ae80c6): L(y,x \ e,x) \ y = a(x,x \ e,y)
Proof. We have e * y = y by Axiom 1. Then
(L2) (x * (x \ e)) * y = y
by Axiom 4. We have a(x,x \ e,y) = (x * ((x \ e) * y)) \ ((x * (x \ e)) * y) by Definition 1. Then a(x,x \ e,y) = (x * ((x \ e) * y)) \ y by (L2). Hence we are done by Proposition 56.
Theorem 97 (prov9_127a481e20): L(R(x,w,w \ u),y,z) * u = (L(x,y,z) * w) * (w \ u)
Proof. We have R(L(x,y,z),w,w \ u) * u = (L(x,y,z) * w) * (w \ u) by Theorem 25. Hence we are done by Axiom 10.
Theorem 98 (prov9_7fed2c3e64): (x * y) * T((x * y) \ x,z) = x * (y * T(y \ e,z))
Proof. We have (x * y) * L(T(y \ e,z),y,x) = x * (y * T(y \ e,z)) by Proposition 52. Hence we are done by Theorem 30.
Theorem 99 (prov9_391f83e2d8): L(x \ e,x,y) \ (((y * x) \ y) * z) = z
Proof. We have L(x \ e,x,y) * z = ((y * x) \ y) * z by Theorem 42. Hence we are done by Proposition 2.
Theorem 100 (prov9_0689f190c2): L(x \ e,x,y) \ z = ((y * x) \ y) \ z
Proof. We have L(x \ e,x,y) * (((y * x) \ y) \ z) = z by Theorem 43. Hence we are done by Proposition 2.
Theorem 101 (prov9_cdd65ce29a): L(T(x \ e,x),x,e / x) = x \ e
Proof. We have T(e / x,x) = x \ e by Proposition 47. Hence we are done by Theorem 44.
Theorem 102 (prov9_331ecda5a6): (e / x) \ T(e / x,y) = x * T(x \ e,y)
Proof. We have (e / x) \ L(T(x \ e,y),x,e / x) = x * T(x \ e,y) by Theorem 71. Hence we are done by Theorem 44.
Theorem 103 (prov9_6a3c7b964e): L(T(x \ T(y,z),w),x,z) = T((z * x) \ (y * z),w)
Proof. We have L(T(x \ T(y,z),w),x,z) = T(L(x \ T(y,z),x,z),w) by Axiom 8. Hence we are done by Proposition 73.
Theorem 104 (prov9_b800c9183a): L(T((y \ x) \ z,w),y \ x,y) = T(x \ (y * z),w)
Proof. We have L(T((y \ x) \ z,w),y \ x,y) = T(L((y \ x) \ z,y \ x,y),w) by Axiom 8. Hence we are done by Theorem 60.
Theorem 105 (prov9_d1e1edaf24): T(R(x / y,y,y \ z),z) = z \ (x * (y \ z))
Proof. We have T((x * (y \ z)) / z,z) = z \ (x * (y \ z)) by Proposition 47. Hence we are done by Theorem 66.
Theorem 106 (prov9_68580de142): T((x * (y \ z)) / z,y) = R(y \ x,y,y \ z)
Proof. We have T(R(x / y,y,y \ z),y) = R(y \ x,y,y \ z) by Proposition 60. Hence we are done by Theorem 66.
Theorem 107 (prov9_e3322c047c): R(T(x / (z / y),w),z / y,y) = T((x * y) / z,w)
Proof. We have R(T(x / (z / y),w),z / y,y) = T(R(x / (z / y),z / y,y),w) by Axiom 9. Hence we are done by Theorem 67.
Theorem 108 (prov9_1436ef532a): (e / x) * ((x \ e) * x) = x \ e
Proof. We have L(T(x \ e,x),x,e / x) = x \ e by Theorem 101. Hence we are done by Theorem 72.
Theorem 109 (prov9_76614f1639): (e / x) * K(x \ e,x) = x \ e
Proof. We have (e / x) * ((x \ e) * x) = x \ e by Theorem 108. Hence we are done by Proposition 76.
Theorem 110 (prov9_47f3889a7e): T(((y * x) * z) / (y * z),y) = R(x,y,z)
Proof. We have T(((y * x) * z) / (y * z),y) = R(y \ (y * x),y,z) by Proposition 64. Hence we are done by Axiom 3.
Theorem 111 (prov9_0e9586f77e): (y / x) * (x / y) = (x / y) * R((x / y) \ e,x / y,y)
Proof. We have (y / ((x / y) * y)) * (x / y) = (x / y) * R((x / y) \ e,x / y,y) by Theorem 90. Hence we are done by Axiom 6.
Theorem 112 (prov9_ee78192c46): R(x,y,x) * (y * x) = (y * x) * T(x,y)
Proof. We have T(((y * x) * x) / (y * x),y) * (y * x) = (y * x) * T(x,y) by Theorem 47. Hence we are done by Theorem 110.
Theorem 113 (prov9_062c221162): T(y,x) = R(T(y,x * y),x,y)
Proof. We have
(L1) (x * y) * T(y,x) = R(y,x,y) * (x * y)
by Theorem 112. We have (x * y) * R(T(y,x * y),x,y) = R(y,x,y) * (x * y) by Proposition 59. Hence we are done by (L1) and Proposition 7.
Theorem 114 (prov9_c153b90e79): (x * y) * T(y,x) = (y * x) * y
Proof. We have R(y,x,y) * (x * y) = (y * x) * y by Proposition 54. Hence we are done by Theorem 112.
Theorem 115 (prov9_b0dbc61333): T(y,x) * (x * y) = (T(y,x * y) * x) * y
Proof. We have R(T(y,x * y),x,y) * (x * y) = (T(y,x * y) * x) * y by Proposition 54. Hence we are done by Theorem 113.
Theorem 116 (prov9_4ef0a14bfe): K(x,e / x) = T(x,e / x) / x
Proof. We have e * T(x,e / x) = T(x,e / x) by Axiom 1. Then
(L3) T(x,e / x) * (T(x,e / x) \ (e * T(x,e / x))) = T(x,e / x)
by Axiom 4. We have T(x,e / x) * (T(x,e / x) \ (e * T(x,e / x))) = e * T(x,e / x) by Axiom 4. Then ((T(x,e / x) * (T(x,e / x) \ (e * T(x,e / x)))) / x) * x = e * T(x,e / x) by Axiom 6. Then (T(x,e / x) / x) * x = e * T(x,e / x) by (L3). Then
(L7) (T(x,e / x) / x) * x = ((e / x) * x) * T(x,e / x)
by Axiom 6. We have ((e / x) * x) * T(x,e / x) = (x * (e / x)) * x by Theorem 114. Then
(L8) (T(x,e / x) / x) * x = (x * (e / x)) * x
by (L7).
K(x,e / x)
=x * (e / x) by Proposition 77
=T(x,e / x) / x by (L8), Proposition 10

Hence we are done.
Theorem 117 (prov9_cf7ad52a1f): T(x,e / x) = R(x,e / x,x)
Proof. We have
(L1) ((e / x) * x) * T(x,e / x) = (x * (e / x)) * x
by Theorem 114. We have (e / x) * x = e by Axiom 6. Then
(L3) e * T(x,e / x) = (x * (e / x)) * x
by (L1). We have e * ((x * (e / x)) * x) = (x * (e / x)) * x by Axiom 1. Then e * T(x,e / x) = e * ((x * (e / x)) * x) by (L3). Then
(L6) T(x,e / x) = (x * (e / x)) * x
by Proposition 9. We have (x * (e / x)) * x = R(x,e / x,x) by Proposition 69. Hence we are done by (L6).
Theorem 118 (prov9_ad2cde29e6): R(T(x,y),e / x,x) = T(T(x,e / x),y)
Proof. We have R(T(x,y),e / x,x) = T(R(x,e / x,x),y) by Axiom 9. Hence we are done by Theorem 117.
Theorem 119 (prov9_11f7f75fc0): T(T(y,e / y),x) / y = T(y,x) * (e / y)
Proof. We have R(T(y,x),e / y,y) / y = T(y,x) * (e / y) by Theorem 74. Hence we are done by Theorem 118.
Theorem 120 (prov9_37271ca2fe): L(e / x,x,y) * x = x * ((y * x) \ y)
Proof. We have L(e / x,x,y) * x = x * L(x \ e,x,y) by Theorem 85. Hence we are done by Proposition 78.
Theorem 121 (prov9_984a73927e): (x * ((y * x) \ y)) / x = L(e / x,x,y)
Proof. We have L(e / x,x,y) * x = x * ((y * x) \ y) by Theorem 120. Hence we are done by Proposition 1.
Theorem 122 (prov9_a3cfca03fd): (y * L(x,z,w)) / y = L((y * x) / y,z,w)
Proof. We have L((y * x) / y,z,w) * y = y * L(x,z,w) by Theorem 86. Hence we are done by Proposition 1.
Theorem 123 (prov9_89c0416af2): (y * R(x,z,w)) / y = R((y * x) / y,z,w)
Proof. We have R((y * x) / y,z,w) * y = y * R(x,z,w) by Theorem 87. Hence we are done by Proposition 1.
Theorem 124 (prov9_a2ea4b73e7): R((w * x) / w,y,z) \ (w * R(x,y,z)) = w
Proof. We have R((w * x) / w,y,z) * w = w * R(x,y,z) by Theorem 87. Hence we are done by Proposition 2.
Theorem 125 (prov9_f0b77cd812): z * ((y * (z \ x)) / y) = ((y * (x / z)) / y) * z
Proof. We have z * ((y * T(x / z,z)) / y) = ((y * (x / z)) / y) * z by Theorem 93. Hence we are done by Proposition 47.
Theorem 126 (prov9_8ee96f4fa6): (e / (e / x)) * (x * (e / x)) = x
Proof. We have (e / (e / x)) * (((e / x) \ e) * (e / x)) = (e / x) \ e by Theorem 108. Then
(L2) (e / (e / x)) * (x * (e / x)) = (e / x) \ e
by Proposition 25. We have (e / x) \ e = x by Proposition 25. Hence we are done by (L2).
Theorem 127 (prov9_060fe89d9e): L((z * (x \ e)) / z,x,y) = (z * ((y * x) \ y)) / z
Proof. We have L((z * (x \ e)) / z,x,y) = (z * L(x \ e,x,y)) / z by Theorem 122. Hence we are done by Proposition 78.
Theorem 128 (prov9_1789ba83ae): L((w * ((y \ x) \ z)) / w,y \ x,y) = (w * (x \ (y * z))) / w
Proof. We have L((w * ((y \ x) \ z)) / w,y \ x,y) = (w * L((y \ x) \ z,y \ x,y)) / w by Theorem 122. Hence we are done by Theorem 60.
Theorem 129 (prov9_cb63617dda): ((y * R(x,z,w)) / y) * (z * w) = (((y * x) / y) * z) * w
Proof. We have R((y * x) / y,z,w) * (z * w) = (((y * x) / y) * z) * w by Proposition 54. Hence we are done by Theorem 123.
Theorem 130 (prov9_cb0204c575): x \ ((x * (e / y)) * y) = a(x,e / y,y)
Proof. We have (x * e) \ ((x * (e / y)) * y) = a(x,e / y,y) by Theorem 1. Hence we are done by Axiom 2.
Theorem 131 (prov9_efe67e5ac9): x \ R(x,e / y,y) = a(x,e / y,y)
Proof. We have x \ ((x * (e / y)) * y) = a(x,e / y,y) by Theorem 130. Hence we are done by Proposition 69.
Theorem 132 (prov9_942d806a38): T(y,x) \ T(T(y,e / y),x) = a(T(y,x),e / y,y)
Proof. We have T(y,x) \ R(T(y,x),e / y,y) = a(T(y,x),e / y,y) by Theorem 131. Hence we are done by Theorem 118.
Theorem 133 (prov9_9b13b9dc70): (x * y) * R((x * y) \ x,z,w) = x * (y * R(y \ e,z,w))
Proof. We have (x * y) * L(R(y \ e,z,w),y,x) = x * (y * R(y \ e,z,w)) by Proposition 52. Hence we are done by Theorem 31.
Theorem 134 (prov9_d1b753bd4e): L(e / y,y,x) = L((x * y) \ x,y,e / y)
Proof. We have L(L(y \ e,y,e / y),y,x) = L((x * y) \ x,y,e / y) by Theorem 32. Hence we are done by Theorem 35.
Theorem 135 (prov9_f8b654076d): (x * y) * L((x * y) \ x,z,w) = x * (y * L(y \ e,z,w))
Proof. We have (x * y) * L(L(y \ e,z,w),y,x) = x * (y * L(y \ e,z,w)) by Proposition 52. Hence we are done by Theorem 32.
Theorem 136 (prov9_ff3afc4f63): (y * x) * T((y * x) \ z,w) = y * (x * T(x \ (y \ z),w))
Proof. We have (y * x) * T(L(x \ (y \ z),x,y),w) = y * (x * T(x \ (y \ z),w)) by Theorem 55. Hence we are done by Proposition 70.
Theorem 137 (prov9_07343dad2a): T((x * w) / (z * w),y) * (z * w) = (T(x / z,y) * z) * w
Proof. We have T(R(x / z,z,w),y) * (z * w) = (T(x / z,y) * z) * w by Proposition 65. Hence we are done by Proposition 55.
Theorem 138 (prov9_c0ab77d0a4): T(x / (z * w),y) * (z * w) = (T((x / w) / z,y) * z) * w
Proof. We have T(R((x / w) / z,z,w),y) * (z * w) = (T((x / w) / z,y) * z) * w by Proposition 65. Hence we are done by Proposition 74.
Theorem 139 (prov9_54e5e6df3d): x * T(x \ (x / y),z) = (x / y) * (y * T(y \ e,z))
Proof. We have x * L(T(y \ e,z),y,x / y) = (x / y) * (y * T(y \ e,z)) by Theorem 53. Hence we are done by Theorem 75.
Theorem 140 (prov9_cad5b44a33): x \ T(x * y,z) = (x \ e) * T((x \ e) \ y,z)
Proof. We have L(T((x \ e) \ y,z),x \ e,x) = T(L((x \ e) \ y,x \ e,x),z) by Axiom 8. Then
(L2) L(T((x \ e) \ y,z),x \ e,x) = T(x * y,z)
by Theorem 62. We have x \ L(T((x \ e) \ y,z),x \ e,x) = (x \ e) * T((x \ e) \ y,z) by Proposition 57. Hence we are done by (L2).
Theorem 141 (prov9_78e4122e11): T(x * z,y) / z = T(x / (e / z),y) * (e / z)
Proof. We have R(T(x / (e / z),y),e / z,z) = T(R(x / (e / z),e / z,z),y) by Axiom 9. Then
(L2) R(T(x / (e / z),y),e / z,z) = T(x * z,y)
by Theorem 68. We have R(T(x / (e / z),y),e / z,z) / z = T(x / (e / z),y) * (e / z) by Theorem 74. Hence we are done by (L2).
Theorem 142 (prov9_84a4c0cc25): R(T(x / y,z),y,y \ z) = z \ (x * (y \ z))
Proof. We have R(T(x / y,z),y,y \ z) = T(R(x / y,y,y \ z),z) by Axiom 9. Hence we are done by Theorem 105.
Theorem 143 (prov9_cb63b250cb): T(R(z \ x,z,z \ y),y) = T(y \ (x * (z \ y)),z)
Proof. We have T(T((x * (z \ y)) / y,z),y) = T(y \ (x * (z \ y)),z) by Proposition 51. Hence we are done by Theorem 106.
Theorem 144 (prov9_f9b4e2a624): ((z * R(e / x,x,y)) / z) * (x * y) = (x * y) * ((z * ((x * y) \ y)) / z)
Proof. We have ((z * (y / (x * y))) / z) * (x * y) = (x * y) * ((z * ((x * y) \ y)) / z) by Theorem 125. Hence we are done by Proposition 79.
Theorem 145 (prov9_c36fa51902): (x \ e) * T((x \ e) \ e,y) = x \ T(x,y)
Proof. We have (x \ e) * T((x \ e) \ e,y) = x \ T(x * e,y) by Theorem 140. Hence we are done by Axiom 2.
Theorem 146 (prov9_f0c1e52706): y \ T(y,x) = T(y,x) * (y \ e)
Proof. We have (y \ e) * T((y \ e) \ e,x) = T(y,x) * (y \ e) by Theorem 48. Hence we are done by Theorem 145.
Theorem 147 (prov9_375108167d): T(e / (e / y),x) * (e / y) = T(y,x) / y
Proof. We have T(e / (e / y),x) * (e / y) = T(e * y,x) / y by Theorem 141. Hence we are done by Axiom 1.
Theorem 148 (prov9_312da32083): ((e / y) \ x) * (e / y) = T(x * y,e / y) / y
Proof. We have T(x / (e / y),e / y) * (e / y) = T(x * y,e / y) / y by Theorem 141. Hence we are done by Proposition 47.
Theorem 149 (prov9_d1a3ff397b): T(x,y) / x = (e / x) * T((e / x) \ e,y)
Proof. We have T(e / (e / x),y) * (e / x) = (e / x) * T((e / x) \ e,y) by Proposition 61. Hence we are done by Theorem 147.
Theorem 150 (prov9_d021e0e45e): T(x,y) / x = (e / x) * T(x,y)
Proof. We have (e / x) * T(T(e / (e / x),e / x),y) = T(e / (e / x),y) * (e / x) by Proposition 50. Then (e / x) * T(x,y) = T(e / (e / x),y) * (e / x) by Theorem 14. Hence we are done by Theorem 147.
Theorem 151 (prov9_0a9a7c43fe): (x / y) \ (y \ x) = (y \ x) * ((x / y) \ e)
Proof. We have T(x / y,y) * ((x / y) \ e) = (x / y) \ T(x / y,y) by Theorem 146. Then (y \ x) * ((x / y) \ e) = (x / y) \ T(x / y,y) by Proposition 47. Hence we are done by Proposition 47.
Theorem 152 (prov9_fd73ba5036): T(x,e / y) / y = ((e / y) \ (x / y)) * (e / y)
Proof. We have T((x / y) * y,e / y) / y = ((e / y) \ (x / y)) * (e / y) by Theorem 148. Hence we are done by Axiom 6.
Theorem 153 (prov9_6385279d78): (x \ y) / (y / x) = (e / (y / x)) * (x \ y)
Proof. We have (e / (y / x)) * T(y / x,x) = T(y / x,x) / (y / x) by Theorem 150. Then (e / (y / x)) * (x \ y) = T(y / x,x) / (y / x) by Proposition 47. Hence we are done by Proposition 47.
Theorem 154 (prov9_26a09a15e0): (x / y) * K(y \ e,y) = x * T(x \ (x / y),y)
Proof. We have T(x \ (x / y),y) = L(T(y \ e,y),y,x / y) by Theorem 75. Then
(L2) T(x \ (x / y),y) = x \ ((x / y) * ((y \ e) * y))
by Theorem 95. We have x * (x \ ((x / y) * ((y \ e) * y))) = (x / y) * ((y \ e) * y) by Axiom 4. Then x * T(x \ (x / y),y) = (x / y) * ((y \ e) * y) by (L2). Hence we are done by Proposition 76.
Theorem 155 (prov9_58b58aebe3): (y * x) * L((y * x) \ z,w,u) = y * (x * L(x \ (y \ z),w,u))
Proof. We have (y * x) * L(L(x \ (y \ z),x,y),w,u) = y * (x * L(x \ (y \ z),w,u)) by Theorem 57. Hence we are done by Proposition 70.
Theorem 156 (prov9_13eb00bba5): K(x \ e,x) * L(x \ y,x,x \ e) = (x \ e) * y
Proof. We have K(x \ e,x) * L(x \ y,x,x \ e) = (x \ e) * (x * (x \ y)) by Theorem 59. Hence we are done by Axiom 4.
Theorem 157 (prov9_acbffce604): L((x * u) / (w * u),y,z) * (w * u) = (L(x / w,y,z) * w) * u
Proof. We have L(R(x / w,w,u),y,z) * (w * u) = (L(x / w,y,z) * w) * u by Theorem 65. Hence we are done by Proposition 55.
Theorem 158 (prov9_4f616351df): (e / x) \ R((e / x) * y,z,w) = x * R(x \ y,z,w)
Proof. We have (e / x) \ L(R(x \ y,z,w),x,e / x) = x * R(x \ y,z,w) by Theorem 71. Hence we are done by Theorem 81.
Theorem 159 (prov9_b171e33d17): (e / x) \ L(e / x,y,z) = x * L(x \ e,y,z)
Proof. We have (e / x) \ L(L(x \ e,x,e / x),y,z) = x * L(x \ e,y,z) by Theorem 82. Hence we are done by Theorem 35.
Theorem 160 (prov9_b551bcd8e5): L(z \ e,x,y) / (z \ e) = L(e / z,x,y) * z
Proof. We have L(R(e / z,z,z \ e),x,y) / (z \ e) = L(e / z,x,y) * z by Theorem 83. Hence we are done by Theorem 41.
Theorem 161 (prov9_daad63abf1): L(x \ e,y,z) / (x \ e) = x * L(x \ e,y,z)
Proof. We have L(e / x,y,z) * x = x * L(x \ e,y,z) by Theorem 85. Hence we are done by Theorem 160.
Theorem 162 (prov9_da97a1ddef): L(x \ e,x,y) / (x \ e) = x * ((y * x) \ y)
Proof. We have L(e / x,x,y) * x = x * ((y * x) \ y) by Theorem 120. Hence we are done by Theorem 160.
Theorem 163 (prov9_2688cd75fc): (x * ((y * x) \ y)) * (x \ e) = L(x \ e,x,y)
Proof. We have (L(x \ e,x,y) / (x \ e)) * (x \ e) = L(x \ e,x,y) by Axiom 6. Hence we are done by Theorem 162.
Theorem 164 (prov9_628758e6c5): L(T(z,e / z),x,y) / z = L(z,x,y) * (e / z)
Proof. We have L(R(z,e / z,z),x,y) / z = L(z,x,y) * (e / z) by Theorem 84. Hence we are done by Theorem 117.
Theorem 165 (prov9_11f5db8948): L(z,x,y) = (e / z) \ (L(z,x,y) / z)
Proof. We have L(T(z,e / z),x,y) / z = L(z,x,y) * (e / z) by Theorem 164. Then
(L2) T(L(z,x,y),e / z) / z = L(z,x,y) * (e / z)
by Axiom 8. We have ((e / z) \ (L(z,x,y) / z)) * (e / z) = T(L(z,x,y),e / z) / z by Theorem 152. Hence we are done by (L2) and Proposition 8.
Theorem 166 (prov9_8df985dcf9): (e / z) * L(z,x,y) = L(z,x,y) / z
Proof. We have (e / z) * ((e / z) \ (L(z,x,y) / z)) = L(z,x,y) / z by Axiom 4. Hence we are done by Theorem 165.
Theorem 167 (prov9_8216679cd7): T(e / z,L(z,x,y)) = L(z,x,y) \ (L(z,x,y) / z)
Proof. We have T(e / z,L(z,x,y)) = L(z,x,y) \ ((e / z) * L(z,x,y)) by Definition 3. Hence we are done by Theorem 166.
Theorem 168 (prov9_ad78465b3a): y * T(y \ (x * z),w) = x * ((x \ y) * T((x \ y) \ z,w))
Proof. We have y * L(T((x \ y) \ z,w),x \ y,x) = x * ((x \ y) * T((x \ y) \ z,w)) by Theorem 52. Hence we are done by Theorem 104.
Theorem 169 (prov9_3f97b61083): T((x * w) / z,y) * z = (T(x / (z / w),y) * (z / w)) * w
Proof. We have R(T(x / (z / w),y),z / w,w) * z = (T(x / (z / w),y) * (z / w)) * w by Theorem 23. Hence we are done by Theorem 107.
Theorem 170 (prov9_4168391a12): (x * y) * ((z * ((x * y) \ x)) / z) = x * (y * ((z * (y \ e)) / z))
Proof. We have (x * y) * L((z * (y \ e)) / z,y,x) = x * (y * ((z * (y \ e)) / z)) by Proposition 52. Hence we are done by Theorem 127.
Theorem 171 (prov9_5b2e8b9b9c): R(T(z \ x,y),z,z \ y) = T(y \ (x * (z \ y)),z)
Proof. We have R(T(z \ x,y),z,z \ y) = T(R(z \ x,z,z \ y),y) by Axiom 9. Hence we are done by Theorem 143.
Theorem 172 (prov9_d017bad0ce): y * (T(y,x) * z) = T(y,x) * (y * z)
Proof. We have (y * z) * L(T(T(y,z),x),z,y) = y * (z * T(T(y,z),x)) by Proposition 52. Then
(L2) (y * z) * T(T(y,y * z),x) = y * (z * T(T(y,z),x))
by Theorem 78. We have (y * z) * T(T(y,y * z),x) = T(y,x) * (y * z) by Proposition 50. Then y * (z * T(T(y,z),x)) = T(y,x) * (y * z) by (L2). Hence we are done by Proposition 50.
Theorem 173 (prov9_2f8c653304): y \ (T(y,x) * z) = T(y,x) * (y \ z)
Proof. We have T(y,x) \ (T(y,x) * z) = z by Axiom 3. Then
(L3) T(y,x) \ (y * (y \ (T(y,x) * z))) = z
by Axiom 4. We have T(y,x) * (T(y,x) \ (y * (y \ (T(y,x) * z)))) = y * (y \ (T(y,x) * z)) by Axiom 4. Then T(y,x) * (y * (y \ (T(y,x) \ (y * (y \ (T(y,x) * z)))))) = y * (y \ (T(y,x) * z)) by Axiom 4. Then
(L6) T(y,x) * (y * (y \ z)) = y * (y \ (T(y,x) * z))
by (L3). We have T(y,x) * (y * (y \ z)) = y * (T(y,x) * (y \ z)) by Theorem 172. Then y * (y \ (T(y,x) * z)) = y * (T(y,x) * (y \ z)) by (L6). Hence we are done by Proposition 9.
Theorem 174 (prov9_5b931eafca): y * (T(y,x) \ z) = T(y,x) \ (y * z)
Proof. We have y \ (y * z) = z by Axiom 3. Then
(L3) y \ (T(y,x) * (T(y,x) \ (y * z))) = z
by Axiom 4. We have y * (y \ (T(y,x) * (T(y,x) \ (y * z)))) = T(y,x) * (T(y,x) \ (y * z)) by Axiom 4. Then y * (T(y,x) * (T(y,x) \ (y \ (T(y,x) * (T(y,x) \ (y * z)))))) = T(y,x) * (T(y,x) \ (y * z)) by Axiom 4. Then
(L6) y * (T(y,x) * (T(y,x) \ z)) = T(y,x) * (T(y,x) \ (y * z))
by (L3). We have T(y,x) * (y * (T(y,x) \ z)) = y * (T(y,x) * (T(y,x) \ z)) by Theorem 172. Hence we are done by (L6) and Proposition 7.
Theorem 175 (prov9_d18167fcf7): y * (T(y,x) \ e) = T(y,x) \ y
Proof. We have y \ (y * e) = e by Axiom 3. Then y \ (T(y,x) * (T(y,x) \ (y * e))) = e by Axiom 4. Then
(L4) y \ (T(y,x) * (T(y,x) \ y)) = e
by Axiom 2. We have y * (y \ (T(y,x) * (T(y,x) \ y))) = T(y,x) * (T(y,x) \ y) by Axiom 4. Then y * e = T(y,x) * (T(y,x) \ y) by (L4). Then
(L7) T(y,x) * (T(y,x) \ y) = y * (T(y,x) * (T(y,x) \ e))
by Axiom 4. We have T(y,x) * (y * (T(y,x) \ e)) = y * (T(y,x) * (T(y,x) \ e)) by Theorem 172. Hence we are done by (L7) and Proposition 7.
Theorem 176 (prov9_22505d012e): T(y,x) * K(y,e / y) = y * (T(y,x) * (e / y))
Proof. We have T(y,x) * (y * (e / y)) = y * (T(y,x) * (e / y)) by Theorem 172. Hence we are done by Proposition 77.
Theorem 177 (prov9_61c094842b): x * (T(x,y) \ y) = T(y,T(x,y))
Proof. We have T(x,y) \ (x * y) = T(y,T(x,y)) by Theorem 13. Hence we are done by Theorem 174.
Theorem 178 (prov9_a5199d4736): (y / x) * ((x \ y) \ x) = T(x,x \ y)
Proof. We have (y / x) * (T(y / x,x) \ x) = T(x,T(y / x,x)) by Theorem 177. Then (y / x) * ((x \ y) \ x) = T(x,T(y / x,x)) by Proposition 47. Hence we are done by Proposition 47.
Theorem 179 (prov9_a217471464): ((x * y) / x) * (y \ x) = T(x,y)
Proof. We have ((x * y) / x) * ((x \ (x * y)) \ x) = T(x,x \ (x * y)) by Theorem 178. Then ((x * y) / x) * (y \ x) = T(x,x \ (x * y)) by Axiom 3. Hence we are done by Axiom 3.
Theorem 180 (prov9_b8b206a4c7): (x \ y) \ (y / x) = (y / x) * ((x \ y) \ e)
Proof. We have (y / x) * (T(y / x,x) \ e) = T(y / x,x) \ (y / x) by Theorem 175. Then (y / x) * ((x \ y) \ e) = T(y / x,x) \ (y / x) by Proposition 47. Hence we are done by Proposition 47.
Theorem 181 (prov9_6b886c6794): T(T(y,x * y) * x,y) = T(y,x) * T(x,y)
Proof. We have T(T(y,x * y) * x,y) = y \ ((T(y,x * y) * x) * y) by Definition 3. Then T(T(y,x * y) * x,y) = y \ (T(y,x) * (x * y)) by Theorem 115. Then T(y,x) * (y \ (x * y)) = T(T(y,x * y) * x,y) by Theorem 173. Hence we are done by Definition 3.
Theorem 182 (prov9_a4eb3c24ee): (x / y) \ (x * T(x \ (x / y),z)) = y * T(y \ e,z)
Proof. We have (x / y) * (y * T(y \ e,z)) = x * T(x \ (x / y),z) by Theorem 139. Hence we are done by Proposition 2.
Theorem 183 (prov9_f012d5c869): (T(e / x,z) * x) * (x \ y) = y * T(y \ (x \ y),z)
Proof. We have R(T(e / x,z),x,x \ y) = T(R(e / x,x,x \ y),z) by Axiom 9. Then
(L2) R(T(e / x,z),x,x \ y) = T((x \ y) / y,z)
by Theorem 37. We have R(T(e / x,z),x,x \ y) * y = (T(e / x,z) * x) * (x \ y) by Theorem 25. Then
(L4) T((x \ y) / y,z) * y = (T(e / x,z) * x) * (x \ y)
by (L2). We have T((x \ y) / y,z) * y = y * T(y \ (x \ y),z) by Proposition 61. Hence we are done by (L4).
Theorem 184 (prov9_6ba9e4a510): K((x * y) \ x,x * y) = y * T(y \ e,x * y)
Proof. We have (x * y) * L(T(y \ e,x * y),y,x) = L(y \ e,y,x) * (x * y) by Proposition 58. Then
(L2) (x * y) * L(T(y \ e,x * y),y,x) = ((x * y) \ x) * (x * y)
by Proposition 78. We have (x * y) * L(T(y \ e,x * y),y,x) = x * (y * T(y \ e,x * y)) by Proposition 52. Then
(L4) ((x * y) \ x) * (x * y) = x * (y * T(y \ e,x * y))
by (L2). We have x * K((x * y) \ x,x * y) = ((x * y) \ x) * (x * y) by Theorem 17. Then x * K((x * y) \ x,x * y) = x * (y * T(y \ e,x * y)) by (L4). Hence we are done by Proposition 9.
Theorem 185 (prov9_0d67d2e91b): K((x * y) \ x,x * y) / y = T(e / y,x * y)
Proof. We have (y * T(y \ e,x * y)) / y = T(e / y,x * y) by Theorem 49. Hence we are done by Theorem 184.
Theorem 186 (prov9_8d57873c75): ((y * x) / y) \ x = x * (((y * x) / y) \ e)
Proof. We have (y \ (y * x)) * (((y * x) / y) \ e) = ((y * x) / y) \ (y \ (y * x)) by Theorem 151. Then x * (((y * x) / y) \ e) = ((y * x) / y) \ (y \ (y * x)) by Axiom 3. Hence we are done by Axiom 3.
Theorem 187 (prov9_7c86bc7b3b): (e / x) \ R(e / x,y,z) = x * R(x \ e,y,z)
Proof. We have (e / x) \ R((e / x) * e,y,z) = x * R(x \ e,y,z) by Theorem 158. Hence we are done by Axiom 2.
Theorem 188 (prov9_4e0f671c3d): (y / (x * y)) * x = (e / x) \ R(e / x,x,y)
Proof. We have (y / (x * y)) * x = x * R(x \ e,x,y) by Theorem 90. Hence we are done by Theorem 187.
Theorem 189 (prov9_2102bd0c36): (x / (y * x)) / ((x / (y * x)) * y) = e / y
Proof. We have (e / y) \ R(e / y,y,x) = (x / (y * x)) * y by Theorem 188. Then
(L2) (e / y) \ (x / (y * x)) = (x / (y * x)) * y
by Proposition 79. We have (x / (y * x)) / ((e / y) \ (x / (y * x))) = e / y by Proposition 24. Hence we are done by (L2).
Theorem 190 (prov9_560b674293): (e / z) \ L(z \ e,x,y) = L(z \ e,x,y) * z
Proof. We have (e / z) \ L(L(T(z \ e,z),z,e / z),x,y) = z * L(T(z \ e,z),x,y) by Theorem 82. Then
(L2) (e / z) \ L(z \ e,x,y) = z * L(T(z \ e,z),x,y)
by Theorem 101. We have z * L(T(z \ e,z),x,y) = L(z \ e,x,y) * z by Proposition 58. Hence we are done by (L2).
Theorem 191 (prov9_8f9cbb5fbd): R(e / (e / z),x,y) * (e / z) = R(z,x,y) / z
Proof. We have R(R(e / (e / z),x,y),e / z,z) / z = R(e / (e / z),x,y) * (e / z) by Theorem 74. Then R(R(e / (e / z),e / z,z),x,y) / z = R(e / (e / z),x,y) * (e / z) by Axiom 12. Then R(z / e,x,y) / z = R(e / (e / z),x,y) * (e / z) by Theorem 38. Hence we are done by Proposition 27.
Theorem 192 (prov9_48bc89bb88): R(x,y,z) / x = (e / x) * R(x,y,z)
Proof. We have (e / x) * R(T(e / (e / x),e / x),y,z) = R(e / (e / x),y,z) * (e / x) by Proposition 59. Then (e / x) * R(x,y,z) = R(e / (e / x),y,z) * (e / x) by Theorem 14. Hence we are done by Theorem 191.
Theorem 193 (prov9_48d7ae2b56): z * R(z \ e,x,y) = R(z \ e,x,y) / (z \ e)
Proof. We have (e / (z \ e)) * R(z \ e,x,y) = R(z \ e,x,y) / (z \ e) by Theorem 192. Hence we are done by Proposition 24.
Theorem 194 (prov9_47e1e09ded): (x * z) * T((x * z) \ z,y) = (x * T(x \ e,y)) * z
Proof. We have R(T(e / x,y),x,z) * (x * z) = (T(e / x,y) * x) * z by Proposition 54. Then
(L2) T(z / (x * z),y) * (x * z) = (T(e / x,y) * x) * z
by Theorem 39. We have T(z / (x * z),y) * (x * z) = (x * z) * T((x * z) \ z,y) by Proposition 61. Then (T(e / x,y) * x) * z = (x * z) * T((x * z) \ z,y) by (L2). Hence we are done by Proposition 61.
Theorem 195 (prov9_df0fc1f937): y \ ((x * y) / x) = ((x * y) / x) * (y \ e)
Proof. We have ((x * y) / x) * ((x \ (x * y)) \ e) = (x \ (x * y)) \ ((x * y) / x) by Theorem 180. Then ((x * y) / x) * (y \ e) = (x \ (x * y)) \ ((x * y) / x) by Axiom 3. Hence we are done by Axiom 3.
Theorem 196 (prov9_b192646899): x * T(x \ e,y) = K(y \ (y / x),y)
Proof. We have (y / x) \ (y * T(y \ (y / x),y)) = K(y \ (y / x),y) by Theorem 46. Hence we are done by Theorem 182.
Theorem 197 (prov9_c5438141cb): x \ K(y \ (y / x),y) = T(x \ e,y)
Proof. We have x * T(x \ e,y) = K(y \ (y / x),y) by Theorem 196. Hence we are done by Proposition 2.
Theorem 198 (prov9_a9bdc6724e): K(y \ (y / (x \ e)),y) = x \ T(x,y)
Proof. We have (x \ e) * T((x \ e) \ e,y) = x \ T(x,y) by Theorem 145. Hence we are done by Theorem 196.
Theorem 199 (prov9_2e3bc568bd): K(x \ (x / (e / y)),x) = T(y,x) / y
Proof. We have (e / y) * T((e / y) \ e,x) = T(y,x) / y by Theorem 149. Hence we are done by Theorem 196.
Theorem 200 (prov9_fdf69105ba): K(x \ (x / (((x * y) / x) \ e)),x) = ((x * y) / x) \ y
Proof. We have K(x \ (x / (((x * y) / x) \ e)),x) = ((x * y) / x) \ T((x * y) / x,x) by Theorem 198. Hence we are done by Theorem 7.
Theorem 201 (prov9_2d6f0b919a): K(y \ (y / (e / x)),y) * x = T(x,y)
Proof. We have (T(x,y) / x) * x = T(x,y) by Axiom 6. Hence we are done by Theorem 199.
Theorem 202 (prov9_774d43c2d3): K((x / y) \ e,x / y) * y = x * T(x \ y,x / y)
Proof. We have T(R(e / (x / y),x / y,y),x / y) = R((x / y) \ e,x / y,y) by Proposition 60. Then
(L2) T(y / x,x / y) = R((x / y) \ e,x / y,y)
by Theorem 38. We have R((x / y) \ e,x / y,y) * x = (((x / y) \ e) * (x / y)) * y by Theorem 23. Then
(L4) T(y / x,x / y) * x = (((x / y) \ e) * (x / y)) * y
by (L2). We have T(y / x,x / y) * x = x * T(x \ y,x / y) by Proposition 61. Then (((x / y) \ e) * (x / y)) * y = x * T(x \ y,x / y) by (L4). Hence we are done by Proposition 76.
Theorem 203 (prov9_a06014c62d): T(y,y * T(y \ e,x)) = y
Proof. We have (y * y) * T((y * y) \ y,x) = y * (y * T(y \ e,x)) by Theorem 98. Then (y * T(y \ e,x)) * y = y * (y * T(y \ e,x)) by Theorem 194. Hence we are done by Theorem 11.
Theorem 204 (prov9_49726cdcf0): T(x,(x \ e) * x) = x
Proof. We have T(x,x * T(x \ e,x)) = x by Theorem 203. Hence we are done by Proposition 46.
Theorem 205 (prov9_e38ec061dc): T(y \ e,y \ T(y,x)) = y \ e
Proof. We have T(y \ e,(y \ e) * T((y \ e) \ e,x)) = y \ e by Theorem 203. Hence we are done by Theorem 145.
Theorem 206 (prov9_f829c0ceb3): T(e / y,T(y,x) / y) = e / y
Proof. We have T(e / y,(e / y) * T((e / y) \ e,x)) = e / y by Theorem 203. Hence we are done by Theorem 149.
Theorem 207 (prov9_a69214de59): R(x,x \ e,x) = T(x,x \ e)
Proof. We have R(T(x,(x \ e) * x),x \ e,x) = T(x,x \ e) by Theorem 113. Hence we are done by Theorem 204.
Theorem 208 (prov9_3a3c9a39ee): T(x,x \ e) \ e = T(x \ e,x)
Proof. We have
(L1) R(x,x \ e,x) \ x = (x \ e) * x
by Theorem 45.
x * (T(x,x \ e) \ e)
=T(x,x \ e) \ x by Theorem 175
=(x \ e) * x by (L1), Theorem 207
=x * T(x \ e,x) by Proposition 46

Then x * (T(x,x \ e) \ e) = x * T(x \ e,x). Hence we are done by Proposition 9.
Theorem 209 (prov9_742ea65dd3): ((x \ e) \ e) \ e = T(x \ e,x)
Proof. We have T(x,x \ e) \ e = T(x \ e,x) by Theorem 208. Hence we are done by Proposition 49.
Theorem 210 (prov9_1cecad55d3): T(x,e / x) = (x \ e) \ e
Proof. We have T((e / x) \ e,e / x) = (((e / x) \ e) \ e) \ e by Theorem 209. Then T((e / x) \ e,e / x) = (x \ e) \ e by Proposition 25. Hence we are done by Proposition 25.
Theorem 211 (prov9_e2e54b86e3): T(x,x \ e) = T(x,e / x)
Proof. We have T(x,x \ e) = (x \ e) \ e by Proposition 49. Hence we are done by Theorem 210.
Theorem 212 (prov9_6cea34753d): ((x \ e) \ e) / x = K(x,e / x)
Proof. We have T(x,e / x) / x = K(x,e / x) by Theorem 116. Hence we are done by Theorem 210.
Theorem 213 (prov9_889fc2cb68): L((z \ e) \ e,x,y) / z = L(z,x,y) * (e / z)
Proof. We have L(T(z,e / z),x,y) / z = L(z,x,y) * (e / z) by Theorem 164. Hence we are done by Theorem 210.
Theorem 214 (prov9_6540da20a4): R(x,e / x,x) = T(x,x \ e)
Proof. We have R(x,e / x,x) = T(x,e / x) by Theorem 117. Hence we are done by Theorem 211.
Theorem 215 (prov9_88c0400226): T(x,x \ e) / x = K(x,e / x)
Proof. We have T(x,e / x) / x = K(x,e / x) by Theorem 116. Hence we are done by Theorem 211.
Theorem 216 (prov9_68384f14a9): T(T(y,y \ e),x) / y = T(y,x) * (e / y)
Proof. We have T(T(y,e / y),x) / y = T(y,x) * (e / y) by Theorem 119. Hence we are done by Theorem 211.
Theorem 217 (prov9_1260f9d8c8): T(y,x) \ T(T(y,y \ e),x) = a(T(y,x),e / y,y)
Proof. We have T(y,x) \ T(T(y,e / y),x) = a(T(y,x),e / y,y) by Theorem 132. Hence we are done by Theorem 211.
Theorem 218 (prov9_68834d4656): x \ T(x,x \ e) = a(x,e / x,x)
Proof. We have x \ R(x,e / x,x) = a(x,e / x,x) by Theorem 131. Hence we are done by Theorem 214.
Theorem 219 (prov9_990057a3f3): K(x,e / x) = (x \ e) \ (e / x)
Proof. We have (e / x) * T(x,x \ e) = T(x,x \ e) / x by Theorem 150. Then (e / x) * ((x \ e) \ e) = T(x,x \ e) / x by Proposition 49. Then ((x \ e) \ e) / x = (e / x) * ((x \ e) \ e) by Proposition 49. Then (x \ e) \ (e / x) = ((x \ e) \ e) / x by Theorem 180. Hence we are done by Theorem 212.
Theorem 220 (prov9_f41b6769d8): x \ e = R(e / x,x,e / x)
Proof. We have (e / x) / ((x \ e) \ (e / x)) = x \ e by Proposition 24. Then
(L2) (e / x) / K(x,e / x) = x \ e
by Theorem 219. We have R(e / x,x,e / x) = (e / x) / (x * (e / x)) by Proposition 79. Then R(e / x,x,e / x) = (e / x) / K(x,e / x) by Proposition 77. Hence we are done by (L2).
Theorem 221 (prov9_513c4ed4eb): R(y \ e,y,x) = R(x / (y * x),y,e / y)
Proof. We have R(R(e / y,y,e / y),y,x) = R(x / (y * x),y,e / y) by Theorem 40. Hence we are done by Theorem 220.
Theorem 222 (prov9_f31aa2e2c0): T(T(y,x),y \ e) = T(T(y,x),e / y)
Proof. We have T(T(y,e / y),x) = T(T(y,x),e / y) by Axiom 7. Then
(L2) T((y \ e) \ e,x) = T(T(y,x),e / y)
by Theorem 210. We have T(T(y,x),y \ e) = T((y \ e) \ e,x) by Theorem 18. Hence we are done by (L2).
Theorem 223 (prov9_797e228c7e): K(y,x) = T(K(y,x),y \ e)
Proof. We have T(y \ e,y \ T(y,x)) = y \ e by Theorem 205. Then
(L2) T(y \ T(y,x),y \ e) = y \ T(y,x)
by Proposition 21. We have L(T(y \ T(y,x),y \ e),y,x) = T((x * y) \ (y * x),y \ e) by Theorem 103. Then
(L4) L(y \ T(y,x),y,x) = T((x * y) \ (y * x),y \ e)
by (L2). We have L(y \ T(y,x),y,x) = (x * y) \ (y * x) by Proposition 73. Then
(L6) T((x * y) \ (y * x),y \ e) = (x * y) \ (y * x)
by (L4). We have K(y,x) = (x * y) \ (y * x) by Definition 2. Then K(y,x) = T((x * y) \ (y * x),y \ e) by (L6). Hence we are done by Definition 2.
Theorem 224 (prov9_444127c5d7): T(y \ e,K(y,x)) = y \ e
Proof. We have T(K(y,x),y \ e) = K(y,x) by Theorem 223. Hence we are done by Proposition 21.
Theorem 225 (prov9_6d9dec833d): T((z * y) \ z,K(y,x)) = (z * y) \ z
Proof. We have T((z * y) \ z,K(y,x)) = L(T(y \ e,K(y,x)),y,z) by Theorem 30. Then
(L2) T((z * y) \ z,K(y,x)) = L(y \ e,y,z)
by Theorem 224. We have L(y \ e,y,z) = (z * y) \ z by Proposition 78. Hence we are done by (L2).
Theorem 226 (prov9_2e66034661): T(K(y,z),(x * y) \ x) = K(y,z)
Proof. We have T((x * y) \ x,K(y,z)) = (x * y) \ x by Theorem 225. Hence we are done by Proposition 21.
Theorem 227 (prov9_a06b986e74): z * (R(z,x,y) * w) = R(z,x,y) * (z * w)
Proof. We have (z * w) * R(T(z,z * w),x,y) = R(z,x,y) * (z * w) by Proposition 59. Then
(L2) (z * w) * R(L(T(z,w),w,z),x,y) = R(z,x,y) * (z * w)
by Proposition 63. We have (z * w) * R(L(T(z,w),w,z),x,y) = z * (w * R(T(z,w),x,y)) by Theorem 56. Then R(z,x,y) * (z * w) = z * (w * R(T(z,w),x,y)) by (L2). Hence we are done by Proposition 59.
Theorem 228 (prov9_ae42be8c20): z \ (R(z,x,y) * w) = R(z,x,y) * (z \ w)
Proof. We have R(z,x,y) \ (R(z,x,y) * w) = w by Axiom 3. Then
(L3) R(z,x,y) \ (z * (z \ (R(z,x,y) * w))) = w
by Axiom 4. We have R(z,x,y) * (R(z,x,y) \ (z * (z \ (R(z,x,y) * w)))) = z * (z \ (R(z,x,y) * w)) by Axiom 4. Then R(z,x,y) * (z * (z \ (R(z,x,y) \ (z * (z \ (R(z,x,y) * w)))))) = z * (z \ (R(z,x,y) * w)) by Axiom 4. Then
(L6) R(z,x,y) * (z * (z \ w)) = z * (z \ (R(z,x,y) * w))
by (L3). We have R(z,x,y) * (z * (z \ w)) = z * (R(z,x,y) * (z \ w)) by Theorem 227. Then z * (z \ (R(z,x,y) * w)) = z * (R(z,x,y) * (z \ w)) by (L6). Hence we are done by Proposition 9.
Theorem 229 (prov9_a9fd5ec9f9): x \ ((x * y) * z) = R(x,y,z) * (x \ (y * z))
Proof. We have x \ (R(x,y,z) * (y * z)) = R(x,y,z) * (x \ (y * z)) by Theorem 228. Hence we are done by Proposition 54.
Theorem 230 (prov9_4e7d956345): R(y,x,y) * T(x,y) = T(y * x,y)
Proof. We have T(y * x,y) = y \ ((y * x) * y) by Definition 3. Then T(y * x,y) = R(y,x,y) * (y \ (x * y)) by Theorem 229. Hence we are done by Definition 3.
Theorem 231 (prov9_a86280d287): R(y,x,y) \ T(y * x,y) = T(x,y)
Proof. We have R(y,x,y) * T(x,y) = T(y * x,y) by Theorem 230. Hence we are done by Proposition 2.
Theorem 232 (prov9_42907a26d5): ((y * (x * z)) / y) / z = ((y * (x / (e / z))) / y) * (e / z)
Proof. We have R((y * (x / (e / z))) / y,e / z,z) = (y * R(x / (e / z),e / z,z)) / y by Theorem 123. Then
(L2) R((y * (x / (e / z))) / y,e / z,z) = (y * (x * z)) / y
by Theorem 68. We have R((y * (x / (e / z))) / y,e / z,z) / z = ((y * (x / (e / z))) / y) * (e / z) by Theorem 74. Hence we are done by (L2).
Theorem 233 (prov9_3600149a46): x * ((z * y) \ z) = x * L(y \ e,y,z)
Proof. We have (e / x) \ L((z * y) \ z,x,e / x) = x * ((z * y) \ z) by Theorem 71. Then
(L2) (e / x) \ L(L(y \ e,x,e / x),y,z) = x * ((z * y) \ z)
by Theorem 32. We have (e / x) \ L(L(y \ e,x,e / x),y,z) = x * L(y \ e,y,z) by Theorem 82. Hence we are done by (L2).
Theorem 234 (prov9_cd07360ef7): L(x \ e,x,y) / ((y * x) \ y) = e
Proof. We have e * L(x \ e,x,y) = L(x \ e,x,y) by Axiom 1. Then
(L3) L(x \ e,x,y) * (L(x \ e,x,y) \ (e * L(x \ e,x,y))) = L(x \ e,x,y)
by Axiom 4. We have L(x \ e,x,y) * (L(x \ e,x,y) \ (e * L(x \ e,x,y))) = e * L(x \ e,x,y) by Axiom 4. Then ((L(x \ e,x,y) * (L(x \ e,x,y) \ (e * L(x \ e,x,y)))) / ((y * x) \ y)) * ((y * x) \ y) = e * L(x \ e,x,y) by Axiom 6. Then
(L6) (L(x \ e,x,y) / ((y * x) \ y)) * ((y * x) \ y) = e * L(x \ e,x,y)
by (L3). We have (L(x \ e,x,y) / ((y * x) \ y)) * L(x \ e,x,y) = (L(x \ e,x,y) / ((y * x) \ y)) * ((y * x) \ y) by Theorem 233. Hence we are done by (L6) and Proposition 8.
Theorem 235 (prov9_d998a6f3fa): x * (z * T(z \ (x \ z),y)) = (x * T(x \ e,y)) * z
Proof. We have (x * z) * T((x * z) \ z,y) = (x * T(x \ e,y)) * z by Theorem 194. Hence we are done by Theorem 136.
Theorem 236 (prov9_0eb1a6f94a): x * ((y \ (x \ y)) * y) = (x * T(x \ e,y)) * y
Proof. We have x * (y * T(y \ (x \ y),y)) = (x * T(x \ e,y)) * y by Theorem 235. Hence we are done by Proposition 46.
Theorem 237 (prov9_f5ecec4811): (T(y / x,w) * x) * z = (x * z) * T((x * z) \ (y * z),w)
Proof. We have T((y * z) / (x * z),w) * (x * z) = (x * z) * T((x * z) \ (y * z),w) by Proposition 61. Hence we are done by Theorem 137.
Theorem 238 (prov9_5e3d8ae7d5): (x * T(y,z)) * y = (x * y) * T(y,z)
Proof. We have T(((x * y) * y) / (x * y),z) * (x * y) = (x * y) * T(y,z) by Theorem 47. Then (T((x * y) / x,z) * x) * y = (x * y) * T(y,z) by Theorem 137. Hence we are done by Theorem 47.
Theorem 239 (prov9_aaf28e3c4b): (x * z) / T(z,y) = (x / T(z,y)) * z
Proof. We have (x * z) / z = x by Axiom 5. Then
(L3) (((x * z) / T(z,y)) * T(z,y)) / z = x
by Axiom 6. We have ((((x * z) / T(z,y)) * T(z,y)) / z) * z = ((x * z) / T(z,y)) * T(z,y) by Axiom 6. Then ((((((x * z) / T(z,y)) * T(z,y)) / z) / T(z,y)) * T(z,y)) * z = ((x * z) / T(z,y)) * T(z,y) by Axiom 6. Then
(L6) ((x / T(z,y)) * T(z,y)) * z = ((x * z) / T(z,y)) * T(z,y)
by (L3). We have ((x / T(z,y)) * T(z,y)) * z = ((x / T(z,y)) * z) * T(z,y) by Theorem 238. Then ((x * z) / T(z,y)) * T(z,y) = ((x / T(z,y)) * z) * T(z,y) by (L6). Hence we are done by Proposition 10.
Theorem 240 (prov9_49c7860b44): (x * (z \ y)) * (y / z) = (x * (y / z)) * T(y / z,z)
Proof. We have (x * T(y / z,z)) * (y / z) = (x * (y / z)) * T(y / z,z) by Theorem 238. Hence we are done by Proposition 47.
Theorem 241 (prov9_400e27a280): (x * (y / z)) * (z \ y) = (x * (z \ y)) * (y / z)
Proof. We have (x * (y / z)) * T(y / z,z) = (x * (z \ y)) * (y / z) by Theorem 240. Hence we are done by Proposition 47.
Theorem 242 (prov9_13e5c8ed0a): T(e / (e / x),x \ e) = x
Proof. We have ((e / x) * T(x,x \ e)) / (e / x) = T(e / (e / x),x \ e) by Theorem 51. Then
(L2) ((e / x) * ((x \ e) \ e)) / (e / x) = T(e / (e / x),x \ e)
by Proposition 49. We have ((e / x) * T(x,e / x)) / (e / x) = x by Proposition 48. Then ((e / x) * ((x \ e) \ e)) / (e / x) = x by Theorem 210. Hence we are done by (L2).
Theorem 243 (prov9_5e226df13f): T(y,x) = T(T(e / (e / y),x),y \ e)
Proof. We have T(T(e / (e / y),y \ e),x) = T(T(e / (e / y),x),y \ e) by Axiom 7. Hence we are done by Theorem 242.
Theorem 244 (prov9_e99a5bdd5a): K(x,x \ e) = T(x,x \ e) / x
Proof. We have T(e / (e / x),x \ e) = (x \ e) \ ((e / (e / x)) * (x \ e)) by Definition 3. Then
(L2) T(e / (e / x),x \ e) = (x \ e) \ ((x \ e) / (e / x))
by Theorem 153. We have K((x \ e) \ ((x \ e) / (e / x)),x \ e) = T(x,x \ e) / x by Theorem 199. Then K(T(e / (e / x),x \ e),x \ e) = T(x,x \ e) / x by (L2). Hence we are done by Theorem 242.
Theorem 245 (prov9_fd07c6e1f2): x * (e / x) = (e / (e / (e / x))) * x
Proof. We have T(e / (e / (e / x)),(e / x) \ e) = e / x by Theorem 242. Then
(L2) T(e / (e / (e / x)),x) = e / x
by Proposition 25. We have x * T(e / (e / (e / x)),x) = (e / (e / (e / x))) * x by Proposition 46. Hence we are done by (L2).
Theorem 246 (prov9_183b179b43): K(x,x \ e) * x = T(x,x \ e)
Proof. We have (T(x,x \ e) / x) * x = T(x,x \ e) by Axiom 6. Hence we are done by Theorem 244.
Theorem 247 (prov9_7e1bad9b53): ((x \ e) \ e) / x = K(x,x \ e)
Proof. We have T(x,x \ e) / x = K(x,x \ e) by Theorem 244. Hence we are done by Proposition 49.
Theorem 248 (prov9_d65e145d1a): K(x,x \ e) = K(x,e / x)
Proof. We have T(x,x \ e) / x = K(x,e / x) by Theorem 215. Hence we are done by Theorem 244.
Theorem 249 (prov9_91ccedcc76): K(x \ e,x) = K(x \ e,(x \ e) \ e)
Proof. We have K(x \ e,e / (x \ e)) = K(x \ e,(x \ e) \ e) by Theorem 248. Hence we are done by Proposition 24.
Theorem 250 (prov9_411bf01791): K(x,x \ e) = x * (e / x)
Proof. We have K(x,e / x) = x * (e / x) by Proposition 77. Hence we are done by Theorem 248.
Theorem 251 (prov9_0b4cc0c63d): x \ K(x,x \ e) = e / x
Proof. We have x \ K(x,e / x) = e / x by Theorem 22. Hence we are done by Theorem 248.
Theorem 252 (prov9_3014da99f4): (e / (e / x)) * K(x,x \ e) = x
Proof. We have (e / (e / x)) * (x * (e / x)) = x by Theorem 126. Then (e / (e / x)) * K(x,e / x) = x by Proposition 77. Hence we are done by Theorem 248.
Theorem 253 (prov9_505770952a): K((e / x) \ e,e / x) = K(x,x \ e)
Proof. We have K((e / x) \ e,e / x) = ((e / x) \ e) * (e / x) by Proposition 76. Then K((e / x) \ e,e / x) = x * (e / x) by Proposition 25. Hence we are done by Theorem 250.
Theorem 254 (prov9_33ec4f1d7a): K(x \ e,x) = (x \ e) / (e / x)
Proof. We have (x \ e) * T(e / (e / x),x \ e) = (e / (e / x)) * (x \ e) by Proposition 46. Then
(L2) (x \ e) * x = (e / (e / x)) * (x \ e)
by Theorem 242. We have (e / (e / x)) * (x \ e) = (x \ e) / (e / x) by Theorem 153. Then
(L4) (x \ e) * x = (x \ e) / (e / x)
by (L2). We have K(x \ e,x) = (x \ e) * x by Proposition 76. Hence we are done by (L4).
Theorem 255 (prov9_00b8dc5e0f): K(x \ e,x) * (e / x) = x \ e
Proof. We have ((x \ e) / (e / x)) * (e / x) = x \ e by Axiom 6. Hence we are done by Theorem 254.
Theorem 256 (prov9_095c329c86): K(x \ e,x) \ (x \ e) = e / x
Proof. We have ((x \ e) / (e / x)) \ (x \ e) = e / x by Proposition 25. Hence we are done by Theorem 254.
Theorem 257 (prov9_ac9c58c2c6): K(x \ e,x) = x \ (e / (e / x))
Proof. We have (e / (e / x)) * (((e / x) \ e) \ e) = ((e / x) \ e) \ (e / (e / x)) by Theorem 180. Then (e / (e / x)) * (x \ e) = ((e / x) \ e) \ (e / (e / x)) by Proposition 25. Then
(L3) x \ (e / (e / x)) = (e / (e / x)) * (x \ e)
by Proposition 25. We have (e / (e / x)) * (x \ e) = (x \ e) / (e / x) by Theorem 153. Then x \ (e / (e / x)) = (x \ e) / (e / x) by (L3). Hence we are done by Theorem 254.
Theorem 258 (prov9_c1942da1cc): K(e / x,e / (e / x)) = K(x \ e,x)
Proof. We have (((e / x) \ e) \ e) / (e / x) = K(e / x,e / (e / x)) by Theorem 212. Then (x \ e) / (e / x) = K(e / x,e / (e / x)) by Proposition 25. Hence we are done by Theorem 254.
Theorem 259 (prov9_8502421812): (x \ e) * L(y,e / x,K(x \ e,x)) = K(x \ e,x) * ((e / x) * y)
Proof. We have (K(x \ e,x) * (e / x)) * L(y,e / x,K(x \ e,x)) = K(x \ e,x) * ((e / x) * y) by Proposition 52. Hence we are done by Theorem 255.
Theorem 260 (prov9_1f319920e6): L((e / x) \ y,e / x,K(x \ e,x)) = (x \ e) \ (K(x \ e,x) * y)
Proof. We have L((e / x) \ y,e / x,K(x \ e,x)) = (K(x \ e,x) * (e / x)) \ (K(x \ e,x) * y) by Proposition 53. Hence we are done by Theorem 255.
Theorem 261 (prov9_9c510f6c21): e / x = L(x \ e,x,x \ e)
Proof. We have K(x \ e,x) \ (x \ e) = L(x \ e,x,x \ e) by Theorem 36. Hence we are done by Theorem 256.
Theorem 262 (prov9_84f090ac8a): x * K(x \ e,x) = e / (e / x)
Proof. We have x * (x \ (e / (e / x))) = e / (e / x) by Axiom 4. Hence we are done by Theorem 257.
Theorem 263 (prov9_6d29c88d91): e / (e / x) = L(x,x \ e,x)
Proof. We have x * K(x \ e,x) = L(x,x \ e,x) by Theorem 70. Hence we are done by Theorem 262.
Theorem 264 (prov9_44c7a0453c): L(T(x,y),x \ e,x) = T(e / (e / x),y)
Proof. We have L(T(x,y),x \ e,x) = T(L(x,x \ e,x),y) by Axiom 8. Hence we are done by Theorem 263.
Theorem 265 (prov9_81583f5384): x * (e / x) = x / (e / (e / x))
Proof. We have (e / (e / (e / x))) * ((e / x) \ e) = ((e / x) \ e) / (e / (e / x)) by Theorem 153. Then (e / (e / (e / x))) * x = ((e / x) \ e) / (e / (e / x)) by Proposition 25. Then x / (e / (e / x)) = (e / (e / (e / x))) * x by Proposition 25. Hence we are done by Theorem 245.
Theorem 266 (prov9_4f558c5aa7): L(x,e / x,x) = e / (e / x)
Proof. We have K(x,e / x) = x * (e / x) by Proposition 77. Then
(L2) K(x,e / x) = x / (e / (e / x))
by Theorem 265. We have (x / (e / (e / x))) * (e / (e / x)) = x by Axiom 6. Then
(L4) K(x,e / x) * (e / (e / x)) = x
by (L2). We have K(x,e / x) * (K(x,e / x) \ x) = x by Axiom 4. Then K(x,e / x) * L(x,e / x,x) = x by Theorem 76. Then K(x,e / x) * L(x,e / x,x) = K(x,e / x) * (e / (e / x)) by (L4). Hence we are done by Proposition 9.
Theorem 267 (prov9_326aec52aa): x = T(x,K(x,e / x))
Proof. We have T(e / (e / x),x \ e) = x by Theorem 242. Then
(L2) T(L(x,e / x,x),x \ e) = x
by Theorem 266. We have L(T(x,x \ e),e / x,x) = T(L(x,e / x,x),x \ e) by Axiom 8. Then
(L4) L(T(x,x \ e),e / x,x) = x
by (L2). We have L(T(x,e / x),e / x,x) = T(x,x * (e / x)) by Proposition 63. Then L(T(x,x \ e),e / x,x) = T(x,x * (e / x)) by Theorem 211. Then L(T(x,x \ e),e / x,x) = T(x,K(x,e / x)) by Proposition 77. Hence we are done by (L4).
Theorem 268 (prov9_81482e27b6): T(x,K(x,x \ e)) = x
Proof. We have T(x,K(x,e / x)) = x by Theorem 267. Hence we are done by Theorem 248.
Theorem 269 (prov9_9759f534c5): K(x,e / x) = a(x,e / x,x)
Proof. We have x \ ((x * (e / x)) * x) = a(x,e / x,x) by Theorem 130. Then
(L4) x \ (K(x,e / x) * x) = a(x,e / x,x)
by Proposition 77. We have T(K(x,e / x),x) = x \ (K(x,e / x) * x) by Definition 3. Then
(L6) T(K(x,e / x),x) = a(x,e / x,x)
by (L4). We have T(x,K(x,e / x)) = x by Theorem 267. Then T(K(x,e / x),x) = K(x,e / x) by Proposition 21. Hence we are done by (L6).
Theorem 270 (prov9_26ea453676): x \ T(x,x \ e) = K(x,e / x)
Proof. We have x \ T(x,x \ e) = a(x,e / x,x) by Theorem 218. Hence we are done by Theorem 269.
Theorem 271 (prov9_916fd5e3c9): x \ T(x,x \ e) = K(x,x \ e)
Proof. We have K(x,e / x) = K(x,x \ e) by Theorem 248. Hence we are done by Theorem 270.
Theorem 272 (prov9_fd01c8974f): K(x \ e,x) = K(e / x,(e / x) \ e)
Proof. We have K(e / x,e / (e / x)) = K(e / x,(e / x) \ e) by Theorem 248. Hence we are done by Theorem 258.
Theorem 273 (prov9_85b3a78a03): T(e / (e / y),x) / ((y \ e) * T(y,x)) = y
Proof. We have L(T(y,x),y \ e,y) / ((y \ e) * T(y,x)) = y by Theorem 69. Hence we are done by Theorem 264.
Theorem 274 (prov9_259ae5c872): K(x \ e,x) * L(e / x,y,z) = (x \ e) * (x * L(x \ e,y,z))
Proof. We have L(L(x \ e,x,x \ e),y,z) = L(L(x \ e,y,z),x,x \ e) by Axiom 11. Then
(L2) L(e / x,y,z) = L(L(x \ e,y,z),x,x \ e)
by Theorem 261. We have K(x \ e,x) * L(L(x \ e,y,z),x,x \ e) = (x \ e) * (x * L(x \ e,y,z)) by Theorem 59. Hence we are done by (L2).
Theorem 275 (prov9_3c312f6969): K(y,y \ e) * T(y,x) = y * (T(y,x) * (e / y))
Proof. We have K(y,e / y) * T(T(y,K(y,e / y)),x) = T(y,x) * K(y,e / y) by Proposition 50. Then
(L2) K(y,e / y) * T(y,x) = T(y,x) * K(y,e / y)
by Theorem 267. We have T(y,x) * K(y,e / y) = y * (T(y,x) * (e / y)) by Theorem 176. Then K(y,e / y) * T(y,x) = y * (T(y,x) * (e / y)) by (L2). Hence we are done by Theorem 248.
Theorem 276 (prov9_aaaaf32dd9): z * (L(z,x,y) * w) = L(z,x,y) * (z * w)
Proof. We have (z * w) * L(L(T(z,w),w,z),x,y) = z * (w * L(T(z,w),x,y)) by Theorem 57. Then
(L2) (z * w) * L(T(z,z * w),x,y) = z * (w * L(T(z,w),x,y))
by Proposition 63. We have (z * w) * L(T(z,z * w),x,y) = L(z,x,y) * (z * w) by Proposition 58. Then z * (w * L(T(z,w),x,y)) = L(z,x,y) * (z * w) by (L2). Hence we are done by Proposition 58.
Theorem 277 (prov9_7e6de80c2c): z * (L(z,x,y) \ w) = L(z,x,y) \ (z * w)
Proof. We have z \ (z * w) = w by Axiom 3. Then
(L3) z \ (L(z,x,y) * (L(z,x,y) \ (z * w))) = w
by Axiom 4. We have z * (z \ (L(z,x,y) * (L(z,x,y) \ (z * w)))) = L(z,x,y) * (L(z,x,y) \ (z * w)) by Axiom 4. Then z * (L(z,x,y) * (L(z,x,y) \ (z \ (L(z,x,y) * (L(z,x,y) \ (z * w)))))) = L(z,x,y) * (L(z,x,y) \ (z * w)) by Axiom 4. Then
(L6) z * (L(z,x,y) * (L(z,x,y) \ w)) = L(z,x,y) * (L(z,x,y) \ (z * w))
by (L3). We have L(z,x,y) * (z * (L(z,x,y) \ w)) = z * (L(z,x,y) * (L(z,x,y) \ w)) by Theorem 276. Hence we are done by (L6) and Proposition 7.
Theorem 278 (prov9_87c679f2bd): z * (L(z,x,y) \ e) = L(z,x,y) \ z
Proof. We have z \ (z * e) = e by Axiom 3. Then z \ (L(z,x,y) * (L(z,x,y) \ (z * e))) = e by Axiom 4. Then
(L4) z \ (L(z,x,y) * (L(z,x,y) \ z)) = e
by Axiom 2. We have z * (z \ (L(z,x,y) * (L(z,x,y) \ z))) = L(z,x,y) * (L(z,x,y) \ z) by Axiom 4. Then z * e = L(z,x,y) * (L(z,x,y) \ z) by (L4). Then
(L7) L(z,x,y) * (L(z,x,y) \ z) = z * (L(z,x,y) * (L(z,x,y) \ e))
by Axiom 4. We have L(z,x,y) * (z * (L(z,x,y) \ e)) = z * (L(z,x,y) * (L(z,x,y) \ e)) by Theorem 276. Hence we are done by (L7) and Proposition 7.
Theorem 279 (prov9_74a9d50a35): L(z,x,y) * K(z,z \ e) = z * (L(z,x,y) * (e / z))
Proof. We have L(z,x,y) * (z * (e / z)) = z * (L(z,x,y) * (e / z)) by Theorem 276. Hence we are done by Theorem 250.
Theorem 280 (prov9_167e15ce3d): x \ (L(x,y,z) \ x) = L(x,y,z) \ e
Proof. We have x * (L(x,y,z) \ e) = L(x,y,z) \ x by Theorem 278. Hence we are done by Proposition 2.
Theorem 281 (prov9_2fc614b1dd): x * (L(x,y,z) * (e / x)) = K(x,x \ e) * L(x,y,z)
Proof. We have K(x,x \ e) * L(T(x,K(x,x \ e)),y,z) = L(x,y,z) * K(x,x \ e) by Proposition 58. Then K(x,x \ e) * L(x,y,z) = L(x,y,z) * K(x,x \ e) by Theorem 268. Hence we are done by Theorem 279.
Theorem 282 (prov9_690768235c): z * T(e / z,L(z,x,y)) = K(z,L(z,x,y) / z)
Proof. We have L(z,x,y) \ (z * (L(z,x,y) / z)) = K(z,L(z,x,y) / z) by Theorem 3. Then z * (L(z,x,y) \ (L(z,x,y) / z)) = K(z,L(z,x,y) / z) by Theorem 277. Hence we are done by Theorem 167.
Theorem 283 (prov9_f7b6da6cac): y * ((y \ x) * T((y \ x) \ e,z)) = x * T(x \ y,z)
Proof. We have y * ((y \ x) * T((y \ x) \ e,z)) = x * T(x \ (y * e),z) by Theorem 168. Hence we are done by Axiom 2.
Theorem 284 (prov9_584ffb9386): y * K(z \ (z / (y \ x)),z) = x * T(x \ y,z)
Proof. We have y * ((y \ x) * T((y \ x) \ e,z)) = x * T(x \ y,z) by Theorem 283. Hence we are done by Theorem 196.
Theorem 285 (prov9_762a2098e8): x \ (y * T(y \ x,z)) = K(z \ (z / (x \ y)),z)
Proof. We have x * K(z \ (z / (x \ y)),z) = y * T(y \ x,z) by Theorem 284. Hence we are done by Proposition 2.
Theorem 286 (prov9_8de3568474): K(z \ (z / (x / y)),z) * y = x * T(x \ y,z)
Proof. We have (T(e / (x / y),z) * (x / y)) * y = T((e * y) / x,z) * x by Theorem 169. Then
(L2) (T(e / (x / y),z) * (x / y)) * y = T(y / x,z) * x
by Axiom 1. We have T(y / x,z) * x = x * T(x \ y,z) by Proposition 61. Then (T(e / (x / y),z) * (x / y)) * y = x * T(x \ y,z) by (L2). Then ((x / y) * T((x / y) \ e,z)) * y = x * T(x \ y,z) by Proposition 61. Hence we are done by Theorem 196.
Theorem 287 (prov9_2d64f57221): (x * T(x \ y,z)) / y = K(z \ (z / (x / y)),z)
Proof. We have K(z \ (z / (x / y)),z) * y = x * T(x \ y,z) by Theorem 286. Hence we are done by Proposition 1.
Theorem 288 (prov9_9203ae02ac): K((x * (y / z)) \ x,x * (y / z)) = (y * T(y \ z,x * (y / z))) / z
Proof. We have K((x * (y / z)) \ ((x * (y / z)) / (y / z)),x * (y / z)) = (y * T(y \ z,x * (y / z))) / z by Theorem 287. Hence we are done by Axiom 5.
Theorem 289 (prov9_43f439c228): (x * ((y * (x \ e)) / y)) * y = T(y,R(e / x,x,y))
Proof. We have T(y,R(e / x,x,y)) = ((y * R(e / x,x,y)) / y) * (R(e / x,x,y) \ y) by Theorem 179. Then
(L2) T(y,R(e / x,x,y)) = ((y * R(e / x,x,y)) / y) * (x * y)
by Proposition 80. We have ((y * R(e / x,x,y)) / y) * (x * y) = (((y * (e / x)) / y) * x) * y by Theorem 129. Then T(y,R(e / x,x,y)) = (((y * (e / x)) / y) * x) * y by (L2). Hence we are done by Theorem 125.
Theorem 290 (prov9_35ac413abf): ((y * x) / y) / x = (e / x) * ((y * ((e / x) \ e)) / y)
Proof. We have ((y * (e / (e / x))) / y) * (e / x) = ((y * (e * x)) / y) / x by Theorem 232. Then
(L2) ((y * (e / (e / x))) / y) * (e / x) = ((y * x) / y) / x
by Axiom 1. We have ((y * (e / (e / x))) / y) * (e / x) = (e / x) * ((y * ((e / x) \ e)) / y) by Theorem 125. Hence we are done by (L2).
Theorem 291 (prov9_aa4f49d38f): (x * w) * T((x * w) \ (y * w),z) = (x * T(x \ y,z)) * w
Proof. We have (x * w) * T((x * w) \ (y * w),z) = (T(y / x,z) * x) * w by Theorem 237. Hence we are done by Proposition 61.
Theorem 292 (prov9_dbeafef9be): (x * z) / (y / (y / z)) = (x / (y / (y / z))) * z
Proof. We have (x * z) / z = x by Axiom 5. Then
(L5) (((x * z) / (y / (y / z))) * (y / (y / z))) / z = x
by Axiom 6. We have ((((x * z) / (y / (y / z))) * (y / (y / z))) / z) * z = ((x * z) / (y / (y / z))) * (y / (y / z)) by Axiom 6. Then ((((((x * z) / (y / (y / z))) * (y / (y / z))) / z) / (y / (y / z))) * (y / (y / z))) * z = ((x * z) / (y / (y / z))) * (y / (y / z)) by Axiom 6. Then
(L8) ((x / (y / (y / z))) * (y / (y / z))) * z = ((x * z) / (y / (y / z))) * (y / (y / z))
by (L5). We have ((x / (y / (y / z))) * (y / (y / z))) * T(y / (y / z),y / z) = ((x / (y / (y / z))) * ((y / z) \ y)) * (y / (y / z)) by Theorem 240. Then ((x / (y / (y / z))) * (y / (y / z))) * z = ((x / (y / (y / z))) * ((y / z) \ y)) * (y / (y / z)) by Theorem 14. Then ((x / (y / (y / z))) * z) * (y / (y / z)) = ((x / (y / (y / z))) * (y / (y / z))) * z by Proposition 25. Then ((x * z) / (y / (y / z))) * (y / (y / z)) = ((x / (y / (y / z))) * z) * (y / (y / z)) by (L8). Hence we are done by Proposition 10.
Theorem 293 (prov9_0cc95e5d7e): (T((z / x) / y,w) * y) * x = y * (x * T(x \ (y \ z),w))
Proof. We have T(z / (y * x),w) * (y * x) = (y * x) * T((y * x) \ z,w) by Proposition 61. Then
(L2) (T((z / x) / y,w) * y) * x = (y * x) * T((y * x) \ z,w)
by Theorem 138. We have (y * x) * T((y * x) \ z,w) = y * (x * T(x \ (y \ z),w)) by Theorem 136. Hence we are done by (L2).
Theorem 294 (prov9_ebdc42562c): (x * w) * L((x * w) \ w,y,z) = (x * L(x \ e,y,z)) * w
Proof. We have L(R(e / x,x,w),y,z) * (x * w) = (L(e / x,y,z) * x) * w by Theorem 65. Then
(L2) L(w / (x * w),y,z) * (x * w) = (L(e / x,y,z) * x) * w
by Proposition 79. We have L(w / (x * w),y,z) * (x * w) = (x * w) * L((x * w) \ w,y,z) by Theorem 85. Then (L(e / x,y,z) * x) * w = (x * w) * L((x * w) \ w,y,z) by (L2). Hence we are done by Theorem 85.
Theorem 295 (prov9_e2033c7406): K(y \ (y / T(x,z)),y) = z \ ((x * T(x \ e,y)) * z)
Proof. We have K(y \ (y / (z \ (x * z))),y) = z \ ((x * z) * T((x * z) \ z,y)) by Theorem 285. Then K(y \ (y / T(x,z)),y) = z \ ((x * z) * T((x * z) \ z,y)) by Definition 3. Hence we are done by Theorem 194.
Theorem 296 (prov9_889b5a7516): T(x * T(x \ e,z),y) = K(z \ (z / T(x,y)),z)
Proof. We have T(x * T(x \ e,z),y) = y \ ((x * T(x \ e,z)) * y) by Definition 3. Hence we are done by Theorem 295.
Theorem 297 (prov9_9b237d6193): (x * R(x \ e,y,z)) * x = x * (x * R(x \ e,y,z))
Proof. We have (x * x) * R((x * x) \ x,y,z) = x * (x * R(x \ e,y,z)) by Theorem 133. Hence we are done by Proposition 84.
Theorem 298 (prov9_eb96e1b2b8): T(z,z * R(z \ e,x,y)) = z
Proof. We have (z * R(z \ e,x,y)) * z = z * (z * R(z \ e,x,y)) by Theorem 297. Hence we are done by Theorem 11.
Theorem 299 (prov9_c205cf50df): T(y,(x / (y * x)) * y) = y
Proof. We have T(y,y * R(y \ e,y,x)) = y by Theorem 298. Hence we are done by Theorem 90.
Theorem 300 (prov9_1c1364a365): T(y / (x * y),x * (y / (x * y))) = y / (x * y)
Proof. We have T(y / (x * y),(y / (x * y)) * R((y / (x * y)) \ e,y / (x * y),x * y)) = y / (x * y) by Theorem 298. Then T(y / (x * y),((x * y) / y) * (y / (x * y))) = y / (x * y) by Theorem 111. Hence we are done by Axiom 5.
Theorem 301 (prov9_b2368c2943): (x * T(x \ y,w)) * z = x * (z * T(z \ (x \ (y * z)),w))
Proof. We have (x * z) * T((x * z) \ (y * z),w) = x * (z * T(z \ (x \ (y * z)),w)) by Theorem 136. Hence we are done by Theorem 291.
Theorem 302 (prov9_74b7ed23f9): (y * T(y \ (z / x),w)) * x = y * (x * T(x \ (y \ z),w))
Proof. We have (T((z / x) / y,w) * y) * x = y * (x * T(x \ (y \ z),w)) by Theorem 293. Hence we are done by Proposition 61.
Theorem 303 (prov9_ae9ce2cada): (x * L(x \ e,y,z)) * x = x * (x * L(x \ e,y,z))
Proof. We have (x * x) * L((x * x) \ x,y,z) = x * (x * L(x \ e,y,z)) by Theorem 135. Hence we are done by Theorem 294.
Theorem 304 (prov9_0188a9a9ac): T(z,z * L(z \ e,x,y)) = z
Proof. We have (z * L(z \ e,x,y)) * z = z * (z * L(z \ e,x,y)) by Theorem 303. Hence we are done by Theorem 11.
Theorem 305 (prov9_1ac9ff191c): T(y,y * ((x * y) \ x)) = y
Proof. We have T(y,y * L(y \ e,y,x)) = y by Theorem 304. Hence we are done by Proposition 78.
Theorem 306 (prov9_9df8fcb99d): T(x \ y,(x \ y) * (y \ x)) = x \ y
Proof. We have T(x \ y,(x \ y) * L((x \ y) \ e,x \ y,x)) = x \ y by Theorem 304. Hence we are done by Theorem 28.
Theorem 307 (prov9_7e62347910): T(T(y,x) * ((T(y,x) / y) \ (e / y)),T(y,x)) = T(y,x) * ((T(y,x) / y) \ (e / y))
Proof. We have T(T(y,x),T(y,x) * (((e / y) * T(y,x)) \ (e / y))) = T(y,x) by Theorem 305. Then T(T(y,x),T(y,x) * ((T(y,x) / y) \ (e / y))) = T(y,x) by Theorem 150. Hence we are done by Proposition 21.
Theorem 308 (prov9_11beba6916): T(T(y,z) \ e,y \ T(y,x)) = T(y,z) \ e
Proof. We have T(e,z) = z \ (e * z) by Definition 3. Then
(L2) T(e,z) = z \ z
by Axiom 1. We have z \ z = e by Proposition 28. Then
(L4) T(e,z) = e
by (L2). We have T(y,z) \ K((y \ T(y,x)) \ ((y \ T(y,x)) / T(y,z)),y \ T(y,x)) = T(T(y,z) \ e,y \ T(y,x)) by Theorem 197. Then T(y,z) \ T(y * T(y \ e,y \ T(y,x)),z) = T(T(y,z) \ e,y \ T(y,x)) by Theorem 296. Then T(y,z) \ T(y * (y \ e),z) = T(T(y,z) \ e,y \ T(y,x)) by Theorem 205. Then T(T(y,z) \ e,y \ T(y,x)) = T(y,z) \ T(e,z) by Axiom 4. Hence we are done by (L4).
Theorem 309 (prov9_36e36e3db4): z * T(z \ T(x,z),y) = T(z * T(z \ x,y),z)
Proof. We have z * (z * T(z \ (z \ (x * z)),y)) = (z * T(z \ x,y)) * z by Theorem 301. Then T(z * T(z \ x,y),z) = z * T(z \ (z \ (x * z)),y) by Theorem 11. Hence we are done by Definition 3.
Theorem 310 (prov9_7a83ca4d6a): T((y \ x) * y,y) = (y \ T(x,y)) * y
Proof. We have y * T(y \ T(x,y),y) = (y \ T(x,y)) * y by Proposition 46. Then T(y * T(y \ x,y),y) = (y \ T(x,y)) * y by Theorem 309. Hence we are done by Proposition 46.
Theorem 311 (prov9_9eaf06f89d): T((y \ x) * y,y) / y = y \ T(x,y)
Proof. We have (y \ T(x,y)) * y = T((y \ x) * y,y) by Theorem 310. Hence we are done by Proposition 1.
Theorem 312 (prov9_53dea67a46): y * (T((y \ x) * y,y) / y) = T(x,y)
Proof. We have y * (y \ T(x,y)) = T(x,y) by Axiom 4. Hence we are done by Theorem 311.
Theorem 313 (prov9_a618d95d6a): T(x,y) / y = y \ T(y * (x / y),y)
Proof. We have (x / y) * y = x by Axiom 6. Then
(L2) (y \ (y * (x / y))) * y = x
by Axiom 3. We have T((y \ (y * (x / y))) * y,y) / y = y \ T(y * (x / y),y) by Theorem 311. Hence we are done by (L2).
Theorem 314 (prov9_5cb7c3c16e): y * (T(x,y) / y) = T(y * (x / y),y)
Proof. We have (x / y) * y = x by Axiom 6. Then
(L2) (y \ (y * (x / y))) * y = x
by Axiom 3. We have y * (T((y \ (y * (x / y))) * y,y) / y) = T(y * (x / y),y) by Theorem 312. Hence we are done by (L2).
Theorem 315 (prov9_4d25fc9889): ((x / y) * K(y \ e,y)) * y = x * K(y \ e,y)
Proof. We have (x * T(x \ (x / y),y)) * y = x * (y * T(y \ (x \ x),y)) by Theorem 302. Then ((x / y) * K(y \ e,y)) * y = x * (y * T(y \ (x \ x),y)) by Theorem 154. Then ((x / y) * K(y \ e,y)) * y = x * (y * T(y \ e,y)) by Proposition 28. Then ((x / y) * K(y \ e,y)) * y = x * ((y \ e) * y) by Proposition 46. Hence we are done by Proposition 76.
Theorem 316 (prov9_ad5a27eee5): (x * L(x \ e,z,w)) * (x \ y) = y * L(y \ (x \ y),z,w)
Proof. We have L((x \ y) / y,z,w) * y = y * L(y \ (x \ y),z,w) by Theorem 85. Then
(L2) L(R(e / x,x,x \ y),z,w) * y = y * L(y \ (x \ y),z,w)
by Theorem 37. We have L(R(e / x,x,x \ y),z,w) * y = (L(e / x,z,w) * x) * (x \ y) by Theorem 97. Then y * L(y \ (x \ y),z,w) = (L(e / x,z,w) * x) * (x \ y) by (L2). Hence we are done by Theorem 85.
Theorem 317 (prov9_0e6358f08b): y * L(y \ (x \ y),x,x \ e) = K(x,x \ e) * (x \ y)
Proof. We have (x * L(x \ e,x,x \ e)) * (x \ y) = y * L(y \ (x \ y),x,x \ e) by Theorem 316. Then (x * (e / x)) * (x \ y) = y * L(y \ (x \ y),x,x \ e) by Theorem 261. Hence we are done by Theorem 250.
Theorem 318 (prov9_8f5549ec85): y * ((w * (y \ (x * z))) / w) = x * ((x \ y) * ((w * ((x \ y) \ z)) / w))
Proof. We have y * L((w * ((x \ y) \ z)) / w,x \ y,x) = x * ((x \ y) * ((w * ((x \ y) \ z)) / w)) by Theorem 52. Hence we are done by Theorem 128.
Theorem 319 (prov9_cbb3ebdaae): (e / (x / y)) \ (y / x) = (y / x) * (x / y)
Proof. We have (e / (x / y)) \ R(e / (x / y),x / y,y) = (x / y) * R((x / y) \ e,x / y,y) by Theorem 187. Then
(L2) (e / (x / y)) \ (y / x) = (x / y) * R((x / y) \ e,x / y,y)
by Theorem 38. We have (x / y) * R((x / y) \ e,x / y,y) = (y / x) * (x / y) by Theorem 111. Hence we are done by (L2).
Theorem 320 (prov9_e93bb87524): (e / (y / x)) * ((x / y) * (y / x)) = x / y
Proof. We have (e / (y / x)) * ((e / (y / x)) \ (x / y)) = x / y by Axiom 4. Hence we are done by Theorem 319.
Theorem 321 (prov9_ac9cc63298): (x \ y) * (y \ x) = (e / (y \ x)) \ (x \ y)
Proof. We have (e / (y \ x)) \ L((y \ x) \ e,y \ x,y) = L((y \ x) \ e,y \ x,y) * (y \ x) by Theorem 190. Then (e / (y \ x)) \ (x \ y) = L((y \ x) \ e,y \ x,y) * (y \ x) by Theorem 28. Hence we are done by Theorem 28.
Theorem 322 (prov9_a1bc9612d3): (e / (y \ x)) * ((x \ y) * (y \ x)) = x \ y
Proof. We have (e / (y \ x)) * ((e / (y \ x)) \ (x \ y)) = x \ y by Axiom 4. Hence we are done by Theorem 321.
Theorem 323 (prov9_199c11aebe): ((y * x) \ y) * x = (e / x) \ ((y * x) \ y)
Proof. We have (e / (y \ (y * x))) \ ((y * x) \ y) = ((y * x) \ y) * (y \ (y * x)) by Theorem 321. Then (e / x) \ ((y * x) \ y) = ((y * x) \ y) * (y \ (y * x)) by Axiom 3. Hence we are done by Axiom 3.
Theorem 324 (prov9_caf79ed64b): T(y \ (K(x,x \ e) * (x \ y)),x) = y \ (x \ y)
Proof. We have R(T(x \ K(x,x \ e),y),x,x \ y) = T(y \ (K(x,x \ e) * (x \ y)),x) by Theorem 171. Then
(L2) R(T(e / x,y),x,x \ y) = T(y \ (K(x,x \ e) * (x \ y)),x)
by Theorem 251. We have R(T(e / x,y),x,x \ y) = y \ (e * (x \ y)) by Theorem 142. Then T(y \ (K(x,x \ e) * (x \ y)),x) = y \ (e * (x \ y)) by (L2). Hence we are done by Axiom 1.
Theorem 325 (prov9_843556d846): x * (w * L(w \ (x \ w),y,z)) = (x * L(x \ e,y,z)) * w
Proof. We have (x * w) * L((x * w) \ w,y,z) = (x * L(x \ e,y,z)) * w by Theorem 294. Hence we are done by Theorem 155.
Theorem 326 (prov9_9eefa3c341): x * (K(x,x \ e) * (x \ y)) = (x * (e / x)) * y
Proof. We have x * (y * L(y \ (x \ y),x,x \ e)) = (x * L(x \ e,x,x \ e)) * y by Theorem 325. Then x * (y * L(y \ (x \ y),x,x \ e)) = (x * (e / x)) * y by Theorem 261. Hence we are done by Theorem 317.
Theorem 327 (prov9_e1147ed0b2): x * (K(x,x \ e) * (x \ y)) = K(x,x \ e) * y
Proof. We have x * (K(x,x \ e) * (x \ y)) = (x * (e / x)) * y by Theorem 326. Hence we are done by Theorem 250.
Theorem 328 (prov9_92fe16dcf1): K(y,y \ e) * (y \ T(y,x)) = T(y,x) * (e / y)
Proof. We have
(L1) y * (K(y,y \ e) * (y \ T(y,x))) = K(y,y \ e) * T(y,x)
by Theorem 327. We have y * (T(y,x) * (e / y)) = K(y,y \ e) * T(y,x) by Theorem 275. Hence we are done by (L1) and Proposition 7.
Theorem 329 (prov9_9eee5d672e): (L(y / x,w,u) * x) * z = (x * z) * L((x * z) \ (y * z),w,u)
Proof. We have L((y * z) / (x * z),w,u) * (x * z) = (x * z) * L((x * z) \ (y * z),w,u) by Theorem 85. Hence we are done by Theorem 157.
Theorem 330 (prov9_0fa01c4137): (x / y) \ (x * ((z * (x \ (x / y))) / z)) = y * ((z * (y \ e)) / z)
Proof. We have L((z * (y \ e)) / z,y,x / y) = (z * L(y \ e,y,x / y)) / z by Theorem 122. Then
(L2) L((z * (y \ e)) / z,y,x / y) = (z * (x \ (x / y))) / z
by Theorem 29. We have x * L((z * (y \ e)) / z,y,x / y) = (x / y) * (y * ((z * (y \ e)) / z)) by Theorem 53. Then x * ((z * (x \ (x / y))) / z) = (x / y) * (y * ((z * (y \ e)) / z)) by (L2). Hence we are done by Proposition 2.
Theorem 331 (prov9_a7e524f257): K(y,y \ e) * ((e / y) \ x) = x * T(x \ ((e / y) \ x),y \ e)
Proof. We have (T(e / (e / y),y \ e) * (e / y)) * ((e / y) \ x) = x * T(x \ ((e / y) \ x),y \ e) by Theorem 183. Then (y * (e / y)) * ((e / y) \ x) = x * T(x \ ((e / y) \ x),y \ e) by Theorem 242. Hence we are done by Theorem 250.
Theorem 332 (prov9_c4c7154ce5): (x * u) * L((x * u) \ (y * u),z,w) = (x * L(x \ y,z,w)) * u
Proof. We have (x * u) * L((x * u) \ (y * u),z,w) = (L(y / x,z,w) * x) * u by Theorem 329. Hence we are done by Theorem 85.
Theorem 333 (prov9_6a420bac28): x * ((y * (x \ e)) / y) = K(y,(y / x) / y)
Proof. We have (y / x) \ (y * ((y * (y \ (y / x))) / y)) = x * ((y * (x \ e)) / y) by Theorem 330. Then
(L2) (y / x) \ (y * ((y / x) / y)) = x * ((y * (x \ e)) / y)
by Axiom 4. We have (y / x) \ (y * ((y / x) / y)) = K(y,(y / x) / y) by Theorem 3. Hence we are done by (L2).
Theorem 334 (prov9_bbcc303d26): T(x * R(e / x,y,z),x) = x * R(e / x,y,z)
Proof. We have x * (x * R(x \ e,y,z)) = (x * R(x \ e,y,z)) * x by Theorem 297. Then
(L2) T(x * R(x \ e,y,z),x) = x * R(x \ e,y,z)
by Theorem 11. We have T(x * ((x * R(x \ e,y,z)) / x),x) = x * (T(x * R(x \ e,y,z),x) / x) by Theorem 314. Then T(x * ((x * R(x \ e,y,z)) / x),x) = x * ((x * R(x \ e,y,z)) / x) by (L2). Then T(x * ((x * R(x \ e,y,z)) / x),x) = x * R(e / x,y,z) by Theorem 88. Hence we are done by Theorem 88.
Theorem 335 (prov9_f81fb582c3): T(x * (y / (x * y)),x) = x * (y / (x * y))
Proof. We have T(x * R(e / x,x,y),x) = x * R(e / x,x,y) by Theorem 334. Then T(x * (y / (x * y)),x) = x * R(e / x,x,y) by Proposition 79. Hence we are done by Proposition 79.
Theorem 336 (prov9_bcc886d420): T(y,y * (x / (y * x))) = y
Proof. We have T(y * (x / (y * x)),y) = y * (x / (y * x)) by Theorem 335. Hence we are done by Proposition 21.
Theorem 337 (prov9_c09cbc75f7): T(x / y,(x / y) * (y / x)) = x / y
Proof. We have T(x / y,(x / y) * (y / ((x / y) * y))) = x / y by Theorem 336. Hence we are done by Axiom 6.
Theorem 338 (prov9_7f109e9c10): (x \ e) * (K(x \ e,x) * ((x \ e) \ y)) = K(x \ e,x) * y
Proof. We have (x \ e) * (K(x \ e,(x \ e) \ e) * ((x \ e) \ y)) = K(x \ e,(x \ e) \ e) * y by Theorem 327. Then (x \ e) * (K(x \ e,x) * ((x \ e) \ y)) = K(x \ e,(x \ e) \ e) * y by Theorem 249. Hence we are done by Theorem 249.
Theorem 339 (prov9_522162d599): (x * (y * L(y \ e,z,w))) * y = (x * y) * (y * L(y \ e,z,w))
Proof. We have ((x * y) * y) * L(((x * y) * y) \ (x * y),z,w) = (x * y) * (y * L(y \ e,z,w)) by Theorem 135. Then ((x * y) * L((x * y) \ x,z,w)) * y = (x * y) * (y * L(y \ e,z,w)) by Theorem 332. Hence we are done by Theorem 135.
Theorem 340 (prov9_ed6189b1b0): (((x * y) \ x) * y) \ ((x * y) \ x) = e / y
Proof. We have K((((x * y) \ x) * (x \ (x * y))) \ ((((x * y) \ x) * (x \ (x * y))) / (x \ (x * y))),((x * y) \ x) * (x \ (x * y))) = x \ ((x * y) * T((x * y) \ x,((x * y) \ x) * (x \ (x * y)))) by Theorem 285. Then K((((x * y) \ x) * (x \ (x * y))) \ ((x * y) \ x),((x * y) \ x) * (x \ (x * y))) = x \ ((x * y) * T((x * y) \ x,((x * y) \ x) * (x \ (x * y)))) by Axiom 5. Then
(L3) K((((x * y) \ x) * (x \ (x * y))) \ ((x * y) \ x),((x * y) \ x) * (x \ (x * y))) = x \ ((x * y) * ((x * y) \ x))
by Theorem 306. We have x \ (x * e) = e by Axiom 3. Then x \ ((x * y) * ((x * y) \ (x * e))) = e by Axiom 4. Then x \ ((x * y) * ((x * y) \ x)) = e by Axiom 2. Then
(L7) K((((x * y) \ x) * (x \ (x * y))) \ ((x * y) \ x),((x * y) \ x) * (x \ (x * y))) = e
by (L3). We have K((((x * y) \ x) * (x \ (x * y))) \ ((x * y) \ x),((x * y) \ x) * (x \ (x * y))) / (x \ (x * y)) = T(e / (x \ (x * y)),((x * y) \ x) * (x \ (x * y))) by Theorem 185. Then
(L9) e / (x \ (x * y)) = T(e / (x \ (x * y)),((x * y) \ x) * (x \ (x * y)))
by (L7). We have T(e / (x \ (x * y)),((x * y) \ x) * (x \ (x * y))) = (((x * y) \ x) * (x \ (x * y))) \ ((e / (x \ (x * y))) * (((x * y) \ x) * (x \ (x * y)))) by Definition 3. Then T(e / (x \ (x * y)),((x * y) \ x) * (x \ (x * y))) = (((x * y) \ x) * (x \ (x * y))) \ ((x * y) \ x) by Theorem 322. Then e / (x \ (x * y)) = (((x * y) \ x) * (x \ (x * y))) \ ((x * y) \ x) by (L9). Then (((x * y) \ x) * y) \ ((x * y) \ x) = e / (x \ (x * y)) by Axiom 3. Hence we are done by Axiom 3.
Theorem 341 (prov9_2a455796cf): ((x * y) \ x) / (e / y) = ((x * y) \ x) * y
Proof. We have ((x * y) \ x) / ((((x * y) \ x) * y) \ ((x * y) \ x)) = ((x * y) \ x) * y by Proposition 24. Hence we are done by Theorem 340.
Theorem 342 (prov9_4a60bdf7b9): L(y \ e,y,(x * y) \ x) = e / y
Proof. We have L(y \ e,y,(x * y) \ x) = (((x * y) \ x) * y) \ ((x * y) \ x) by Proposition 78. Hence we are done by Theorem 340.
Theorem 343 (prov9_9a65075e73): (x * z) * ((y * ((x * z) \ z)) / y) = (x * ((y * (x \ e)) / y)) * z
Proof. We have ((y * R(e / x,x,z)) / y) * (x * z) = (((y * (e / x)) / y) * x) * z by Theorem 129. Then (x * z) * ((y * ((x * z) \ z)) / y) = (((y * (e / x)) / y) * x) * z by Theorem 144. Hence we are done by Theorem 125.
Theorem 344 (prov9_bc9f984ae7): T(x * ((z * (x \ e)) / z),y) = K(z,(z / T(x,y)) / z)
Proof. We have y * ((y \ (x * y)) * ((z * ((y \ (x * y)) \ e)) / z)) = (x * y) * ((z * ((x * y) \ (y * e))) / z) by Theorem 318. Then y * K(z,(z / (y \ (x * y))) / z) = (x * y) * ((z * ((x * y) \ (y * e))) / z) by Theorem 333. Then y * K(z,(z / (y \ (x * y))) / z) = (x * y) * ((z * ((x * y) \ y)) / z) by Axiom 2. Then y * K(z,(z / T(x,y)) / z) = (x * y) * ((z * ((x * y) \ y)) / z) by Definition 3. Then (x * ((z * (x \ e)) / z)) * y = y * K(z,(z / T(x,y)) / z) by Theorem 343. Hence we are done by Theorem 11.
Theorem 345 (prov9_29eee8e85c): T(x * ((z * (x \ e)) / z),y) = T(x,y) * ((z * (T(x,y) \ e)) / z)
Proof. We have K(z,(z / T(x,y)) / z) = T(x,y) * ((z * (T(x,y) \ e)) / z) by Theorem 333. Hence we are done by Theorem 344.
Theorem 346 (prov9_193165ce9e): L(e / x,T(x,y),e / x) = (T(x,y) * (e / x)) / T(x,y)
Proof. We have T((T(x,y) * (e / x)) / T(x,y),(e / x) * T(x,y)) = L(e / x,T(x,y),e / x) by Theorem 92. Then
(L2) T((T(x,y) * (e / x)) / T(x,y),T(x,y) / x) = L(e / x,T(x,y),e / x)
by Theorem 150. We have (T(x,y) * T(e / x,T(x,y) / x)) / T(x,y) = T((T(x,y) * (e / x)) / T(x,y),T(x,y) / x) by Theorem 50. Then (T(x,y) * (e / x)) / T(x,y) = T((T(x,y) * (e / x)) / T(x,y),T(x,y) / x) by Theorem 206. Hence we are done by (L2).
Theorem 347 (prov9_283ee7ef8d): T(x,y / (x * y)) = T(x,R(x \ e,x,y))
Proof. We have R(T(y / (x * y),x * (y / (x * y))),x,y / (x * y)) = T(y / (x * y),x) by Theorem 113. Then
(L5) R(y / (x * y),x,y / (x * y)) = T(y / (x * y),x)
by Theorem 300. We have T(y / (x * y),x) = R(x \ e,x,y) by Theorem 89. Then
(L7) R(y / (x * y),x,y / (x * y)) = R(x \ e,x,y)
by (L5). We have R(y / (x * y),x,y / (x * y)) \ T((y / (x * y)) * x,y / (x * y)) = T(x,y / (x * y)) by Theorem 231. Then
(L9) R(x \ e,x,y) \ T((y / (x * y)) * x,y / (x * y)) = T(x,y / (x * y))
by (L7). We have T(y / (x * y),(y / (x * y)) * ((x * y) / y)) = y / (x * y) by Theorem 337. Then T(y / (x * y),(y / (x * y)) * x) = y / (x * y) by Axiom 5. Then T((y / (x * y)) * x,y / (x * y)) = (y / (x * y)) * x by Proposition 21. Then
(L10) R(x \ e,x,y) \ ((y / (x * y)) * x) = T(x,y / (x * y))
by (L9). We have T(x,R(x \ e,x,y)) = R(x \ e,x,y) \ (x * R(x \ e,x,y)) by Definition 3. Then T(x,R(x \ e,x,y)) = R(x \ e,x,y) \ ((y / (x * y)) * x) by Theorem 90. Hence we are done by (L10).
Theorem 348 (prov9_6a90e0da80): ((y \ e) * (y * L(y \ e,y,x))) * y = (y * ((x * y) \ x)) * K(y \ e,y)
Proof. We have T(e / y,(y \ e) * y) = L(y \ e,y,y \ e) by Theorem 91. Then T(e / y,K(y \ e,y)) = L(y \ e,y,y \ e) by Proposition 76. Then
(L3) T(e / y,K(y \ e,y)) = e / y
by Theorem 261. We have K(y \ e,y) * L(T(e / y,K(y \ e,y)),y,x) = L(e / y,y,x) * K(y \ e,y) by Proposition 58. Then K(y \ e,y) * L(e / y,y,x) = L(e / y,y,x) * K(y \ e,y) by (L3). Then
(L6) L(e / y,y,x) * K(y \ e,y) = (y \ e) * (y * L(y \ e,y,x))
by Theorem 274. We have (((y * ((x * y) \ x)) / y) * K(y \ e,y)) * y = (y * ((x * y) \ x)) * K(y \ e,y) by Theorem 315. Then (L(e / y,y,x) * K(y \ e,y)) * y = (y * ((x * y) \ x)) * K(y \ e,y) by Theorem 121. Hence we are done by (L6).
Theorem 349 (prov9_55c1d37b6b): T(x * R(x \ e,z,w),y) = T(x,y) * R(T(x,y) \ e,z,w)
Proof. We have (x * y) * L(R(T(x,y) \ e,z,w),T(x,y),y) = y * (T(x,y) * R(T(x,y) \ e,z,w)) by Theorem 58. Then (x * y) * R((y * T(x,y)) \ y,z,w) = y * (T(x,y) * R(T(x,y) \ e,z,w)) by Theorem 31. Then
(L3) (x * y) * R((x * y) \ y,z,w) = y * (T(x,y) * R(T(x,y) \ e,z,w))
by Proposition 46. We have (x * y) * R((x * y) \ y,z,w) = (x * R(x \ e,z,w)) * y by Proposition 84. Then y * (T(x,y) * R(T(x,y) \ e,z,w)) = (x * R(x \ e,z,w)) * y by (L3). Hence we are done by Theorem 11.
Theorem 350 (prov9_e94a18853e): w * (T(x,w) * L(T(x,w) \ e,y,z)) = (x * L(x \ e,y,z)) * w
Proof. We have (x * w) * L(L(T(x,w) \ e,y,z),T(x,w),w) = w * (T(x,w) * L(T(x,w) \ e,y,z)) by Theorem 58. Then (x * w) * L((w * T(x,w)) \ w,y,z) = w * (T(x,w) * L(T(x,w) \ e,y,z)) by Theorem 32. Then
(L3) (x * w) * L((x * w) \ w,y,z) = w * (T(x,w) * L(T(x,w) \ e,y,z))
by Proposition 46. We have (x * w) * L((x * w) \ w,y,z) = (x * L(x \ e,y,z)) * w by Theorem 294. Hence we are done by (L3).
Theorem 351 (prov9_1604252226): T(x * L(x \ e,z,w),y) = T(x,y) * L(T(x,y) \ e,z,w)
Proof. We have y * (T(x,y) * L(T(x,y) \ e,z,w)) = (x * L(x \ e,z,w)) * y by Theorem 350. Hence we are done by Theorem 11.
Theorem 352 (prov9_35e680c856): e / ((y \ x) / x) = (y * ((y \ x) / x)) \ y
Proof. We have T(y,y * ((y \ x) / (y * (y \ x)))) = y by Theorem 336. Then
(L2) T(y,y * ((y \ x) / x)) = y
by Axiom 4. We have T(T(y,y * ((y \ x) / x)),y \ x) = T((y \ x) \ x,y * ((y \ x) / x)) by Theorem 18. Then
(L4) T(y,y \ x) = T((y \ x) \ x,y * ((y \ x) / x))
by (L2). We have K((y * ((y \ x) / x)) \ y,y * ((y \ x) / x)) = ((y \ x) * T((y \ x) \ x,y * ((y \ x) / x))) / x by Theorem 288. Then K((y * ((y \ x) / x)) \ y,y * ((y \ x) / x)) = ((y \ x) * T(y,y \ x)) / x by (L4). Then
(L7) K((y * ((y \ x) / x)) \ y,y * ((y \ x) / x)) = x / x
by Theorem 8. We have x / x = e by Proposition 29. Then
(L9) K((y * ((y \ x) / x)) \ y,y * ((y \ x) / x)) = e
by (L7). We have K((y * ((y \ x) / x)) \ y,y * ((y \ x) / x)) / ((y \ x) / x) = T(e / ((y \ x) / x),y * ((y \ x) / x)) by Theorem 185. Then
(L11) e / ((y \ x) / x) = T(e / ((y \ x) / x),y * ((y \ x) / x))
by (L9). We have x / (y \ x) = y by Proposition 24. Then (e / ((y \ x) / x)) * ((x / (y \ x)) * ((y \ x) / x)) = y by Theorem 320. Then
(L14) (e / ((y \ x) / x)) * (y * ((y \ x) / x)) = y
by Proposition 24. We have T(e / ((y \ x) / x),y * ((y \ x) / x)) = (y * ((y \ x) / x)) \ ((e / ((y \ x) / x)) * (y * ((y \ x) / x))) by Definition 3. Then T(e / ((y \ x) / x),y * ((y \ x) / x)) = (y * ((y \ x) / x)) \ y by (L14). Hence we are done by (L11).
Theorem 353 (prov9_3fd94e8927): ((y / x) * (x / y)) \ (y / x) = e / (x / y)
Proof. We have ((y / x) * (((y / x) \ y) / y)) \ (y / x) = e / (((y / x) \ y) / y) by Theorem 352. Then ((y / x) * (x / y)) \ (y / x) = e / (((y / x) \ y) / y) by Proposition 25. Hence we are done by Proposition 25.
Theorem 354 (prov9_a8578e35b7): ((x / y) * (y / x)) * (e / (y / x)) = x / y
Proof. We have ((x / y) * (y / x)) * (((x / y) * (y / x)) \ (x / y)) = x / y by Axiom 4. Hence we are done by Theorem 353.
Theorem 355 (prov9_4ab3ac8daa): K(e / (x / y),(y / x) * (x / y)) = e
Proof. We have T(T(y / x,(y / x) * (x / y)),x) = T(x \ y,(y / x) * (x / y)) by Proposition 51. Then
(L2) T(y / x,x) = T(x \ y,(y / x) * (x / y))
by Theorem 337. We have T(y / x,x) = x \ y by Proposition 47. Then
(L4) T(x \ y,(y / x) * (x / y)) = x \ y
by (L2). We have K(((y / x) * (x / y)) \ (y / x),(y / x) * (x / y)) = (x * T(x \ y,(y / x) * (x / y))) / y by Theorem 288. Then K(((y / x) * (x / y)) \ (y / x),(y / x) * (x / y)) = (x * (x \ y)) / y by (L4). Then
(L7) K(((y / x) * (x / y)) \ (y / x),(y / x) * (x / y)) = y / y
by Axiom 4. We have y / y = e by Proposition 29. Then K(((y / x) * (x / y)) \ (y / x),(y / x) * (x / y)) = e by (L7). Hence we are done by Theorem 353.
Theorem 356 (prov9_023826e049): K(e / y,(x / (y * x)) * y) = e
Proof. We have K(e / ((y * x) / x),(x / (y * x)) * ((y * x) / x)) = e by Theorem 355. Then K(e / y,(x / (y * x)) * ((y * x) / x)) = e by Axiom 5. Hence we are done by Axiom 5.
Theorem 357 (prov9_cd0786d361): e / x = L(x \ e,x,y / (x * y))
Proof. We have K(e / x,(y / (x * y)) * x) = e by Theorem 356. Then
(L2) T(e / x,(y / (x * y)) * x) = e / x
by Proposition 23. We have
(L3) T((y / (x * y)) / ((y / (x * y)) * x),(y / (x * y)) * x) = ((y / (x * y)) * x) \ (y / (x * y))
by Proposition 47.
L(x \ e,x,y / (x * y))
=((y / (x * y)) * x) \ (y / (x * y)) by Proposition 78
=T(e / x,(y / (x * y)) * x) by (L3), Theorem 189

Then L(x \ e,x,y / (x * y)) = T(e / x,(y / (x * y)) * x). Hence we are done by (L2).
Theorem 358 (prov9_6963e86b30): R(x \ e,x,y) * K(x,x \ e) = y / (x * y)
Proof. We have ((y / (x * y)) * x) * (((y / (x * y)) * x) \ (y / (x * y))) = y / (x * y) by Axiom 4. Then ((y / (x * y)) * x) * L(x \ e,x,y / (x * y)) = y / (x * y) by Proposition 78. Then
(L3) ((y / (x * y)) * x) * (e / x) = y / (x * y)
by Theorem 357. We have
(L4) R(y / (x * y),x,e / x) \ (((y / (x * y)) * x) * (e / x)) = x * (e / x)
by Theorem 6.
K(x,x \ e)
=x * (e / x) by Theorem 250
=R(x \ e,x,y) \ (((y / (x * y)) * x) * (e / x)) by (L4), Theorem 221

Then K(x,x \ e) = R(x \ e,x,y) \ (((y / (x * y)) * x) * (e / x)). Then
(L8) R(x \ e,x,y) \ (y / (x * y)) = K(x,x \ e)
by (L3). We have R(x \ e,x,y) * (R(x \ e,x,y) \ (y / (x * y))) = y / (x * y) by Axiom 4. Hence we are done by (L8).
Theorem 359 (prov9_1ed608f50d): T(e / (e / y),R(e / y,y,x)) = y
Proof. We have L(y \ e,y,x / (y * x)) = e / y by Theorem 357. Then
(L2) L(y \ e,y,R(e / y,y,x)) = e / y
by Proposition 79. We have R(e / y,y,x) / L(y \ e,y,R(e / y,y,x)) = R(e / y,y,x) * y by Theorem 33. Then
(L4) R(e / y,y,x) / (e / y) = R(e / y,y,x) * y
by (L2). We have (e / (e / y)) * R(e / y,y,x) = R(e / y,y,x) / (e / y) by Theorem 192. Then R(e / y,y,x) * y = (e / (e / y)) * R(e / y,y,x) by (L4). Hence we are done by Theorem 11.
Theorem 360 (prov9_3b25f1da4d): T(e / (e / y),x / (y * x)) = y
Proof. We have T(e / (e / y),R(e / y,y,x)) = y by Theorem 359. Hence we are done by Proposition 79.
Theorem 361 (prov9_143d61b3db): T(x,x \ e) = T(x,R(e / x,x,y))
Proof. We have T(T(e / (e / x),R(e / x,x,y)),x \ e) = T(x,R(e / x,x,y)) by Theorem 243. Hence we are done by Theorem 359.
Theorem 362 (prov9_310565e2f4): T(x,x \ e) = T(x,y / (x * y))
Proof. We have T(T(e / (e / x),y / (x * y)),x \ e) = T(x,y / (x * y)) by Theorem 243. Hence we are done by Theorem 360.
Theorem 363 (prov9_147ece8c71): T(y,(y \ x) / x) = T(y,y \ e)
Proof. We have T(y,(y \ x) / (y * (y \ x))) = T(y,y \ e) by Theorem 362. Hence we are done by Axiom 4.
Theorem 364 (prov9_0e797df8ad): T(x,(y * x) \ y) / x = K(x,(y * x) \ y)
Proof. We have T(x,(y * x) \ y) = K(((y * x) \ y) \ (((y * x) \ y) / (e / x)),(y * x) \ y) * x by Theorem 201. Then T(x,(y * x) \ y) = K(((y * x) \ y) \ (((y * x) \ y) * x),(y * x) \ y) * x by Theorem 341. Then K(x,(y * x) \ y) * x = T(x,(y * x) \ y) by Axiom 3. Hence we are done by Proposition 1.
Theorem 365 (prov9_f3db3d7109): K(x,y) = ((e / y) * K(x,y)) / (e / y)
Proof. We have ((e / y) * (e / y)) * ((x * (((e / y) * (e / y)) \ (e / y))) / x) = (e / y) * ((e / y) * ((x * ((e / y) \ e)) / x)) by Theorem 170. Then ((e / y) * ((x * ((e / y) \ e)) / x)) * (e / y) = (e / y) * ((e / y) * ((x * ((e / y) \ e)) / x)) by Theorem 343. Then (((x * y) / x) / y) * (e / y) = (e / y) * ((e / y) * ((x * ((e / y) \ e)) / x)) by Theorem 290. Then (e / y) * (((x * y) / x) / y) = (((x * y) / x) / y) * (e / y) by Theorem 290. Then
(L5) ((e / y) * (((x * y) / x) / y)) / (e / y) = ((x * y) / x) / y
by Proposition 1. We have R(((e / y) * (((x * y) / x) / y)) / (e / y),y,x) = ((e / y) * R(((x * y) / x) / y,y,x)) / (e / y) by Theorem 123. Then R(((e / y) * (((x * y) / x) / y)) / (e / y),y,x) = ((e / y) * ((x * y) / (y * x))) / (e / y) by Proposition 74. Then
(L8) R(((x * y) / x) / y,y,x) = ((e / y) * ((x * y) / (y * x))) / (e / y)
by (L5). We have R(((x * y) / x) / y,y,x) = (x * y) / (y * x) by Proposition 74. Then
(L10) ((e / y) * ((x * y) / (y * x))) / (e / y) = (x * y) / (y * x)
by (L8). We have T(((e / y) * ((x * y) / (y * x))) / (e / y),y * x) = ((e / y) * ((y * x) \ (x * y))) / (e / y) by Theorem 94. Then
(L12) T((x * y) / (y * x),y * x) = ((e / y) * ((y * x) \ (x * y))) / (e / y)
by (L10). We have T((x * y) / (y * x),y * x) = (y * x) \ (x * y) by Proposition 47. Then
(L14) ((e / y) * ((y * x) \ (x * y))) / (e / y) = (y * x) \ (x * y)
by (L12). We have K(x,y) = (y * x) \ (x * y) by Definition 2. Then K(x,y) = ((e / y) * ((y * x) \ (x * y))) / (e / y) by (L14). Hence we are done by Definition 2.
Theorem 366 (prov9_1164529ab5): T(K(x,y),e / y) = K(x,y)
Proof. We have T(((e / y) * K(x,y)) / (e / y),e / y) = K(x,y) by Theorem 7. Hence we are done by Theorem 365.
Theorem 367 (prov9_d1d3a85f48): T(e / y,K(x,y)) = e / y
Proof. We have T(K(x,y),e / y) = K(x,y) by Theorem 366. Hence we are done by Proposition 21.
Theorem 368 (prov9_8cf6b15ed2): K(y,K(x,y \ e)) = e
Proof. We have T(e / (y \ e),K(x,y \ e)) = e / (y \ e) by Theorem 367. Then K(e / (y \ e),K(x,y \ e)) = e by Proposition 22. Hence we are done by Proposition 24.
Theorem 369 (prov9_934cad4608): T(y,K(x,y \ e)) = y
Proof. We have K(y,K(x,y \ e)) = e by Theorem 368. Hence we are done by Proposition 23.
Theorem 370 (prov9_0a77a0dfd4): T(z,x) = T(T(z,x),K(y,z \ e))
Proof. We have T(T(z,K(y,z \ e)),x) = T(T(z,x),K(y,z \ e)) by Axiom 7. Hence we are done by Theorem 369.
Theorem 371 (prov9_268d7fa75e): T(K(y,z \ e),T(z,x)) = K(y,z \ e)
Proof. We have T(T(z,x),K(y,z \ e)) = T(z,x) by Theorem 370. Hence we are done by Proposition 21.
Theorem 372 (prov9_ddc6a6fbbd): T(T(y,z),x \ T(x,y \ e)) = T(y,z)
Proof. We have T(T(y,z),K((y \ e) \ ((y \ e) / (x \ e)),y \ e)) = T(y,z) by Theorem 370. Hence we are done by Theorem 198.
Theorem 373 (prov9_d0dcd54a69): (z / (y * z)) * K(x,y) = K(x,y) * (z / (y * z))
Proof. We have ((e / y) * K(x,y)) / K(x,y) = e / y by Axiom 5. Then ((((e / y) * K(x,y)) / (e / y)) * (e / y)) / K(x,y) = e / y by Axiom 6. Then
(L3) (K(x,y) * (e / y)) / K(x,y) = e / y
by Theorem 365. We have R((K(x,y) * (e / y)) / K(x,y),y,z) \ (K(x,y) * R(e / y,y,z)) = K(x,y) by Theorem 124. Then
(L5) R(e / y,y,z) \ (K(x,y) * R(e / y,y,z)) = K(x,y)
by (L3). We have T(K(x,y),R(e / y,y,z)) = R(e / y,y,z) \ (K(x,y) * R(e / y,y,z)) by Definition 3. Then T(K(x,y),R(e / y,y,z)) = K(x,y) by (L5). Then
(L8) T(K(x,y),z / (y * z)) = K(x,y)
by Proposition 79. We have (z / (y * z)) * T(K(x,y),z / (y * z)) = K(x,y) * (z / (y * z)) by Proposition 46. Hence we are done by (L8).
Theorem 374 (prov9_41570cf21e): (x \ e) / (x \ T(x,y)) = T(x,y) \ e
Proof. We have T(x \ e,(x \ e) * T((x \ e) \ e,y)) = x \ e by Theorem 203. Then
(L2) T(x \ e,T(x,y) * (x \ e)) = x \ e
by Theorem 48. We have R(T(x \ e,T(x,y) * (x \ e)),T(x,y),x \ e) = T(x \ e,T(x,y)) by Theorem 113. Then
(L4) R(x \ e,T(x,y),x \ e) = T(x \ e,T(x,y))
by (L2). We have ((x \ e) / (T(x,y) * (x \ e))) * T(x,y) = T(x,y) * R(T(x,y) \ e,T(x,y),x \ e) by Theorem 90. Then
(L6) ((x \ e) / (x \ T(x,y))) * T(x,y) = T(x,y) * R(T(x,y) \ e,T(x,y),x \ e)
by Theorem 146. We have T(x,y) * R(T(x,y) \ e,T(x,y),x \ e) = T(x * R(x \ e,T(x,y),x \ e),y) by Theorem 349. Then ((x \ e) / (x \ T(x,y))) * T(x,y) = T(x * R(x \ e,T(x,y),x \ e),y) by (L6). Then
(L9) T(x * T(x \ e,T(x,y)),y) = ((x \ e) / (x \ T(x,y))) * T(x,y)
by (L4). We have T(x * T(x \ e,T(x,y)),y) = K(T(x,y) \ (T(x,y) / T(x,y)),T(x,y)) by Theorem 296. Then
(L11) T(x * T(x \ e,T(x,y)),y) = K(T(x,y) \ e,T(x,y))
by Proposition 29. We have K(T(x,y) \ e,T(x,y)) = (T(x,y) \ e) * T(x,y) by Proposition 76. Then T(x * T(x \ e,T(x,y)),y) = (T(x,y) \ e) * T(x,y) by (L11). Hence we are done by (L9) and Proposition 8.
Theorem 375 (prov9_c148e12e54): (x \ T(x,y)) \ (x \ e) = T(x,y) \ e
Proof. We have T((x \ e) / (x \ T(x,y)),x \ T(x,y)) = (x \ T(x,y)) \ (x \ e) by Proposition 47. Then
(L2) T(T(x,y) \ e,x \ T(x,y)) = (x \ T(x,y)) \ (x \ e)
by Theorem 374. We have T(T(x,y) \ e,x \ T(x,y)) = T(x,y) \ e by Theorem 308. Hence we are done by (L2).
Theorem 376 (prov9_9974252beb): (x * T(x \ e,y)) \ x = T(e / x,y) \ e
Proof. We have ((e / x) \ T(e / x,y)) \ ((e / x) \ e) = T(e / x,y) \ e by Theorem 375. Then (x * T(x \ e,y)) \ ((e / x) \ e) = T(e / x,y) \ e by Theorem 102. Hence we are done by Proposition 25.
Theorem 377 (prov9_79307cf28f): x / (T(e / x,y) \ e) = x * T(x \ e,y)
Proof. We have x / ((x * T(x \ e,y)) \ x) = x * T(x \ e,y) by Proposition 24. Hence we are done by Theorem 376.
Theorem 378 (prov9_eb0a9e40d7): L(x \ e,x,y) = (x \ e) * (x * L(x \ e,x,y))
Proof. We have
(L1) (e / x) * (K(e / x,(e / x) \ e) * ((e / x) \ L(e / x,x,y))) = K(e / x,(e / x) \ e) * L(e / x,x,y)
by Theorem 327. We have (e / x) * (L(e / x,x,y) * (e / (e / x))) = K(e / x,(e / x) \ e) * L(e / x,x,y) by Theorem 281. Then
(L3) K(e / x,(e / x) \ e) * ((e / x) \ L(e / x,x,y)) = L(e / x,x,y) * (e / (e / x))
by (L1) and Proposition 7. We have L(((e / x) \ e) \ e,x,y) / (e / x) = L(e / x,x,y) * (e / (e / x)) by Theorem 213. Then L(x \ e,x,y) / (e / x) = L(e / x,x,y) * (e / (e / x)) by Proposition 25. Then K(e / x,(e / x) \ e) * ((e / x) \ L(e / x,x,y)) = L(x \ e,x,y) / (e / x) by (L3). Then K(x \ e,x) * ((e / x) \ L(e / x,x,y)) = L(x \ e,x,y) / (e / x) by Theorem 272. Then
(L8) K(x \ e,x) * (x * L(x \ e,x,y)) = L(x \ e,x,y) / (e / x)
by Theorem 159. We have ((x \ e) * (x * L(x \ e,x,y))) * x = ((x \ e) * x) * (x * L(x \ e,x,y)) by Theorem 339. Then (x * ((y * x) \ y)) * K(x \ e,x) = ((x \ e) * x) * (x * L(x \ e,x,y)) by Theorem 348. Then K(x \ e,x) * (x * L(x \ e,x,y)) = (x * ((y * x) \ y)) * K(x \ e,x) by Proposition 76. Then
(L12) L(x \ e,x,y) / (e / x) = (x * ((y * x) \ y)) * K(x \ e,x)
by (L8). We have (((y * x) \ y) * x) * ((((y * x) \ y) * x) \ ((y * x) \ y)) = (y * x) \ y by Axiom 4. Then
(L14) (((y * x) \ y) * x) * (e / x) = (y * x) \ y
by Theorem 340.
L(x \ e,x,y)
=(y * x) \ y by Proposition 78
=(L(x \ e,x,y) * x) * (e / x) by (L14), Proposition 78

Then L(x \ e,x,y) = (L(x \ e,x,y) * x) * (e / x). Then L(x \ e,x,y) / (e / x) = L(x \ e,x,y) * x by Proposition 1. Then
(L19) (x * ((y * x) \ y)) * K(x \ e,x) = L(x \ e,x,y) * x
by (L12). We have ((x \ e) * (x * L(x \ e,x,y))) * x = (x * ((y * x) \ y)) * K(x \ e,x) by Theorem 348. Hence we are done by (L19) and Proposition 8.
Theorem 379 (prov9_798c41821d): (y * x) \ y = (x \ e) * (x * ((y * x) \ y))
Proof. We have L(x \ e,x,y) = (y * x) \ y by Proposition 78. Then (x \ e) * (x * L(x \ e,x,y)) = (y * x) \ y by Theorem 378. Hence we are done by Proposition 78.
Theorem 380 (prov9_ffcad76aea): (x \ e) \ ((y * x) \ y) = x * ((y * x) \ y)
Proof. We have (x \ e) * (x * ((y * x) \ y)) = (y * x) \ y by Theorem 379. Hence we are done by Proposition 2.
Theorem 381 (prov9_9633b04252): e = K(x \ e,x * ((y * x) \ y))
Proof. We have ((y * x) \ y) \ (((y * x) \ y) * e) = e by Axiom 3. Then ((y * x) \ y) \ ((x \ e) * ((x \ e) \ (((y * x) \ y) * e))) = e by Axiom 4. Then
(L3) ((y * x) \ y) \ ((x \ e) * ((x \ e) \ ((y * x) \ y))) = e
by Axiom 2. We have ((y * x) \ y) * (((y * x) \ y) \ ((x \ e) * ((x \ e) \ ((y * x) \ y)))) = (x \ e) * ((x \ e) \ ((y * x) \ y)) by Axiom 4. Then ((y * x) \ y) * e = (x \ e) * ((x \ e) \ ((y * x) \ y)) by (L3). Then
(L6) (x \ e) * (x * ((y * x) \ y)) = ((y * x) \ y) * e
by Theorem 380. We have L(x \ e,x,y) / (x \ e) = x * ((y * x) \ y) by Theorem 162. Then
(L8) ((y * x) \ y) / (x \ e) = x * ((y * x) \ y)
by Proposition 78. We have (((y * x) \ y) / (x \ e)) * (x \ e) = (y * x) \ y by Axiom 6. Then
(L10) (x * ((y * x) \ y)) * (x \ e) = (y * x) \ y
by (L8). We have ((x * ((y * x) \ y)) * (x \ e)) * K(x \ e,x * ((y * x) \ y)) = (x \ e) * (x * ((y * x) \ y)) by Proposition 82. Then ((y * x) \ y) * K(x \ e,x * ((y * x) \ y)) = (x \ e) * (x * ((y * x) \ y)) by (L10). Hence we are done by (L6) and Proposition 7.
Theorem 382 (prov9_6c05351db3): e = (y \ e) * T(y,x \ (x / y))
Proof. We have (y \ e) * T(e / (y \ e),L(y \ e,y,x / y)) = K(y \ e,L(y \ e,y,x / y) / (y \ e)) by Theorem 282. Then (y \ e) * T(y,L(y \ e,y,x / y)) = K(y \ e,L(y \ e,y,x / y) / (y \ e)) by Proposition 24. Then K(y \ e,y * L(y \ e,y,x / y)) = (y \ e) * T(y,L(y \ e,y,x / y)) by Theorem 161. Then K(y \ e,y * L(y \ e,y,x / y)) = (y \ e) * T(y,x \ (x / y)) by Theorem 29. Then
(L7) K(y \ e,y * (x \ (x / y))) = (y \ e) * T(y,x \ (x / y))
by Theorem 29. We have K(y \ e,y * (((x / y) * y) \ (x / y))) = e by Theorem 381. Then K(y \ e,y * (x \ (x / y))) = e by Axiom 6. Hence we are done by (L7).
Theorem 383 (prov9_df1a2afe75): T(y,y \ e) = T(y,x \ (x / y))
Proof. We have (y \ e) * T(y,x \ (x / y)) = e by Theorem 382. Hence we are done by Theorem 20.
Theorem 384 (prov9_655c25c0c2): (y \ e) \ e = T(y,x \ (x / y))
Proof. We have (y \ e) * T(y,x \ (x / y)) = e by Theorem 382. Hence we are done by Proposition 2.
Theorem 385 (prov9_b9f06fe295): T(y,(x * y) \ x) = T(y,y \ e)
Proof. We have T(y,(x * y) \ ((x * y) / y)) = T(y,y \ e) by Theorem 383. Hence we are done by Axiom 5.
Theorem 386 (prov9_97698ea455): T(y,(x * y) \ x) = (y \ e) \ e
Proof. We have T(y,(x * y) \ ((x * y) / y)) = (y \ e) \ e by Theorem 384. Hence we are done by Axiom 5.
Theorem 387 (prov9_baeb28dc94): T(x \ y,y \ x) = ((x \ y) \ e) \ e
Proof. We have T(x \ y,y \ (y / (x \ y))) = ((x \ y) \ e) \ e by Theorem 384. Hence we are done by Proposition 24.
Theorem 388 (prov9_90ab4078b6): T(y,L(y \ e,y,x)) = T(y,y \ e)
Proof. We have T(y,(x * y) \ x) = T(y,y \ e) by Theorem 385. Hence we are done by Proposition 78.
Theorem 389 (prov9_2b9cf8d81c): T(y,L(y \ e,y,x)) = (y \ e) \ e
Proof. We have T(y,(x * y) \ x) = (y \ e) \ e by Theorem 386. Hence we are done by Proposition 78.
Theorem 390 (prov9_0905c176b5): x = T(e / (e / x),(y * x) \ y)
Proof. We have K(x \ e,x * ((y * x) \ y)) = ((x * ((y * x) \ y)) * (x \ e)) \ ((x \ e) * (x * ((y * x) \ y))) by Definition 2. Then
(L4) K(x \ e,x * ((y * x) \ y)) = L(x \ e,x,y) \ ((x \ e) * (x * ((y * x) \ y)))
by Theorem 163. We have L(x \ e,x,y) \ ((x \ e) * (x * ((y * x) \ y))) = (x \ e) * (L(x \ e,x,y) \ (x * ((y * x) \ y))) by Theorem 277. Then
(L6) K(x \ e,x * ((y * x) \ y)) = (x \ e) * (L(x \ e,x,y) \ (x * ((y * x) \ y)))
by (L4). We have L(x \ e,x,y) \ (((y * x) \ y) * T(x,(y * x) \ y)) = T(x,(y * x) \ y) by Theorem 99. Then L(x \ e,x,y) \ (x * ((y * x) \ y)) = T(x,(y * x) \ y) by Proposition 46. Then
(L7) (x \ e) * T(x,(y * x) \ y) = K(x \ e,x * ((y * x) \ y))
by (L6). We have T(e / (e / x),(y * x) \ y) / ((x \ e) * T(x,(y * x) \ y)) = x by Theorem 273. Then T(e / (e / x),(y * x) \ y) / K(x \ e,x * ((y * x) \ y)) = x by (L7). Then
(L10) T(e / (e / x),(y * x) \ y) / e = x
by Theorem 381. We have T(e / (e / x),(y * x) \ y) / e = T(e / (e / x),(y * x) \ y) by Proposition 27. Hence we are done by (L10).
Theorem 391 (prov9_0b2a3d5aa4): T(x \ (x / y),y) = ((x \ (x / y)) \ e) \ e
Proof. We have T(x \ (x / y),(x / y) \ x) = ((x \ (x / y)) \ e) \ e by Theorem 387. Hence we are done by Proposition 25.
Theorem 392 (prov9_fdd494ec06): K(y,(x * y) \ x) = K(y,y \ e)
Proof. We have T(y,(x * y) \ x) / y = K(y,(x * y) \ x) by Theorem 364. Then
(L2) ((y \ e) \ e) / y = K(y,(x * y) \ x)
by Theorem 386. We have ((y \ e) \ e) / y = K(y,y \ e) by Theorem 247. Hence we are done by (L2).
Theorem 393 (prov9_792f3dc31f): e / (e / y) = L(y,(x * y) \ x,y)
Proof. We have (((x * y) \ x) * T(e / (e / y),(x * y) \ x)) / ((x * y) \ x) = e / (e / y) by Proposition 48. Then
(L2) (((x * y) \ x) * y) / ((x * y) \ x) = e / (e / y)
by Theorem 390. We have (((x * y) \ x) * T(y,y * ((x * y) \ x))) / ((x * y) \ x) = L(y,(x * y) \ x,y) by Theorem 80. Then (((x * y) \ x) * y) / ((x * y) \ x) = L(y,(x * y) \ x,y) by Theorem 305. Hence we are done by (L2).
Theorem 394 (prov9_2045c7ca6b): L(x \ y,y \ x,x \ y) = e / (e / (x \ y))
Proof. We have L(x \ y,(x * (x \ y)) \ x,x \ y) = e / (e / (x \ y)) by Theorem 393. Hence we are done by Axiom 4.
Theorem 395 (prov9_6cab550d98): e / (e / ((y * x) \ y)) = L(e / x,x,y)
Proof. We have L((y * x) \ y,y \ (y * x),(y * x) \ y) = e / (e / ((y * x) \ y)) by Theorem 394. Then
(L2) L((y * x) \ y,x,(y * x) \ y) = e / (e / ((y * x) \ y))
by Axiom 3. We have L(L(x \ e,x,(y * x) \ y),x,y) = L((y * x) \ y,x,(y * x) \ y) by Theorem 32. Then L(e / x,x,y) = L((y * x) \ y,x,(y * x) \ y) by Theorem 342. Hence we are done by (L2).
Theorem 396 (prov9_ed6a51a134): L(e / y,y,x / y) = e / (e / (x \ (x / y)))
Proof. We have L(e / y,y,x / y) = e / (e / (((x / y) * y) \ (x / y))) by Theorem 395. Hence we are done by Axiom 6.
Theorem 397 (prov9_b117018e24): K(T(x,R(x \ e,y,z)),R(x \ e,y,z)) = x \ T(x,R(x \ e,y,z))
Proof. We have K(R(x \ e,y,z) \ (R(x \ e,y,z) / (x \ e)),R(x \ e,y,z)) = x \ T(x,R(x \ e,y,z)) by Theorem 198. Then K(R(x \ e,y,z) \ (x * R(x \ e,y,z)),R(x \ e,y,z)) = x \ T(x,R(x \ e,y,z)) by Theorem 193. Hence we are done by Definition 3.
Theorem 398 (prov9_9bc031ebc3): a(T(y,x),e / y,y) = y * L(y \ e,T(y,x),e / y)
Proof. We have ((T(y,x) * (e / y)) / T(y,x)) * ((e / y) \ e) = (e / y) \ ((T(y,x) * (e / y)) / T(y,x)) by Theorem 195. Then
(L6) ((T(y,x) * (e / y)) / T(y,x)) * y = (e / y) \ ((T(y,x) * (e / y)) / T(y,x))
by Proposition 25. We have ((T(y,x) * (e / y)) / T(y,x)) * y = y * ((T(y,x) * (y \ e)) / T(y,x)) by Theorem 125. Then
(L8) (e / y) \ ((T(y,x) * (e / y)) / T(y,x)) = y * ((T(y,x) * (y \ e)) / T(y,x))
by (L6). We have (y * ((T(y,x) * (y \ e)) / T(y,x))) * T(y,x) = T(T(y,x),R(e / y,y,T(y,x))) by Theorem 289. Then T(T(y,x),R(e / y,y,T(y,x))) / T(y,x) = y * ((T(y,x) * (y \ e)) / T(y,x)) by Proposition 1. Then (e / y) \ ((T(y,x) * (e / y)) / T(y,x)) = T(T(y,x),R(e / y,y,T(y,x))) / T(y,x) by (L8). Then
(L10) (e / y) \ ((T(T(y,y \ e),x) / y) / T(y,x)) = T(T(y,x),R(e / y,y,T(y,x))) / T(y,x)
by Theorem 216. We have T(T(y,R(e / y,y,T(y,x))),x) = T(T(y,x),R(e / y,y,T(y,x))) by Axiom 7. Then T(T(y,y \ e),x) = T(T(y,x),R(e / y,y,T(y,x))) by Theorem 361. Then
(L11) (e / y) \ ((T(T(y,y \ e),x) / y) / T(y,x)) = T(T(y,y \ e),x) / T(y,x)
by (L10). We have T(y,x) \ T(T(y,e / y),x) = a(T(y,x),e / y,y) by Theorem 132. Then T(y,x) \ T(T(y,x),e / y) = a(T(y,x),e / y,y) by Axiom 7. Then
(L14) T(y,x) \ T(T(y,x),y \ e) = a(T(y,x),e / y,y)
by Theorem 222. We have T(T(y,x),T(y,x) \ T(T(y,x),y \ e)) = T(y,x) by Theorem 372. Then
(L16) T(T(y,x),a(T(y,x),e / y,y)) = T(y,x)
by (L14). We have T(T(y,y \ e),x) / T(T(y,x),T(y,x) \ T(T(y,y \ e),x)) = T(y,x) \ T(T(y,y \ e),x) by Theorem 15. Then
(L18) T(T(y,y \ e),x) / T(T(y,x),a(T(y,x),e / y,y)) = T(y,x) \ T(T(y,y \ e),x)
by Theorem 217. We have T(y,x) \ T(T(y,y \ e),x) = a(T(y,x),e / y,y) by Theorem 217. Then T(T(y,y \ e),x) / T(T(y,x),a(T(y,x),e / y,y)) = a(T(y,x),e / y,y) by (L18). Then T(T(y,y \ e),x) / T(y,x) = a(T(y,x),e / y,y) by (L16). Then (e / y) \ ((T(T(y,y \ e),x) / y) / T(y,x)) = a(T(y,x),e / y,y) by (L11). Then (e / y) \ ((T(y,x) * (e / y)) / T(y,x)) = a(T(y,x),e / y,y) by Theorem 216. Then
(L24) (e / y) \ L(e / y,T(y,x),e / y) = a(T(y,x),e / y,y)
by Theorem 346. We have (e / y) \ L(e / y,T(y,x),e / y) = y * L(y \ e,T(y,x),e / y) by Theorem 159. Hence we are done by (L24).
Theorem 399 (prov9_a01678193b): y * ((y \ T(y,x)) / T(y,x)) = a(T(y,x),e / y,y)
Proof. We have T((T(y,x) * (e / y)) / T(y,x),y) = (T(y,x) * (y \ e)) / T(y,x) by Theorem 94. Then
(L2) T(L(e / y,T(y,x),e / y),y) = (T(y,x) * (y \ e)) / T(y,x)
by Theorem 346. We have T(L(e / y,T(y,x),e / y),y) = L(y \ e,T(y,x),e / y) by Proposition 72. Then (T(y,x) * (y \ e)) / T(y,x) = L(y \ e,T(y,x),e / y) by (L2). Then
(L5) (y \ T(y,x)) / T(y,x) = L(y \ e,T(y,x),e / y)
by Theorem 146. We have y * L(y \ e,T(y,x),e / y) = a(T(y,x),e / y,y) by Theorem 398. Hence we are done by (L5).
Theorem 400 (prov9_48a98d904d): (y \ e) * T(y,x) = K(y \ e,y) * (T(y,x) / y)
Proof. We have ((y \ e) / (e / y)) * ((e / y) * T((e / y) \ e,x)) = (y \ e) * T((y \ e) \ ((y \ e) / (e / y)),x) by Theorem 139. Then K(y \ e,y) * ((e / y) * T((e / y) \ e,x)) = (y \ e) * T((y \ e) \ ((y \ e) / (e / y)),x) by Theorem 254. Then K(y \ e,y) * (T(y,x) / y) = (y \ e) * T((y \ e) \ ((y \ e) / (e / y)),x) by Theorem 149. Then (y \ e) * T((y \ e) \ K(y \ e,y),x) = K(y \ e,y) * (T(y,x) / y) by Theorem 254. Hence we are done by Theorem 21.
Theorem 401 (prov9_58dac3aeb3): T(e / (e / y),x) \ T(y,x) = K(T(y,x),e / y)
Proof. We have K((e / y) \ ((e / y) / ((((e / y) * T(y,x)) / (e / y)) \ e)),e / y) = (((e / y) * T(y,x)) / (e / y)) \ T(y,x) by Theorem 200. Then K((e / y) \ ((e / y) / (T(e / (e / y),x) \ e)),e / y) = (((e / y) * T(y,x)) / (e / y)) \ T(y,x) by Theorem 51. Then K((e / y) \ ((e / y) * T((e / y) \ e,x)),e / y) = (((e / y) * T(y,x)) / (e / y)) \ T(y,x) by Theorem 377. Then
(L6) K((e / y) \ (T(y,x) / y),e / y) = (((e / y) * T(y,x)) / (e / y)) \ T(y,x)
by Theorem 149. We have (e / y) * T(y,x) = T(y,x) / y by Theorem 150. Then (e / y) \ (T(y,x) / y) = T(y,x) by Proposition 2. Then (((e / y) * T(y,x)) / (e / y)) \ T(y,x) = K(T(y,x),e / y) by (L6). Hence we are done by Theorem 51.
Theorem 402 (prov9_0f7760ebb8): K(x / y,y / x) = K(x / y,(x / y) \ e)
Proof. We have T(T(x / y,(y / ((x / y) * y)) * (x / y)) * (y / ((x / y) * y)),x / y) = T(x / y,y / ((x / y) * y)) * T(y / ((x / y) * y),x / y) by Theorem 181. Then T((x / y) * (y / ((x / y) * y)),x / y) = T(x / y,y / ((x / y) * y)) * T(y / ((x / y) * y),x / y) by Theorem 299. Then T(x / y,y / ((x / y) * y)) * R((x / y) \ e,x / y,y) = T((x / y) * (y / ((x / y) * y)),x / y) by Theorem 89. Then
(L4) T(x / y,y / ((x / y) * y)) * R((x / y) \ e,x / y,y) = (x / y) * (y / ((x / y) * y))
by Theorem 335. We have ((x / y) * R((x / y) \ e,x / y,y)) \ (T(x / y,R((x / y) \ e,x / y,y)) * R((x / y) \ e,x / y,y)) = K(T(x / y,R((x / y) \ e,x / y,y)),R((x / y) \ e,x / y,y)) by Theorem 12. Then ((x / y) * R((x / y) \ e,x / y,y)) \ (T(x / y,y / ((x / y) * y)) * R((x / y) \ e,x / y,y)) = K(T(x / y,R((x / y) \ e,x / y,y)),R((x / y) \ e,x / y,y)) by Theorem 347. Then
(L7) ((x / y) * R((x / y) \ e,x / y,y)) \ ((x / y) * (y / ((x / y) * y))) = K(T(x / y,R((x / y) \ e,x / y,y)),R((x / y) \ e,x / y,y))
by (L4). We have K(x / y,y / ((x / y) * y)) = ((y / ((x / y) * y)) * (x / y)) \ ((x / y) * (y / ((x / y) * y))) by Definition 2. Then K(x / y,y / ((x / y) * y)) = ((x / y) * R((x / y) \ e,x / y,y)) \ ((x / y) * (y / ((x / y) * y))) by Theorem 90. Then
(L10) K(T(x / y,R((x / y) \ e,x / y,y)),R((x / y) \ e,x / y,y)) = K(x / y,y / ((x / y) * y))
by (L7). We have K(T(x / y,R((x / y) \ e,x / y,y)),R((x / y) \ e,x / y,y)) = (x / y) \ T(x / y,R((x / y) \ e,x / y,y)) by Theorem 397. Then K(T(x / y,R((x / y) \ e,x / y,y)),R((x / y) \ e,x / y,y)) = (x / y) \ T(x / y,y / ((x / y) * y)) by Theorem 347. Then
(L13) K(x / y,y / ((x / y) * y)) = (x / y) \ T(x / y,y / ((x / y) * y))
by (L10). We have (x / y) \ T(x / y,(x / y) \ e) = K(x / y,(x / y) \ e) by Theorem 271. Then (x / y) \ T(x / y,y / ((x / y) * y)) = K(x / y,(x / y) \ e) by Theorem 362. Then K(x / y,y / ((x / y) * y)) = K(x / y,(x / y) \ e) by (L13). Hence we are done by Axiom 6.
Theorem 403 (prov9_b5772d2eb8): K(x / (y \ x),(y \ x) / x) = K(y,y \ e)
Proof. We have K(x / (y \ x),(x / (y \ x)) \ e) = K(x / (y \ x),(y \ x) / x) by Theorem 402. Then K(y,(x / (y \ x)) \ e) = K(x / (y \ x),(y \ x) / x) by Proposition 24. Hence we are done by Proposition 24.
Theorem 404 (prov9_c7784a37de): ((y * x) \ y) * K(x,x \ e) = L(e / x,x,y)
Proof. We have ((e / x) * T((e / x) \ e,(y * x) \ y)) * ((y * x) \ y) = (e / x) * ((((y * x) \ y) \ ((e / x) \ ((y * x) \ y))) * ((y * x) \ y)) by Theorem 236. Then ((e / x) * T((e / x) \ e,(y * x) \ y)) * ((y * x) \ y) = (e / x) * ((((y * x) \ y) \ (((y * x) \ y) * x)) * ((y * x) \ y)) by Theorem 323. Then (e / x) * ((((y * x) \ y) \ (((y * x) \ y) * x)) * ((y * x) \ y)) = (T(x,(y * x) \ y) / x) * ((y * x) \ y) by Theorem 149. Then
(L4) (e / x) * (x * ((y * x) \ y)) = (T(x,(y * x) \ y) / x) * ((y * x) \ y)
by Axiom 3. We have (e / x) * (x * ((y * x) \ y)) = L((y * x) \ y,x,e / x) by Proposition 67. Then (e / x) * (x * ((y * x) \ y)) = L(e / x,x,y) by Theorem 134. Then (T(x,(y * x) \ y) / x) * ((y * x) \ y) = L(e / x,x,y) by (L4). Then K(x,(y * x) \ y) * ((y * x) \ y) = L(e / x,x,y) by Theorem 364. Then
(L9) K(x,x \ e) * ((y * x) \ y) = L(e / x,x,y)
by Theorem 392. We have ((y * x) \ y) * T(K(x,x \ e),(y * x) \ y) = K(x,x \ e) * ((y * x) \ y) by Proposition 46. Then ((y * x) \ y) * T(K(x,x \ e),(y * x) \ y) = L(e / x,x,y) by (L9). Hence we are done by Theorem 226.
Theorem 405 (prov9_ebe83dc357): (y * x) * K(x,x \ e) = y * T(x,L(x \ e,x,y))
Proof. We have L(e / x,x,y) = e / (e / ((y * x) \ y)) by Theorem 395. Then
(L4) L(e / x,x,y) = e / (e / L(x \ e,x,y))
by Proposition 78. We have L(x \ e,x,y) \ (e / (e / L(x \ e,x,y))) = ((y * x) \ y) \ (e / (e / L(x \ e,x,y))) by Theorem 100. Then K(L(x \ e,x,y) \ e,L(x \ e,x,y)) = ((y * x) \ y) \ (e / (e / L(x \ e,x,y))) by Theorem 257. Then
(L7) K(L(x \ e,x,y) \ e,L(x \ e,x,y)) = ((y * x) \ y) \ L(e / x,x,y)
by (L4). We have ((y * x) \ y) * K(x,x \ e) = L(e / x,x,y) by Theorem 404. Then ((y * x) \ y) \ L(e / x,x,y) = K(x,x \ e) by Proposition 2. Then
(L8) K(L(x \ e,x,y) \ e,L(x \ e,x,y)) = K(x,x \ e)
by (L7). We have y * T(y \ (y * x),L(x \ e,x,y)) = (y * x) * K(L(x \ e,x,y) \ (L(x \ e,x,y) / ((y * x) \ y)),L(x \ e,x,y)) by Theorem 284. Then y * T(x,L(x \ e,x,y)) = (y * x) * K(L(x \ e,x,y) \ (L(x \ e,x,y) / ((y * x) \ y)),L(x \ e,x,y)) by Axiom 3. Then (y * x) * K(L(x \ e,x,y) \ e,L(x \ e,x,y)) = y * T(x,L(x \ e,x,y)) by Theorem 234. Hence we are done by (L8).
Theorem 406 (prov9_cd51c1340a): x * T(y,y \ e) = (x * y) * K(y,y \ e)
Proof. We have x * T(y,L(y \ e,y,x)) = (x * y) * K(y,y \ e) by Theorem 405. Hence we are done by Theorem 388.
Theorem 407 (prov9_8e4eabaca0): x * ((y \ e) \ e) = (x * y) * K(y,y \ e)
Proof. We have x * T(y,L(y \ e,y,x)) = (x * y) * K(y,y \ e) by Theorem 405. Hence we are done by Theorem 389.
Theorem 408 (prov9_680911f7d1): (x * y) \ (x * ((y \ e) \ e)) = K(y,y \ e)
Proof. We have (x * y) * K(y,y \ e) = x * ((y \ e) \ e) by Theorem 407. Hence we are done by Proposition 2.
Theorem 409 (prov9_a424761050): R(y,K(x,x \ e),x) = y
Proof. We have (y * K(x,x \ e)) / K(x,x \ e) = y by Axiom 5. Then
(L3) (((y * K(x,x \ e)) / T(x,x \ e)) * T(x,x \ e)) / K(x,x \ e) = y
by Axiom 6. We have ((((y * K(x,x \ e)) / T(x,x \ e)) * T(x,x \ e)) / K(x,x \ e)) * K(x,x \ e) = ((y * K(x,x \ e)) / T(x,x \ e)) * T(x,x \ e) by Axiom 6. Then ((((((y * K(x,x \ e)) / T(x,x \ e)) * T(x,x \ e)) / K(x,x \ e)) / x) * x) * K(x,x \ e) = ((y * K(x,x \ e)) / T(x,x \ e)) * T(x,x \ e) by Axiom 6. Then
(L6) ((y / x) * x) * K(x,x \ e) = ((y * K(x,x \ e)) / T(x,x \ e)) * T(x,x \ e)
by (L3). We have ((y / x) * x) * K(x,x \ e) = (y / x) * T(x,x \ e) by Theorem 406. Then ((y * K(x,x \ e)) / T(x,x \ e)) * T(x,x \ e) = (y / x) * T(x,x \ e) by (L6). Then
(L8) (y * K(x,x \ e)) / T(x,x \ e) = y / x
by Proposition 10. We have R(y,K(x,x \ e),x) = ((y * K(x,x \ e)) * x) / (K(x,x \ e) * x) by Definition 5. Then
(L10) R(y,K(x,x \ e),x) = ((y * K(x,x \ e)) * x) / T(x,x \ e)
by Theorem 246. We have ((y * K(x,x \ e)) * x) / T(x,x \ e) = ((y * K(x,x \ e)) / T(x,x \ e)) * x by Theorem 239. Then R(y,K(x,x \ e),x) = ((y * K(x,x \ e)) / T(x,x \ e)) * x by (L10). Then
(L13) (y / x) * x = R(y,K(x,x \ e),x)
by (L8). We have (y / x) * x = y by Axiom 6. Hence we are done by (L13).
Theorem 410 (prov9_4ff9017167): R(y,K(x \ e,x),e / x) = y
Proof. We have R(y,K(e / x,(e / x) \ e),e / x) = y by Theorem 409. Hence we are done by Theorem 272.
Theorem 411 (prov9_5bb774b397): x * (y \ e) = (x * K(y \ e,y)) * (e / y)
Proof. We have R(x,K(y \ e,y),e / y) * (K(y \ e,y) * (e / y)) = (x * K(y \ e,y)) * (e / y) by Proposition 54. Then R(x,K(y \ e,y),e / y) * (y \ e) = (x * K(y \ e,y)) * (e / y) by Theorem 255. Hence we are done by Theorem 410.
Theorem 412 (prov9_2a06468ced): (x * (e / y)) * K(y \ e,y) = x * (y \ e)
Proof. We have
(L1) ((x * (e / y)) * K(y \ e,y)) * (e / y) = (x * (e / y)) * (y \ e)
by Theorem 411. We have (x * (y \ e)) * (e / y) = (x * (e / y)) * (y \ e) by Theorem 241. Hence we are done by (L1) and Proposition 8.
Theorem 413 (prov9_6b92f514e0): R(y,e / x,K(x \ e,x)) = y
Proof. We have
(L1) (y * (e / x)) * K(x \ e,x) = y * (x \ e)
by Theorem 412. We have (e / x) * K(x \ e,x) = x \ e by Theorem 109. Then (y * (e / x)) * K(x \ e,x) = y * ((e / x) * K(x \ e,x)) by (L1). Hence we are done by Theorem 64.
Theorem 414 (prov9_599dcddd1b): R(y,e / (e / x),K(x,x \ e)) = y
Proof. We have R(y,e / (e / x),K((e / x) \ e,e / x)) = y by Theorem 413. Hence we are done by Theorem 253.
Theorem 415 (prov9_9968e0583a): x * y = (x * (e / (e / y))) * K(y,y \ e)
Proof. We have R(x,e / (e / y),K(y,y \ e)) * ((e / (e / y)) * K(y,y \ e)) = (x * (e / (e / y))) * K(y,y \ e) by Proposition 54. Then R(x,e / (e / y),K(y,y \ e)) * y = (x * (e / (e / y))) * K(y,y \ e) by Theorem 252. Hence we are done by Theorem 414.
Theorem 416 (prov9_ef9c7faace): x * K(y,y \ e) = (x / (e / (e / y))) * y
Proof. We have ((x / (e / (e / y))) * (e / (e / y))) * K(y,y \ e) = (x / (e / (e / y))) * y by Theorem 415. Hence we are done by Axiom 6.
Theorem 417 (prov9_342cc896be): (x / y) * K(y,y \ e) = x / (e / (e / y))
Proof. We have ((x / y) * y) / (e / (e / y)) = ((x / y) / (e / (e / y))) * y by Theorem 292. Then x / (e / (e / y)) = ((x / y) / (e / (e / y))) * y by Axiom 6. Hence we are done by Theorem 416.
Theorem 418 (prov9_4f52e8dc47): x \ (x / (e / (e / y))) = e / (e / (x \ (x / y)))
Proof. We have x \ ((x / y) * (y * (e / y))) = L(e / y,y,x / y) by Theorem 4. Then x \ ((x / y) * (y * (e / y))) = e / (e / (x \ (x / y))) by Theorem 396. Then x \ ((x / y) * K(y,y \ e)) = e / (e / (x \ (x / y))) by Theorem 250. Hence we are done by Theorem 417.
Theorem 419 (prov9_168471ab1e): (x \ (x / (e / y))) \ e = e / (x \ (x / (y \ e)))
Proof. We have (e / (e / (x \ (x / (y \ e))))) \ e = e / (x \ (x / (y \ e))) by Proposition 25. Then (x \ (x / (e / (e / (y \ e))))) \ e = e / (x \ (x / (y \ e))) by Theorem 418. Hence we are done by Proposition 24.
Theorem 420 (prov9_0ccf242444): ((x \ (x / y)) \ e) \ e = x \ (x / ((y \ e) \ e))
Proof. We have e / (x \ (x / ((y \ e) \ e))) = (x \ (x / (e / (y \ e)))) \ e by Theorem 419. Then
(L2) e / (x \ (x / ((y \ e) \ e))) = (x \ (x / y)) \ e
by Proposition 24. We have (e / (x \ (x / ((y \ e) \ e)))) \ e = x \ (x / ((y \ e) \ e)) by Proposition 25. Hence we are done by (L2).
Theorem 421 (prov9_e47746f6a1): T(x \ (x / y),y) = x \ (x / ((y \ e) \ e))
Proof. We have T(x \ (x / y),y) = ((x \ (x / y)) \ e) \ e by Theorem 391. Hence we are done by Theorem 420.
Theorem 422 (prov9_fc32a37692): x * T(x \ (x / y),y) = x / ((y \ e) \ e)
Proof. We have x * (x \ (x / ((y \ e) \ e))) = x / ((y \ e) \ e) by Axiom 4. Hence we are done by Theorem 421.
Theorem 423 (prov9_e68d7fe7f5): (x / y) * K(y \ e,y) = x / ((y \ e) \ e)
Proof. We have (x / y) * K(y \ e,y) = x * T(x \ (x / y),y) by Theorem 154. Hence we are done by Theorem 422.
Theorem 424 (prov9_2d250499aa): (x / ((y \ e) \ e)) * y = x * K(y \ e,y)
Proof. We have ((x / y) * K(y \ e,y)) * y = x * K(y \ e,y) by Theorem 315. Hence we are done by Theorem 423.
Theorem 425 (prov9_87eed62324): x * K(y \ e,y) = x / K(y,y \ e)
Proof. We have x / (((x / ((y \ e) \ e)) * y) \ ((x / ((y \ e) \ e)) * ((y \ e) \ e))) = (x / ((y \ e) \ e)) * y by Theorem 24. Then x / K(y,y \ e) = (x / ((y \ e) \ e)) * y by Theorem 408. Hence we are done by Theorem 424.
Theorem 426 (prov9_094fb736a2): (y * K(x \ e,x)) * K(x,x \ e) = y
Proof. We have
(L1) ((y * K(x \ e,x)) * K(x,x \ e)) / K(x,x \ e) = ((y * K(x \ e,x)) * K(x,x \ e)) * K(x \ e,x)
by Theorem 425. We have ((y * K(x \ e,x)) * K(x,x \ e)) / K(x,x \ e) = y * K(x \ e,x) by Axiom 5. Then ((y * K(x \ e,x)) * K(x,x \ e)) * K(x \ e,x) = y * K(x \ e,x) by (L1). Hence we are done by Proposition 10.
Theorem 427 (prov9_dfa9a02389): (y / (x * y)) * K(x \ e,x) = R(x \ e,x,y)
Proof. We have
(L1) ((y / (x * y)) * K(x \ e,x)) * K(x,x \ e) = y / (x * y)
by Theorem 426. We have R(x \ e,x,y) * K(x,x \ e) = y / (x * y) by Theorem 358. Hence we are done by (L1) and Proposition 8.
Theorem 428 (prov9_1a4a507e6c): K(y \ e,y) = K(x / (y * x),y)
Proof. We have ((y * x) / x) * x = y * x by Axiom 6. Then ((((y * x) / x) / (e / (x / (y * x)))) * (e / (x / (y * x)))) * x = y * x by Axiom 6. Then ((y / (e / (x / (y * x)))) * (e / (x / (y * x)))) * x = y * x by Axiom 5. Then
(L6) (y * x) / x = (y / (e / (x / (y * x)))) * (e / (x / (y * x)))
by Proposition 1. We have (((y * x) / x) * (x / (y * x))) * (e / (x / (y * x))) = (y * x) / x by Theorem 354. Then ((y * x) / x) * (x / (y * x)) = y / (e / (x / (y * x))) by (L6) and Proposition 8. Then
(L8) y * (x / (y * x)) = y / (e / (x / (y * x)))
by Axiom 5. We have K(y \ (y / (e / (x / (y * x)))),y) = T(x / (y * x),y) / (x / (y * x)) by Theorem 199. Then K(y \ (y / (e / (x / (y * x)))),y) = R(y \ e,y,x) / (x / (y * x)) by Theorem 89. Then K(y \ (y * (x / (y * x))),y) = R(y \ e,y,x) / (x / (y * x)) by (L8). Then
(L12) K(x / (y * x),y) = R(y \ e,y,x) / (x / (y * x))
by Axiom 3. We have (R(y \ e,y,x) / (x / (y * x))) * (x / (y * x)) = R(y \ e,y,x) by Axiom 6. Then K(x / (y * x),y) * (x / (y * x)) = R(y \ e,y,x) by (L12). Then
(L15) (x / (y * x)) * K(x / (y * x),y) = R(y \ e,y,x)
by Theorem 373. We have (x / (y * x)) * K(y \ e,y) = R(y \ e,y,x) by Theorem 427. Hence we are done by (L15) and Proposition 7.
Theorem 429 (prov9_f428d5996e): K(y / x,x / y) = K((x / y) \ e,x / y)
Proof. We have K(y / ((x / y) * y),x / y) = K((x / y) \ e,x / y) by Theorem 428. Hence we are done by Axiom 6.
Theorem 430 (prov9_18f692c668): K(y / x,x / y) * y = x * T(x \ y,x / y)
Proof. We have K((x / y) \ e,x / y) * y = x * T(x \ y,x / y) by Theorem 202. Hence we are done by Theorem 429.
Theorem 431 (prov9_c70558bd6a): T(x,x \ e) * (x \ y) = K(x,x \ e) * y
Proof. We have (x \ y) * T((x \ y) \ y,(x \ y) / y) = T(x,(x \ y) / y) * (x \ y) by Theorem 48. Then K(y / (x \ y),(x \ y) / y) * y = T(x,(x \ y) / y) * (x \ y) by Theorem 430. Then T(x,(x \ y) / y) * (x \ y) = K(x,x \ e) * y by Theorem 403. Hence we are done by Theorem 363.
Theorem 432 (prov9_b140e27690): L(y,x,K(x,x \ e)) = y
Proof. We have
(L1) T(x,x \ e) * (x \ (x * y)) = K(x,x \ e) * (x * y)
by Theorem 431. We have K(x,x \ e) * x = T(x,x \ e) by Theorem 246. Then (K(x,x \ e) * x) * (x \ (x * y)) = K(x,x \ e) * (x * y) by (L1). Then
(L4) L(y,x,K(x,x \ e)) = x \ (x * y)
by Theorem 54. We have x \ (x * y) = y by Axiom 3. Hence we are done by (L4).
Theorem 433 (prov9_98f3703707): L(y,e / x,K(x \ e,x)) = y
Proof. We have L(y,e / x,K(e / x,(e / x) \ e)) = y by Theorem 432. Hence we are done by Theorem 272.
Theorem 434 (prov9_d97a8f81f3): L(x \ y,x,x \ e) = (e / x) * y
Proof.
K(x \ e,x) * L(x \ L(y,e / x,K(x \ e,x)),x,x \ e)
=(x \ e) * L(y,e / x,K(x \ e,x)) by Theorem 156
=K(x \ e,x) * ((e / x) * y) by Theorem 259

Then K(x \ e,x) * L(x \ L(y,e / x,K(x \ e,x)),x,x \ e) = K(x \ e,x) * ((e / x) * y). Then L(x \ L(y,e / x,K(x \ e,x)),x,x \ e) = (e / x) * y by Proposition 9. Hence we are done by Theorem 433.
Theorem 435 (prov9_e94c880984): (e / x) \ y = (x \ e) \ (K(x \ e,x) * y)
Proof. We have L((e / x) \ y,e / x,K(x \ e,x)) = (x \ e) \ (K(x \ e,x) * y) by Theorem 260. Hence we are done by Theorem 433.
Theorem 436 (prov9_a589ff6035): (e / x) \ y = K(x \ e,x) * ((x \ e) \ y)
Proof. We have (x \ e) * (K(x \ e,x) * ((x \ e) \ y)) = K(x \ e,x) * y by Theorem 338. Then
(L2) (x \ e) \ (K(x \ e,x) * y) = K(x \ e,x) * ((x \ e) \ y)
by Proposition 2. We have L((e / x) \ y,e / x,K(x \ e,x)) = (x \ e) \ (K(x \ e,x) * y) by Theorem 260. Then L((e / x) \ y,e / x,K(x \ e,x)) = K(x \ e,x) * ((x \ e) \ y) by (L2). Hence we are done by Theorem 433.
Theorem 437 (prov9_de14c5ff70): ((y \ e) * x) / ((e / y) * x) = K(y \ e,y)
Proof. We have K(y \ e,y) = (y \ e) * y by Proposition 76. Then K(y \ e,y) = ((y \ e) * x) / L(y \ x,y,y \ e) by Theorem 61. Hence we are done by Theorem 434.
Theorem 438 (prov9_9fd3bc0575): K(x \ e,x) * y = (e / x) \ ((x \ e) * y)
Proof. We have K(x \ e,x) * ((x \ e) \ ((x \ e) * y)) = (e / x) \ ((x \ e) * y) by Theorem 436. Hence we are done by Axiom 3.
Theorem 439 (prov9_7b70775554): T(y \ ((e / x) \ y),x \ e) = y \ ((x \ e) \ y)
Proof. We have T(y \ (K(x \ e,(x \ e) \ e) * ((x \ e) \ y)),x \ e) = y \ ((x \ e) \ y) by Theorem 324. Then T(y \ (K(x \ e,x) * ((x \ e) \ y)),x \ e) = y \ ((x \ e) \ y) by Theorem 249. Hence we are done by Theorem 436.
Theorem 440 (prov9_651a864c38): K(x,x \ e) * ((e / x) \ y) = (x \ e) \ y
Proof. We have y * (y \ ((x \ e) \ y)) = (x \ e) \ y by Axiom 4. Then
(L2) y * T(y \ ((e / x) \ y),x \ e) = (x \ e) \ y
by Theorem 439. We have K(x,x \ e) * ((e / x) \ y) = y * T(y \ ((e / x) \ y),x \ e) by Theorem 331. Hence we are done by (L2).
Theorem 441 (prov9_679550e722): (e / x) \ ((x \ e) * (K(x,x \ e) * y)) = y
Proof. We have
(L1) K(x,x \ e) * ((e / x) \ ((x \ e) * (K(x,x \ e) * y))) = (x \ e) \ ((x \ e) * (K(x,x \ e) * y))
by Theorem 440. We have K(x,x \ e) * y = (x \ e) \ ((x \ e) * (K(x,x \ e) * y)) by Axiom 3. Hence we are done by (L1) and Proposition 7.
Theorem 442 (prov9_69e782deba): y = K(x,x \ e) * (K(x \ e,x) * y)
Proof. We have
(L1) K(x \ e,x) * y = (e / x) \ ((x \ e) * (K(x,x \ e) * (K(x \ e,x) * y)))
by Theorem 441. We have K(x \ e,x) * (K(x,x \ e) * (K(x \ e,x) * y)) = (e / x) \ ((x \ e) * (K(x,x \ e) * (K(x \ e,x) * y))) by Theorem 438. Hence we are done by (L1) and Proposition 7.
Theorem 443 (prov9_31d76ffe1b): K(T(y,x),e / y) = K(T(y,x),y \ e)
Proof. We have
(L1) (y \ e) * y = K(y \ e,y)
by Proposition 76. We have (y \ e) * T(K(y \ e,y) / (y \ e),y \ e) = K(y \ e,y) by Theorem 10. Then
(L3) y = T(K(y \ e,y) / (y \ e),y \ e)
by (L1) and Proposition 7. We have (y \ e) * T(T(K(y \ e,y) / (y \ e),y \ e),x) = T(K(y \ e,y) / (y \ e),x) * (y \ e) by Proposition 50. Then
(L5) (y \ e) * T(y,x) = T(K(y \ e,y) / (y \ e),x) * (y \ e)
by (L3). We have ((y \ e) * K(T(y,x),e / y)) * T(((y \ e) * K(T(y,x),e / y)) \ (K(y \ e,y) * K(T(y,x),e / y)),x) = (y \ e) * (K(T(y,x),e / y) * T(K(T(y,x),e / y) \ ((y \ e) \ (K(y \ e,y) * K(T(y,x),e / y))),x)) by Theorem 136. Then (T(K(y \ e,y) / (y \ e),x) * (y \ e)) * K(T(y,x),e / y) = (y \ e) * (K(T(y,x),e / y) * T(K(T(y,x),e / y) \ ((y \ e) \ (K(y \ e,y) * K(T(y,x),e / y))),x)) by Theorem 237. Then (y \ e) * (K(T(y,x),e / y) * T(K(T(y,x),e / y) \ ((y \ e) \ (K(y \ e,y) * K(T(y,x),e / y))),x)) = ((y \ e) * T(y,x)) * K(T(y,x),e / y) by (L5). Then
(L9) (y \ e) * (K(T(y,x),e / y) * T(K(T(y,x),e / y) \ ((e / y) \ K(T(y,x),e / y)),x)) = ((y \ e) * T(y,x)) * K(T(y,x),e / y)
by Theorem 435. We have ((y \ e) * (K(T(y,x),e / y) * T(K(T(y,x),e / y) \ ((e / y) \ K(T(y,x),e / y)),x))) / ((e / y) * (K(T(y,x),e / y) * T(K(T(y,x),e / y) \ ((e / y) \ K(T(y,x),e / y)),x))) = K(y \ e,y) by Theorem 437. Then ((y \ e) * (K(T(y,x),e / y) * T(K(T(y,x),e / y) \ ((e / y) \ K(T(y,x),e / y)),x))) / (((e / y) * T((e / y) \ e,x)) * K(T(y,x),e / y)) = K(y \ e,y) by Theorem 235. Then ((y \ e) * (K(T(y,x),e / y) * T(K(T(y,x),e / y) \ ((e / y) \ K(T(y,x),e / y)),x))) / ((T(y,x) / y) * K(T(y,x),e / y)) = K(y \ e,y) by Theorem 149. Then
(L13) (((y \ e) * T(y,x)) * K(T(y,x),e / y)) / ((T(y,x) / y) * K(T(y,x),e / y)) = K(y \ e,y)
by (L9). We have ((((y \ e) * T(y,x)) * K(T(y,x),e / y)) / ((T(y,x) / y) * K(T(y,x),e / y))) * ((T(y,x) / y) * K(T(y,x),e / y)) = ((y \ e) * T(y,x)) * K(T(y,x),e / y) by Axiom 6. Then
(L15) K(y \ e,y) * ((T(y,x) / y) * K(T(y,x),e / y)) = ((y \ e) * T(y,x)) * K(T(y,x),e / y)
by (L13). We have L(K(T(y,x),e / y),T(y,x) / y,K(y \ e,y)) = (K(y \ e,y) * (T(y,x) / y)) \ (K(y \ e,y) * ((T(y,x) / y) * K(T(y,x),e / y))) by Definition 4. Then L(K(T(y,x),e / y),T(y,x) / y,K(y \ e,y)) = ((y \ e) * T(y,x)) \ (K(y \ e,y) * ((T(y,x) / y) * K(T(y,x),e / y))) by Theorem 400. Then
(L18) ((y \ e) * T(y,x)) \ (((y \ e) * T(y,x)) * K(T(y,x),e / y)) = L(K(T(y,x),e / y),T(y,x) / y,K(y \ e,y))
by (L15). We have ((y \ e) * T(y,x)) \ (((y \ e) * T(y,x)) * K(T(y,x),e / y)) = K(T(y,x),e / y) by Axiom 3. Then
(L20) L(K(T(y,x),e / y),T(y,x) / y,K(y \ e,y)) = K(T(y,x),e / y)
by (L18). We have K(T(y,x),e / y) = ((e / y) * T(y,x)) \ (T(y,x) * (e / y)) by Definition 2. Then
(L25) K(T(y,x),e / y) = (T(y,x) / y) \ (T(y,x) * (e / y))
by Theorem 150. We have L((T(y,x) / y) \ (T(y,x) * (e / y)),T(y,x) / y,K(y \ e,y)) = (K(y \ e,y) * (T(y,x) / y)) \ (K(y \ e,y) * (T(y,x) * (e / y))) by Proposition 53. Then L(K(T(y,x),e / y),T(y,x) / y,K(y \ e,y)) = (K(y \ e,y) * (T(y,x) / y)) \ (K(y \ e,y) * (T(y,x) * (e / y))) by (L25). Then
(L28) ((y \ e) * T(y,x)) \ (K(y \ e,y) * (T(y,x) * (e / y))) = L(K(T(y,x),e / y),T(y,x) / y,K(y \ e,y))
by Theorem 400. We have
(L21) K(y,y \ e) * (K(y \ e,y) * (T(y,x) * (e / y))) = T(y,x) * (e / y)
by Theorem 442. We have K(y,y \ e) * (y \ T(y,x)) = T(y,x) * (e / y) by Theorem 328. Then K(y \ e,y) * (T(y,x) * (e / y)) = y \ T(y,x) by (L21) and Proposition 7. Then
(L29) ((y \ e) * T(y,x)) \ (y \ T(y,x)) = L(K(T(y,x),e / y),T(y,x) / y,K(y \ e,y))
by (L28). We have K(T(y,x),y \ e) = ((y \ e) * T(y,x)) \ (T(y,x) * (y \ e)) by Definition 2. Then K(T(y,x),y \ e) = ((y \ e) * T(y,x)) \ (y \ T(y,x)) by Theorem 146. Then L(K(T(y,x),e / y),T(y,x) / y,K(y \ e,y)) = K(T(y,x),y \ e) by (L29). Hence we are done by (L20).
Theorem 444 (prov9_1f57f17293): T(e / (e / y),x) \ T(y,x) = K(T(y,x),y \ e)
Proof. We have T(e / (e / y),x) \ T(y,x) = K(T(y,x),e / y) by Theorem 401. Hence we are done by Theorem 443.
Theorem 445 (prov9_d9f6fea1f0): T(a(T(y,x),e / y,y),x) = K(T(y,x),y \ e)
Proof. We have
(L3) T(T(y,x) * ((T(y,x) / y) \ (e / y)),T(y,x)) = T(y,x) * ((T(y,x) / y) \ (e / y))
by Theorem 307. Then T(T(y,x) * ((T(y,x) / y) \ (e / y)),T(y,x)) = T(y,x) * (K(x \ (x / (e / y)),x) \ (e / y)) by Theorem 199. Then
(L5) T(y,x) * (K(x \ (x / (e / y)),x) \ (e / y)) = T(y,x) * ((T(y,x) / y) \ (e / y))
by (L3). We have ((e / y) * T((e / y) \ e,x)) \ (e / y) = T(e / (e / y),x) \ e by Theorem 376. Then K(x \ (x / (e / y)),x) \ (e / y) = T(e / (e / y),x) \ e by Theorem 196. Then
(L6) T(y,x) * (T(e / (e / y),x) \ e) = T(y,x) * ((T(y,x) / y) \ (e / y))
by (L5). We have T(y,x) * ((((e / y) * T(y,x)) / (e / y)) \ e) = (((e / y) * T(y,x)) / (e / y)) \ T(y,x) by Theorem 186. Then T(y,x) * (T(e / (e / y),x) \ e) = (((e / y) * T(y,x)) / (e / y)) \ T(y,x) by Theorem 51. Then T(e / (e / y),x) \ T(y,x) = T(y,x) * (T(e / (e / y),x) \ e) by Theorem 51. Then T(y,x) * ((T(y,x) / y) \ (e / y)) = T(e / (e / y),x) \ T(y,x) by (L6). Then
(L11) T(y,x) * ((T(y,x) / y) \ (e / y)) = K(T(y,x),y \ e)
by Theorem 444. We have L(T(y,x) \ e,T(y,x),e / y) = ((e / y) * T(y,x)) \ ((e / y) * e) by Proposition 53. Then
(L13) L(T(y,x) \ e,T(y,x),e / y) = (T(y,x) / y) \ ((e / y) * e)
by Theorem 150. We have T(y,x) * L(T(y,x) \ e,T(y,x),e / y) = T(y * L(y \ e,T(y,x),e / y),x) by Theorem 351. Then T(y,x) * ((T(y,x) / y) \ ((e / y) * e)) = T(y * L(y \ e,T(y,x),e / y),x) by (L13). Then T(y,x) * ((T(y,x) / y) \ (e / y)) = T(y * L(y \ e,T(y,x),e / y),x) by Axiom 2. Then T(y * L(y \ e,T(y,x),e / y),x) = K(T(y,x),y \ e) by (L11). Hence we are done by Theorem 398.
Theorem 446 (prov9_c9ac683a48): T(e / (e / x),y) \ e = e / T(x,y)
Proof. We have L(T(x,y),x \ e,x) \ T(x,y) = a(x,x \ e,T(x,y)) by Theorem 96. Then T(e / (e / x),y) \ T(x,y) = a(x,x \ e,T(x,y)) by Theorem 264. Then
(L13) a(x,x \ e,T(x,y)) = K(T(x,y),e / x)
by Theorem 401. We have T(x,y) \ (L(T(x,y),x \ e,x) \ T(x,y)) = L(T(x,y),x \ e,x) \ e by Theorem 280. Then T(x,y) \ a(x,x \ e,T(x,y)) = L(T(x,y),x \ e,x) \ e by Theorem 96. Then L(T(x,y),x \ e,x) \ e = T(x,y) \ K(T(x,y),e / x) by (L13). Then T(x,y) \ K(T(x,y),e / x) = T(e / (e / x),y) \ e by Theorem 264. Then
(L18) T(x,y) \ K(T(x,y),x \ e) = T(e / (e / x),y) \ e
by Theorem 443. We have T(x,y) \ T(T(x,y) * ((T(x,y) * (T(x,y) \ e)) / T(x,y)),T(x,y)) = T(T(x,y) * (T(x,y) \ e),T(x,y)) / T(x,y) by Theorem 313. Then T(x,y) \ T(T(x * ((T(x,y) * (x \ e)) / T(x,y)),y),T(x,y)) = T(T(x,y) * (T(x,y) \ e),T(x,y)) / T(x,y) by Theorem 345. Then T(x,y) \ T(T(x * ((x \ T(x,y)) / T(x,y)),y),T(x,y)) = T(T(x,y) * (T(x,y) \ e),T(x,y)) / T(x,y) by Theorem 146. Then T(x,y) \ T(T(a(T(x,y),e / x,x),y),T(x,y)) = T(T(x,y) * (T(x,y) \ e),T(x,y)) / T(x,y) by Theorem 399. Then T(x,y) \ T(K(T(x,y),x \ e),T(x,y)) = T(T(x,y) * (T(x,y) \ e),T(x,y)) / T(x,y) by Theorem 445. Then T(T(x,y) * (T(x,y) \ e),T(x,y)) / T(x,y) = T(x,y) \ K(T(x,y),x \ e) by Theorem 371. Then T(T(x,y) * (T(x,y) \ e),T(x,y)) / T(x,y) = T(e / (e / x),y) \ e by (L18). Then
(L20) T(e,T(x,y)) / T(x,y) = T(e / (e / x),y) \ e
by Axiom 4. We have T(e,T(x,y)) = T(x,y) \ (e * T(x,y)) by Definition 3. Then
(L2) T(e,T(x,y)) = T(x,y) \ T(x,y)
by Axiom 1. We have T(x,y) \ T(x,y) = e by Proposition 28. Then T(e,T(x,y)) = e by (L2). Hence we are done by (L20).
Theorem 447 (T_prop): e / (e / T(x,y)) = T(e / (e / x),y)
Proof. We have e / (T(e / (e / x),y) \ e) = T(e / (e / x),y) by Proposition 24. Hence we are done by Theorem 446.
Theorem 448 (prov9_2821f6d306): K(y / x,x) = (x * (y / x)) \ y
Proof. We have K(y / x,x) = (x * (y / x)) \ ((y / x) * x) by Definition 2. Hence we are done by Axiom 6.
Theorem 449 (prov9_d6edc40444): L(z,x \ y,x) = y \ (x * ((x \ y) * z))
Proof. We have L(z,x \ y,x) = (x * (x \ y)) \ (x * ((x \ y) * z)) by Definition 4. Hence we are done by Axiom 4.
Theorem 450 (prov9_97de2591a2): T(x,x) = x
Proof. We have x \ (x * x) = x by Axiom 3. Hence we are done by Definition 3.
Theorem 451 (prov9_a15e2b5c93): T(y,x) = T(T(y,x),y)
Proof. We have T(T(y,y),x) = T(T(y,x),y) by Axiom 7. Hence we are done by Theorem 450.
Theorem 452 (prov9_1a25915906): (y * x) / K(y,x) = x * y
Proof. We have (y * x) / ((x * y) \ (y * x)) = x * y by Proposition 24. Hence we are done by Definition 2.
Theorem 453 (prov9_161ae051f8): (y * (z * x)) / L(x,z,y) = y * z
Proof. We have (y * (z * x)) / ((y * z) \ (y * (z * x))) = y * z by Proposition 24. Hence we are done by Definition 4.
Theorem 454 (prov9_4335b33235): y = x \ (y * T(x,y))
Proof. We have x * y = y * T(x,y) by Proposition 46. Hence we are done by Proposition 2.
Theorem 455 (prov9_63d1257897): K(T(y,x),y) = e
Proof. We have T(T(y,x),y) = T(y,x) by Theorem 451. Hence we are done by Proposition 22.
Theorem 456 (prov9_fdcded2aae): (R(x,y,z) * w) / R(T(x,w),y,z) = w
Proof. We have (R(x,y,z) * w) / T(R(x,y,z),w) = w by Theorem 5. Hence we are done by Axiom 9.
Theorem 457 (prov9_705a2e1fea): K(x,(y * x) / y) = e
Proof. We have K(T((y * x) / y,y),(y * x) / y) = e by Theorem 455. Hence we are done by Theorem 7.
Theorem 458 (prov9_5f178d2871): T(y,(x * y) / x) = y
Proof. We have K(y,(x * y) / x) = e by Theorem 457. Hence we are done by Proposition 23.
Theorem 459 (prov9_cd8c3f5e5a): x * K(y,x / y) = y * (x / y)
Proof. We have ((x / y) * y) * K(y,x / y) = y * (x / y) by Proposition 82. Hence we are done by Axiom 6.
Theorem 460 (prov9_0f91c22ae2): x * y = (y * x) * z → K(x,y) = z
Proof. Assume
(H1) x * y = (y * x) * z.
We have (y * x) * K(x,y) = x * y by Proposition 82. Then (y * x) * K(x,y) = (y * x) * z by (H1). Hence we are done by Proposition 9.
Theorem 461 (prov9_1443eef47c): R(y,x,K(x,y)) = (x * y) / (x * K(x,y))
Proof. We have R(y,x,K(x,y)) = ((y * x) * K(x,y)) / (x * K(x,y)) by Definition 5. Hence we are done by Proposition 82.
Theorem 462 (prov9_3bf0860ad3): x / K(y,y \ x) = (y \ x) * y
Proof. We have (y * (y \ x)) / K(y,y \ x) = (y \ x) * y by Theorem 452. Hence we are done by Axiom 4.
Theorem 463 (prov9_90757da98d): T(x \ y,T(x,x \ y)) = T(x,x \ y) \ y
Proof. We have T(x \ y,T(x,x \ y)) = T(x,x \ y) \ ((x \ y) * T(x,x \ y)) by Definition 3. Hence we are done by Theorem 8.
Theorem 464 (prov9_a6d3f44bb7): ((x \ y) \ y) * z = T(x,x \ y) * z
Proof. We have ((y / T(x,x \ y)) \ y) * z = T(x,x \ y) * z by Theorem 26. Hence we are done by Theorem 15.
Theorem 465 (prov9_e79c7f9ef9): K(x,e / x) \ e = K(e / x,x)
Proof. We have (x * (e / x)) \ e = K(e / x,x) by Theorem 448. Hence we are done by Proposition 77.
Theorem 466 (prov9_b09cbe8b87): z * T(z \ (x * z),y) = T(x,y) * z
Proof. We have z * T(T(x,z),y) = T(x,y) * z by Proposition 50. Hence we are done by Definition 3.
Theorem 467 (prov9_169655b1a3): L(T(x \ z,w),x,y) = T((y * x) \ (y * z),w)
Proof. We have L(T(x \ z,w),x,y) = T(L(x \ z,x,y),w) by Axiom 8. Hence we are done by Proposition 53.
Theorem 468 (prov9_b1836f66ba): R(R(x,w,u),y,z) * (w * u) = (R(x,y,z) * w) * u
Proof. We have R(R(x,y,z),w,u) * (w * u) = (R(x,y,z) * w) * u by Proposition 54. Hence we are done by Axiom 12.
Theorem 469 (prov9_3b2e46961d): R(x,y \ e,y) * K(y \ e,y) = (x * (y \ e)) * y
Proof. We have R(x,y \ e,y) * ((y \ e) * y) = (x * (y \ e)) * y by Proposition 54. Hence we are done by Proposition 76.
Theorem 470 (prov9_cf9bc830e9): R(T(x / y,w),y,z) = T((x * z) / (y * z),w)
Proof. We have R(T(x / y,w),y,z) = T(R(x / y,y,z),w) by Axiom 9. Hence we are done by Proposition 55.
Theorem 471 (prov9_a318c8a853): T(R(y / x,x,z),x * z) = (x * z) \ (y * z)
Proof. We have T((y * z) / (x * z),x * z) = (x * z) \ (y * z) by Proposition 47. Hence we are done by Proposition 55.
Theorem 472 (prov9_dd91d69133): (x * y) / L(z,y / z,x) = x * (y / z)
Proof. We have (x * ((y / z) * z)) / L(z,y / z,x) = x * (y / z) by Theorem 453. Hence we are done by Axiom 6.
Theorem 473 (prov9_b426c1937a): R(T(x / y,z),y,y \ e) = T(x * (y \ e),z)
Proof. We have R(T(x / y,z),y,y \ e) = T(R(x / y,y,y \ e),z) by Axiom 9. Hence we are done by Proposition 71.
Theorem 474 (prov9_4c4d05a423): x \ R(L(y,x \ e,x),z,w) = (x \ e) * R(y,z,w)
Proof. We have x \ L(R(y,z,w),x \ e,x) = (x \ e) * R(y,z,w) by Proposition 57. Hence we are done by Axiom 10.
Theorem 475 (prov9_85444acf3d): (y * L(y \ x,z,w)) / y = L(x / y,z,w)
Proof. We have (y * T(L(x / y,z,w),y)) / y = L(x / y,z,w) by Proposition 48. Hence we are done by Proposition 72.
Theorem 476 (prov9_389f32de4f): T(x / (y * z),y) = R(y \ (x / z),y,z)
Proof. We have T(R((x / z) / y,y,z),y) = R(y \ (x / z),y,z) by Proposition 60. Hence we are done by Proposition 74.
Theorem 477 (prov9_f7a54d912d): T((x \ y) / y,x) = R(x \ e,x,x \ y)
Proof. We have T(R(e / x,x,x \ y),x) = R(x \ e,x,x \ y) by Proposition 60. Hence we are done by Theorem 37.
Theorem 478 (prov9_1703e10f0e): K(x,(y \ z) \ z) = (T(y,y \ z) * x) \ (x * ((y \ z) \ z))
Proof. We have K(x,(y \ z) \ z) = (((y \ z) \ z) * x) \ (x * ((y \ z) \ z)) by Definition 2. Hence we are done by Theorem 464.
Theorem 479 (prov9_8503717dec): (z / y) * (y * T(x,z)) = L(x,y,z / y) * z
Proof. We have z * L(T(x,z),y,z / y) = L(x,y,z / y) * z by Proposition 58. Hence we are done by Theorem 53.
Theorem 480 (prov9_0795b90fd4): x \ ((x \ e) \ e) = K((x \ e) \ e,x \ e)
Proof. We have ((x * K((x \ e) \ e,x \ e)) / K((x \ e) \ e,x \ e)) * K((x \ e) \ e,x \ e) = x * K((x \ e) \ e,x \ e) by Axiom 6. Then ((((x * K((x \ e) \ e,x \ e)) / K((x \ e) \ e,x \ e)) / K((x \ e) \ e,x \ e)) * K((x \ e) \ e,x \ e)) * K((x \ e) \ e,x \ e) = x * K((x \ e) \ e,x \ e) by Axiom 6. Then
(L7) ((x / K((x \ e) \ e,x \ e)) * K((x \ e) \ e,x \ e)) * K((x \ e) \ e,x \ e) = x * K((x \ e) \ e,x \ e)
by Axiom 5. We have (e / (x \ e)) * (x \ e) = e by Axiom 6. Then (((e / (x \ e)) / K((x \ e) \ e,x \ e)) * K((x \ e) \ e,x \ e)) * (x \ e) = e by Axiom 6. Then
(L3) ((x / K((x \ e) \ e,x \ e)) * K((x \ e) \ e,x \ e)) * (x \ e) = e
by Proposition 24.
(x * K((x \ e) \ e,x \ e)) / K((x \ e) \ e,x \ e)
=(x / K((x \ e) \ e,x \ e)) * K((x \ e) \ e,x \ e) by (L7), Proposition 1
=e / (x \ e) by (L3), Proposition 1

Then
(L9) (x * K((x \ e) \ e,x \ e)) / K((x \ e) \ e,x \ e) = e / (x \ e)
. We have x \ (x * K((x \ e) \ e,x \ e)) = K((x \ e) \ e,x \ e) by Axiom 3. Then x \ (((x * K((x \ e) \ e,x \ e)) / K((x \ e) \ e,x \ e)) * K((x \ e) \ e,x \ e)) = K((x \ e) \ e,x \ e) by Axiom 6. Then x \ ((e / (x \ e)) * K((x \ e) \ e,x \ e)) = K((x \ e) \ e,x \ e) by (L9). Hence we are done by Theorem 109.
Theorem 481 (prov9_ebbad55107): T(R(z \ x,z,y),z * y) = T((z * y) \ (x * y),z)
Proof. We have T(T((x * y) / (z * y),z),z * y) = T((z * y) \ (x * y),z) by Proposition 51. Hence we are done by Proposition 64.
Theorem 482 (prov9_33e792c336): z * R(x,z,y) = (((z * x) * y) / (z * y)) * z
Proof. We have z * T(((z * x) * y) / (z * y),z) = (((z * x) * y) / (z * y)) * z by Proposition 46. Hence we are done by Theorem 110.
Theorem 483 (prov9_081c073daa): (y \ z) * ((x * ((y \ z) \ z)) / x) = ((x * y) / x) * (y \ z)
Proof. We have (y \ z) * ((x * T(y,y \ z)) / x) = ((x * y) / x) * (y \ z) by Theorem 93. Hence we are done by Proposition 49.
Theorem 484 (prov9_626f3b0665): T(x \ e,y) / (x \ e) = x * T(x \ e,y)
Proof. We have R(T(e / x,y),x,x \ e) = T(R(e / x,x,x \ e),y) by Axiom 9. Then
(L2) R(T(e / x,y),x,x \ e) = T(x \ e,y)
by Theorem 41. We have R(T(e / x,y),x,x \ e) / (x \ e) = T(e / x,y) * x by Theorem 73. Then
(L4) T(x \ e,y) / (x \ e) = T(e / x,y) * x
by (L2). We have T(e / x,y) * x = x * T(x \ e,y) by Proposition 61. Hence we are done by (L4).
Theorem 485 (prov9_a4bcffd7b0): R((z * (e / x)) / z,x,y) = (z * (y / (x * y))) / z
Proof. We have R((z * (e / x)) / z,x,y) = (z * R(e / x,x,y)) / z by Theorem 123. Hence we are done by Proposition 79.
Theorem 486 (prov9_716e2ec18e): (x * y) * T((x * y) \ (x * z),w) = x * (y * T(y \ z,w))
Proof. We have (x * y) * T(L(y \ z,y,x),w) = x * (y * T(y \ z,w)) by Theorem 55. Hence we are done by Proposition 53.
Theorem 487 (prov9_d5adecb7d2): T((z * w) \ (w * x),y) = L(T(T(z,w) \ x,y),T(z,w),w)
Proof. We have T((w * T(z,w)) \ (w * x),y) = L(T(T(z,w) \ x,y),T(z,w),w) by Theorem 467. Hence we are done by Proposition 46.
Theorem 488 (prov9_9c4b9cc956): R(T(y / x,x * z),x,z) = (x * z) \ (y * z)
Proof. We have T((y * z) / (x * z),x * z) = (x * z) \ (y * z) by Proposition 47. Hence we are done by Theorem 470.
Theorem 489 (prov9_b20834c32f): (z * w) * ((y * ((z * w) \ (x * w))) / y) = ((y * R(x / z,z,w)) / y) * (z * w)
Proof. We have (z * w) * ((y * T(R(x / z,z,w),z * w)) / y) = ((y * R(x / z,z,w)) / y) * (z * w) by Theorem 93. Hence we are done by Theorem 471.
Theorem 490 (prov9_336d25ee1a): T(x * (z \ e),y) / (z \ e) = T(x / z,y) * z
Proof. We have R(T(x / z,y),z,z \ e) / (z \ e) = T(x / z,y) * z by Theorem 73. Hence we are done by Theorem 473.
Theorem 491 (prov9_31c0ce60fa): (z / x) \ (L(y,x,z / x) * z) = x * T(y,z)
Proof. We have (z / x) * (x * T(y,z)) = L(y,x,z / x) * z by Theorem 479. Hence we are done by Proposition 2.
Theorem 492 (prov9_91d2f74312): ((x * T(y,x * y)) * y) / (x * y) = y
Proof. We have x * T(y,x) = y * x by Proposition 46. Then
(L3) x * R(T(y,x * y),x,y) = y * x
by Theorem 113. We have (((x * T(y,x * y)) * y) / (x * y)) * x = x * R(T(y,x * y),x,y) by Theorem 482. Hence we are done by (L3) and Proposition 8.
Theorem 493 (prov9_de8c02f955): T((y / x) * T(x,y),x) = y
Proof. We have ((((y / x) * T(x,(y / x) * x)) * x) / ((y / x) * x)) * ((y / x) * x) = ((y / x) * T(x,(y / x) * x)) * x by Axiom 6. Then x * ((y / x) * x) = ((y / x) * T(x,(y / x) * x)) * x by Theorem 492. Then
(L3) T((y / x) * T(x,(y / x) * x),x) = (y / x) * x
by Theorem 11. We have (y / x) * x = y by Axiom 6. Then T((y / x) * T(x,(y / x) * x),x) = y by (L3). Hence we are done by Axiom 6.
Theorem 494 (prov9_ea51cbe084): T(x * T(x \ y,y),x \ y) = y
Proof. We have T((y / (x \ y)) * T(x \ y,y),x \ y) = y by Theorem 493. Hence we are done by Proposition 24.
Theorem 495 (prov9_0f6bdf03c4): ((x \ y) * y) / (x \ y) = x * T(x \ y,y)
Proof. We have ((x \ y) * T(x * T(x \ y,y),x \ y)) / (x \ y) = x * T(x \ y,y) by Proposition 48. Hence we are done by Theorem 494.
Theorem 496 (prov9_25c61f5a3e): (x \ e) * T((x \ e) \ (x \ y),z) = x \ T(y,z)
Proof. We have (x \ e) * T((x \ e) \ (x \ y),z) = x \ T(x * (x \ y),z) by Theorem 140. Hence we are done by Axiom 4.
Theorem 497 (prov9_92d0ea1dde): T(y * (x \ e),z) / (x \ e) = x * T(x \ y,z)
Proof. We have T(y / x,z) * x = x * T(x \ y,z) by Proposition 61. Hence we are done by Theorem 490.
Theorem 498 (prov9_e684d133c6): T(y * x,z) / x = (e / x) * T((e / x) \ y,z)
Proof. We have T(y / (e / x),z) * (e / x) = (e / x) * T((e / x) \ y,z) by Proposition 61. Hence we are done by Theorem 141.
Theorem 499 (prov9_1c3f354d68): T(((e / x) * y) * x,z) / x = (e / x) * T(y,z)
Proof. We have T(((e / x) * y) / (e / x),z) * (e / x) = (e / x) * T(y,z) by Theorem 47. Hence we are done by Theorem 141.
Theorem 500 (prov9_f349f58cf4): (T(x,y) / x) / (e / x) = T(e / (e / x),y)
Proof. We have T(e / (e / x),y) * (e / x) = T(x,y) / x by Theorem 147. Hence we are done by Proposition 1.
Theorem 501 (prov9_39352c5ad5): T(e / y,T(y,x)) = T(y,x) \ (T(y,x) / y)
Proof. We have T(e / y,T(y,x)) = T(y,x) \ ((e / y) * T(y,x)) by Definition 3. Hence we are done by Theorem 150.
Theorem 502 (prov9_977b21cf30): ((y \ x) / (x / y)) / (e / (x / y)) = T(e / (e / (x / y)),y)
Proof. We have (T(x / y,y) / (x / y)) / (e / (x / y)) = T(e / (e / (x / y)),y) by Theorem 500. Hence we are done by Proposition 47.
Theorem 503 (prov9_dc977daaab): L(x \ e,x,T(x,y) / x) = T(e / x,T(x,y))
Proof. We have L(x \ e,x,T(x,y) / x) = T(x,y) \ (T(x,y) / x) by Theorem 29. Hence we are done by Theorem 501.
Theorem 504 (prov9_42b5084f6e): K(y \ e,y) * (y \ x) = x * T(x \ (y \ x),y)
Proof. We have R(y \ e,y,y \ x) * x = ((y \ e) * y) * (y \ x) by Theorem 25. Then
(L2) T((y \ x) / x,y) * x = ((y \ e) * y) * (y \ x)
by Theorem 477. We have T((y \ x) / x,y) * x = x * T(x \ (y \ x),y) by Proposition 61. Then ((y \ e) * y) * (y \ x) = x * T(x \ (y \ x),y) by (L2). Hence we are done by Proposition 76.
Theorem 505 (prov9_5db365beed): (y \ T(x,z)) / (y \ e) = T((y \ x) / (y \ e),z)
Proof. We have ((y \ e) * T((y \ e) \ (y \ x),z)) / (y \ e) = T((y \ x) / (y \ e),z) by Theorem 49. Hence we are done by Theorem 496.
Theorem 506 (prov9_8546c1597f): (y * T(y \ x,z)) * (y \ e) = T(x * (y \ e),z)
Proof. We have (T(x * (y \ e),z) / (y \ e)) * (y \ e) = T(x * (y \ e),z) by Axiom 6. Hence we are done by Theorem 497.
Theorem 507 (prov9_4a78dd1b5b): ((e / y) * T(x,z)) * y = T(((e / y) * x) * y,z)
Proof. We have (T(((e / y) * x) * y,z) / y) * y = T(((e / y) * x) * y,z) by Axiom 6. Hence we are done by Theorem 499.
Theorem 508 (prov9_f69be58970): T(x * y,x) / y = ((x * y) * x) / (x * y)
Proof. We have (x * y) * T((y * x) / y,x * y) = ((y * x) / y) * (x * y) by Proposition 46. Then
(L2) (x * y) * L(x,y,x) = ((y * x) / y) * (x * y)
by Theorem 92. We have L(((x * y) * x) / (x * y),y,x) * (x * y) = (x * y) * L(x,y,x) by Theorem 86. Then
(L4) (y * x) / y = L(((x * y) * x) / (x * y),y,x)
by (L2) and Proposition 8. We have ((y * x) / y) * y = y * x by Axiom 6. Then
(L6) L(((x * y) * x) / (x * y),y,x) * y = y * x
by (L4). We have y * L(T(((x * y) * x) / (x * y),y),y,x) = L(((x * y) * x) / (x * y),y,x) * y by Proposition 58. Then y * ((x * y) \ (x * ((((x * y) * x) / (x * y)) * y))) = L(((x * y) * x) / (x * y),y,x) * y by Proposition 62. Then
(L9) x = (x * y) \ (x * ((((x * y) * x) / (x * y)) * y))
by (L6) and Proposition 7. We have (x * y) * ((x * y) \ (x * ((((x * y) * x) / (x * y)) * y))) = x * ((((x * y) * x) / (x * y)) * y) by Axiom 4. Then (x * y) * x = x * ((((x * y) * x) / (x * y)) * y) by (L9). Then T(x * y,x) = (((x * y) * x) / (x * y)) * y by Theorem 11. Hence we are done by Proposition 1.
Theorem 509 (prov9_0dfdb526d6): T(T(y * x,y) / x,y * x) = y
Proof. We have T(((y * x) * y) / (y * x),y * x) = y by Theorem 7. Hence we are done by Theorem 508.
Theorem 510 (prov9_421c6c7c73): T(T(x,x / y) / y,x) = x / y
Proof. We have T(T((x / y) * y,x / y) / y,(x / y) * y) = x / y by Theorem 509. Then T(T(x,x / y) / y,(x / y) * y) = x / y by Axiom 6. Hence we are done by Axiom 6.
Theorem 511 (prov9_14cc99566b): (x * (x / y)) / x = T(x,x / y) / y
Proof. We have (x * T(T(x,x / y) / y,x)) / x = T(x,x / y) / y by Proposition 48. Hence we are done by Theorem 510.
Theorem 512 (prov9_8bb85beeba): ((x * (x / y)) / x) * y = T(x,x / y)
Proof. We have (T(x,x / y) / y) * y = T(x,x / y) by Axiom 6. Hence we are done by Theorem 511.
Theorem 513 (prov9_5a2a5a68c9): T(x \ T(y,y / x),y) = x \ y
Proof. We have T(T(T(y,y / x) / x,y),x) = T(x \ T(y,y / x),y) by Proposition 51. Then
(L2) T(y / x,x) = T(x \ T(y,y / x),y)
by Theorem 510. We have T(y / x,x) = x \ y by Proposition 47. Hence we are done by (L2).
Theorem 514 (prov9_3c311180d1): T((x \ y) \ T(y,x),y) = (x \ y) \ y
Proof. We have T((x \ y) \ T(y,y / (x \ y)),y) = (x \ y) \ y by Theorem 513. Hence we are done by Proposition 24.
Theorem 515 (prov9_420d3826fe): T(y,y / x) = x * ((y * (x \ y)) / y)
Proof. We have ((y * (y / x)) / y) * x = x * ((y * (x \ y)) / y) by Theorem 125. Hence we are done by Theorem 512.
Theorem 516 (prov9_c1f6ab5f6e): (x * y) * R((x * y) \ (x * z),w,u) = x * (y * R(y \ z,w,u))
Proof. We have (x * y) * R(L(y \ z,y,x),w,u) = x * (y * R(y \ z,w,u)) by Theorem 56. Hence we are done by Proposition 53.
Theorem 517 (prov9_de0d5e3917): R((x * u) / (w * u),y,z) * (w * u) = (R(x / w,y,z) * w) * u
Proof. We have R(R(x / w,w,u),y,z) * (w * u) = (R(x / w,y,z) * w) * u by Theorem 468. Hence we are done by Proposition 55.
Theorem 518 (prov9_18961a57f6): R(T(z \ x,z * y),z,y) = T((z * y) \ (x * y),z)
Proof. We have R(T(z \ x,z * y),z,y) = T(R(z \ x,z,y),z * y) by Axiom 9. Hence we are done by Theorem 481.
Theorem 519 (prov9_172e98af42): R(T(x,z * y),z,y) = T((z * y) \ ((z * x) * y),z)
Proof. We have R(T(z \ (z * x),z * y),z,y) = T((z * y) \ ((z * x) * y),z) by Theorem 518. Hence we are done by Axiom 3.
Theorem 520 (prov9_a04a999fec): T(z,x) * (y * z) = z * (T(z,x) * T(y,z))
Proof. We have T(z,x) * (z * T(y,z)) = z * (T(z,x) * T(y,z)) by Theorem 172. Hence we are done by Proposition 46.
Theorem 521 (prov9_98e5e28de2): (y \ x) * ((x / y) * z) = (x / y) * (T(x / y,y) * z)
Proof. We have T(x / y,y) * ((x / y) * z) = (x / y) * (T(x / y,y) * z) by Theorem 172. Hence we are done by Proposition 47.
Theorem 522 (prov9_a2ef6a9708): x \ z = T(x,y) * (x \ (T(x,y) \ z))
Proof. We have x \ (T(x,y) * (T(x,y) \ z)) = T(x,y) * (x \ (T(x,y) \ z)) by Theorem 173. Hence we are done by Axiom 4.
Theorem 523 (prov9_5af0847be8): T(y,x) \ K(y,e / y) = y * (T(y,x) \ (e / y))
Proof. We have T(y,x) \ (y * (e / y)) = y * (T(y,x) \ (e / y)) by Theorem 174. Hence we are done by Proposition 77.
Theorem 524 (prov9_866aa8e212): y * (T(y,x) \ (T(y,x) / y)) = K(y,T(y,x) / y)
Proof. We have T(y,x) \ (y * (T(y,x) / y)) = K(y,T(y,x) / y) by Theorem 3. Hence we are done by Theorem 174.
Theorem 525 (prov9_8f063f515d): T(T(y,x),y * (T(y,x) \ z)) = (y * (T(y,x) \ z)) \ (y * z)
Proof. We have T(y,x) * (y * (T(y,x) \ z)) = y * (T(y,x) * (T(y,x) \ z)) by Theorem 172. Then
(L2) T(y,x) * (y * (T(y,x) \ z)) = y * z
by Axiom 4. We have T(T(y,x),y * (T(y,x) \ z)) = (y * (T(y,x) \ z)) \ (T(y,x) * (y * (T(y,x) \ z))) by Definition 3. Hence we are done by (L2).
Theorem 526 (prov9_09304614fe): T(x,y) \ (x \ z) = x \ (T(x,y) \ z)
Proof. We have T(x,y) * (x \ (T(x,y) \ z)) = x \ z by Theorem 522. Hence we are done by Proposition 2.
Theorem 527 (prov9_429cdd8962): T(x,y) * T(T(x,y) \ x,x) = L(x,T(x,y) \ x,T(x,y))
Proof. We have
(L1) x * (T(x,y) * T(T(x,y) \ x,x)) = T(x,y) * ((T(x,y) \ x) * x)
by Theorem 520. We have x * L(x,T(x,y) \ x,T(x,y)) = T(x,y) * ((T(x,y) \ x) * x) by Theorem 52. Hence we are done by (L1) and Proposition 7.
Theorem 528 (prov9_16e58b1572): T(y,z) * (T(y,x) \ e) = T(y,x) \ T(y,z)
Proof. We have y * (T(y,x) \ e) = T(y,x) \ y by Theorem 175. Then
(L2) (T(y,x) \ y) / (T(y,x) \ e) = y
by Proposition 1. We have (T(y,x) \ T(y,z)) / (T(y,x) \ e) = T((T(y,x) \ y) / (T(y,x) \ e),z) by Theorem 505. Then
(L4) (T(y,x) \ T(y,z)) / (T(y,x) \ e) = T(y,z)
by (L2). We have ((T(y,x) \ T(y,z)) / (T(y,x) \ e)) * (T(y,x) \ e) = T(y,x) \ T(y,z) by Axiom 6. Hence we are done by (L4).
Theorem 529 (prov9_407acdd104): y * T(e / y,T(y,x)) = K(y,T(y,x) / y)
Proof. We have y * (T(y,x) \ (T(y,x) / y)) = K(y,T(y,x) / y) by Theorem 524. Hence we are done by Theorem 501.
Theorem 530 (prov9_e9149595d9): ((z \ (y / x)) * z) * x = (z * x) * T((z * x) \ y,z)
Proof. We have R(z \ (y / x),z,x) * (z * x) = ((z \ (y / x)) * z) * x by Proposition 54. Then
(L2) T(y / (z * x),z) * (z * x) = ((z \ (y / x)) * z) * x
by Theorem 476. We have T(y / (z * x),z) * (z * x) = (z * x) * T((z * x) \ y,z) by Proposition 61. Hence we are done by (L2).
Theorem 531 (prov9_7e7171f81c): (x \ e) * R((x \ e) \ e,y,z) = x \ R(x,y,z)
Proof. We have x \ R(L((x \ e) \ e,x \ e,x),y,z) = (x \ e) * R((x \ e) \ e,y,z) by Theorem 474. Then x \ R(e \ x,y,z) = (x \ e) * R((x \ e) \ e,y,z) by Theorem 28. Hence we are done by Proposition 26.
Theorem 532 (prov9_213fe46639): (z \ R(z,x,y)) / R(T(z,z \ e),x,y) = z \ e
Proof. We have (R(z,x,y) * (z \ e)) / R(T(z,z \ e),x,y) = z \ e by Theorem 456. Hence we are done by Proposition 89.
Theorem 533 (prov9_372d6ce6d5): (R(z,x,y) / z) / R(z,x,y) = e / z
Proof. We have (e / z) * R(z,x,y) = R(z,x,y) / z by Theorem 192. Hence we are done by Proposition 1.
Theorem 534 (prov9_409b776d3e): L(R(z,x,y) \ w,R(z,x,y),e / z) = (R(z,x,y) / z) \ ((e / z) * w)
Proof. We have L(R(z,x,y) \ w,R(z,x,y),e / z) = ((e / z) * R(z,x,y)) \ ((e / z) * w) by Proposition 53. Hence we are done by Theorem 192.
Theorem 535 (prov9_d44fed6918): (x / y) * ((y \ x) * z) = (y \ x) * ((x / y) * z)
Proof. We have (x / y) * (T(x / y,y) * z) = (y \ x) * ((x / y) * z) by Theorem 521. Hence we are done by Proposition 47.
Theorem 536 (prov9_97df63eb5d): (x \ y) \ ((y / x) * z) = (y / x) * ((x \ y) \ z)
Proof. We have (y / x) \ ((y / x) * z) = z by Axiom 3. Then
(L3) (y / x) \ ((x \ y) * ((x \ y) \ ((y / x) * z))) = z
by Axiom 4. We have (y / x) * ((y / x) \ ((x \ y) * ((x \ y) \ ((y / x) * z)))) = (x \ y) * ((x \ y) \ ((y / x) * z)) by Axiom 4. Then (y / x) * ((x \ y) * ((x \ y) \ ((y / x) \ ((x \ y) * ((x \ y) \ ((y / x) * z)))))) = (x \ y) * ((x \ y) \ ((y / x) * z)) by Axiom 4. Then
(L6) (y / x) * ((x \ y) * ((x \ y) \ z)) = (x \ y) * ((x \ y) \ ((y / x) * z))
by (L3). We have (y / x) * ((x \ y) * ((x \ y) \ z)) = (x \ y) * ((y / x) * ((x \ y) \ z)) by Theorem 535. Then (x \ y) * ((x \ y) \ ((y / x) * z)) = (x \ y) * ((y / x) * ((x \ y) \ z)) by (L6). Hence we are done by Proposition 9.
Theorem 537 (prov9_7074375b70): (x \ y) \ K(y \ x,y) = T((x \ y) \ e,y)
Proof. We have (x \ y) \ K(y \ (y / (x \ y)),y) = T((x \ y) \ e,y) by Theorem 197. Hence we are done by Proposition 24.
Theorem 538 (prov9_1d91e3c14e): ((y * x) \ y) \ K(x,y) = T(((y * x) \ y) \ e,y)
Proof. We have ((y * x) \ y) \ K(y \ (y * x),y) = T(((y * x) \ y) \ e,y) by Theorem 537. Hence we are done by Axiom 3.
Theorem 539 (prov9_7228c020b4): ((y * x) \ y) * T(((y * x) \ y) \ e,y) = K(x,y)
Proof. We have ((y * x) \ y) * (((y * x) \ y) \ K(x,y)) = K(x,y) by Axiom 4. Hence we are done by Theorem 538.
Theorem 540 (prov9_e9463cc32e): T(L(x \ e,x,y) \ e,y) = ((y * x) \ y) \ K(x,y)
Proof. We have T(((y * x) \ y) \ e,y) = ((y * x) \ y) \ K(x,y) by Theorem 538. Hence we are done by Proposition 78.
Theorem 541 (prov9_de26942095): T(y,y * K(x,y)) = y
Proof. We have T(y,(((y * ((y * x) \ y)) \ y) * y) / ((y * ((y * x) \ y)) \ y)) = y by Theorem 458. Then T(y,(y * ((y * x) \ y)) * T((y * ((y * x) \ y)) \ y,y)) = y by Theorem 495. Then T(y,(y * ((y * x) \ y)) * L(T(((y * x) \ y) \ e,y),(y * x) \ y,y)) = y by Theorem 30. Then T(y,y * (((y * x) \ y) * T(((y * x) \ y) \ e,y))) = y by Proposition 52. Hence we are done by Theorem 539.
Theorem 542 (prov9_41d5954459): L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y) = K(x,y)
Proof. We have L(x \ e,x,y) * (((y * x) \ y) \ K(x,y)) = K(x,y) by Theorem 43. Hence we are done by Theorem 540.
Theorem 543 (prov9_f600fc3b55): (e / y) \ ((y \ x) / x) = ((y \ x) / x) * y
Proof. We have ((y \ x) / (y * (y \ x))) / (((y \ x) / (y * (y \ x))) * y) = e / y by Theorem 189. Then ((y \ x) / x) / (((y \ x) / (y * (y \ x))) * y) = e / y by Axiom 4. Then
(L3) ((y \ x) / x) / (((y \ x) / x) * y) = e / y
by Axiom 4. We have (((y \ x) / x) / (((y \ x) / x) * y)) \ ((y \ x) / x) = ((y \ x) / x) * y by Proposition 25. Hence we are done by (L3).
Theorem 544 (prov9_ebccf2e171): R(x,e / x,x) = (x \ e) \ e
Proof. We have R(x,e / x,x) = T(x,e / x) by Theorem 117. Hence we are done by Theorem 210.
Theorem 545 (prov9_d45a2ebd45): R(z,x,y) \ R(T(z,z \ e),x,y) = a(R(z,x,y),e / z,z)
Proof. We have R(R(z,e / z,z),x,y) = R(R(z,x,y),e / z,z) by Axiom 12. Then R(T(z,e / z),x,y) = R(R(z,x,y),e / z,z) by Theorem 117. Then
(L3) R(R(z,x,y),e / z,z) = R(T(z,z \ e),x,y)
by Theorem 211. We have R(z,x,y) \ R(R(z,x,y),e / z,z) = a(R(z,x,y),e / z,z) by Theorem 131. Hence we are done by (L3).
Theorem 546 (prov9_5817b80f0e): (x \ e) * ((y * ((x \ e) \ e)) / y) = x \ ((y * x) / y)
Proof. We have ((y * x) / y) * (x \ e) = x \ ((y * x) / y) by Theorem 195. Hence we are done by Theorem 483.
Theorem 547 (prov9_894910d14f): y * (z * T(z \ (y \ x),y)) = ((y \ (x / z)) * y) * z
Proof. We have (y * z) * T((y * z) \ x,y) = ((y \ (x / z)) * y) * z by Theorem 530. Hence we are done by Theorem 136.
Theorem 548 (prov9_4b710cfb86): x * (K(x \ e,x) * (x \ y)) = K(x \ e,x) * y
Proof. We have x * (y * T(y \ (x \ y),x)) = ((x \ (y / y)) * x) * y by Theorem 547. Then x * (K(x \ e,x) * (x \ y)) = ((x \ (y / y)) * x) * y by Theorem 504. Then ((x \ e) * x) * y = x * (K(x \ e,x) * (x \ y)) by Proposition 29. Hence we are done by Proposition 76.
Theorem 549 (prov9_281e1c048d): x * (K(x \ e,x) * y) = K(x \ e,x) * (x * y)
Proof. We have x * (K(x \ e,x) * (x \ (x * y))) = K(x \ e,x) * (x * y) by Theorem 548. Hence we are done by Axiom 3.
Theorem 550 (prov9_af4dfabefa): y / T(y,x) = (e / T(y,x)) * y
Proof. We have e * y = y by Axiom 1. Then
(L3) y * (y \ (e * y)) = y
by Axiom 4. We have y * (y \ (e * y)) = e * y by Axiom 4. Then ((y * (y \ (e * y))) / T(y,x)) * T(y,x) = e * y by Axiom 6. Then (y / T(y,x)) * T(y,x) = e * y by (L3). Then
(L7) (y / T(y,x)) * T(y,x) = ((e / T(y,x)) * T(y,x)) * y
by Axiom 6. We have ((e / T(y,x)) * T(y,x)) * y = ((e / T(y,x)) * y) * T(y,x) by Theorem 238. Then (y / T(y,x)) * T(y,x) = ((e / T(y,x)) * y) * T(y,x) by (L7). Hence we are done by Proposition 10.
Theorem 551 (prov9_4b46a9f362): T(e / T(x,y),x) = x \ (x / T(x,y))
Proof. We have T(e / T(x,y),x) = x \ ((e / T(x,y)) * x) by Definition 3. Hence we are done by Theorem 550.
Theorem 552 (prov9_7708945321): K(x,x \ e) * x = (x \ e) \ e
Proof. We have (x * (e / x)) * x = R(x,e / x,x) by Proposition 69. Then K(x,e / x) * x = R(x,e / x,x) by Proposition 77. Then K(x,e / x) * x = (x \ e) \ e by Theorem 544. Hence we are done by Theorem 248.
Theorem 553 (prov9_89ec1fb3f1): (x \ e) * K(x,x \ e) = e / x
Proof. We have (x \ e) * ((x \ e) \ (e / x)) = e / x by Axiom 4. Then (x \ e) * K(x,e / x) = e / x by Theorem 219. Hence we are done by Theorem 248.
Theorem 554 (prov9_66a4ef3718): L(x \ y,x,K(x,x \ e)) = T(x,x \ e) \ (K(x,x \ e) * y)
Proof. We have L(x \ y,x,K(x,x \ e)) = (K(x,x \ e) * x) \ (K(x,x \ e) * y) by Proposition 53. Hence we are done by Theorem 246.
Theorem 555 (prov9_a2ed36a182): K(x,x \ e) = (e / (e / (e / x))) * x
Proof. We have K(x,x \ e) = x * (e / x) by Theorem 250. Hence we are done by Theorem 245.
Theorem 556 (prov9_1520c4b4ea): K(x,x \ e) = x / (e / (e / x))
Proof. We have K(x,x \ e) = x * (e / x) by Theorem 250. Hence we are done by Theorem 265.
Theorem 557 (prov9_87e684f4e1): K(x,x \ e) * (e / (e / x)) = x
Proof. We have (x / (e / (e / x))) * (e / (e / x)) = x by Axiom 6. Hence we are done by Theorem 556.
Theorem 558 (prov9_b1e4f495fb): K(x,x \ e) \ x = e / (e / x)
Proof. We have (x / (e / (e / x))) \ x = e / (e / x) by Proposition 25. Hence we are done by Theorem 556.
Theorem 559 (prov9_5d4d4363c2): K(x,e / x) = x \ ((x \ e) \ e)
Proof. We have x \ R(x,e / x,x) = a(x,e / x,x) by Theorem 131. Then x \ ((x \ e) \ e) = a(x,e / x,x) by Theorem 544. Hence we are done by Theorem 269.
Theorem 560 (prov9_6e24f36e55): K((x \ e) \ e,x \ e) = K(x,e / x)
Proof. We have K((x \ e) \ e,x \ e) = x \ ((x \ e) \ e) by Theorem 480. Hence we are done by Theorem 559.
Theorem 561 (prov9_774387866a): K((x \ e) \ e,x \ e) = K(x,x \ e)
Proof. We have K(x,e / x) = K(x,x \ e) by Theorem 248. Then
(L2) x \ ((x \ e) \ e) = K(x,x \ e)
by Theorem 559. We have K((x \ e) \ e,x \ e) = x \ ((x \ e) \ e) by Theorem 480. Hence we are done by (L2).
Theorem 562 (prov9_b5396f88e6): x * K(x,x \ e) = T(x,x \ e)
Proof. We have x * (x \ T(x,x \ e)) = T(x,x \ e) by Axiom 4. Then x * K(x,e / x) = T(x,x \ e) by Theorem 270. Hence we are done by Theorem 248.
Theorem 563 (prov9_5ec944358b): K(y,y \ e) * ((e / y) \ x) = x * T(x \ ((e / y) \ x),e / y)
Proof. We have K((e / y) \ e,e / y) * ((e / y) \ x) = x * T(x \ ((e / y) \ x),e / y) by Theorem 504. Hence we are done by Theorem 253.
Theorem 564 (prov9_f9b111441e): K(e / x,x) = K(x \ e,x)
Proof. We have K(e / x,(e / x) \ e) = K(x \ e,x) by Theorem 272. Hence we are done by Proposition 25.
Theorem 565 (prov9_948e7c203f): K(x,x \ e) \ e = K(x \ e,x)
Proof. We have K(x,e / x) \ e = K(e / x,x) by Theorem 465. Then K(x,x \ e) \ e = K(e / x,x) by Theorem 248. Hence we are done by Theorem 564.
Theorem 566 (prov9_b6d6f823f3): K(e / (e / x),e / x) = K(x,x \ e)
Proof. We have K((e / x) \ e,e / x) = K(x,x \ e) by Theorem 253. Hence we are done by Theorem 564.
Theorem 567 (prov9_5a5899c447): K(x \ e,x) * T(e / x,y) = (x \ e) * (x * T(x \ e,y))
Proof. We have L(T(x \ e,y),x,x \ e) = T(L(x \ e,x,x \ e),y) by Axiom 8. Then
(L2) L(T(x \ e,y),x,x \ e) = T(e / x,y)
by Theorem 261. We have K(x \ e,x) * L(T(x \ e,y),x,x \ e) = (x \ e) * (x * T(x \ e,y)) by Theorem 59. Hence we are done by (L2).
Theorem 568 (prov9_685c325b06): x \ (y * T(y \ x,z)) = (x \ y) * T((x \ y) \ e,z)
Proof. We have x * ((x \ y) * T((x \ y) \ e,z)) = y * T(y \ x,z) by Theorem 283. Hence we are done by Proposition 2.
Theorem 569 (prov9_7e5bc80534): T(y,y * (T(y,x) \ y)) = y
Proof. We have T(y,y * K(y \ (y / (T(y,x) \ (x \ y))),y)) = y by Theorem 541. Then T(y,y * (T(y,x) \ ((x \ y) * T((x \ y) \ T(y,x),y)))) = y by Theorem 285. Then T(y,y * (T(y,x) \ ((x \ y) * ((x \ y) \ y)))) = y by Theorem 514. Hence we are done by Axiom 4.
Theorem 570 (prov9_d4fe7cc36d): (x * (T(x,y) \ x)) \ (x * x) = T(x,y)
Proof. We have T(T(x,x * (T(x,y) \ x)),y) = T(T(x,y),x * (T(x,y) \ x)) by Axiom 7. Then T(x,y) = T(T(x,y),x * (T(x,y) \ x)) by Theorem 569. Hence we are done by Theorem 525.
Theorem 571 (prov9_2c24db3394): (y * y) / T(y,x) = y * (T(y,x) \ y)
Proof. We have (y * y) / ((y * (T(y,x) \ y)) \ (y * y)) = y * (T(y,x) \ y) by Proposition 24. Hence we are done by Theorem 570.
Theorem 572 (prov9_b253a558ff): y * (T(y,x) \ y) = (y / T(y,x)) * y
Proof. We have (y * y) / T(y,x) = (y / T(y,x)) * y by Theorem 239. Hence we are done by Theorem 571.
Theorem 573 (prov9_5f65df8175): T(x / y,z) \ (y \ x) = (y \ x) * (T(x / y,z) \ e)
Proof. We have T(x / y,y) * (T(x / y,z) \ e) = T(x / y,z) \ T(x / y,y) by Theorem 528. Then (y \ x) * (T(x / y,z) \ e) = T(x / y,z) \ T(x / y,y) by Proposition 47. Hence we are done by Proposition 47.
Theorem 574 (prov9_c87ed036f9): ((x \ y) \ y) * (x * z) = x * (((x \ y) \ y) * z)
Proof. We have (y / (x \ y)) * (((x \ y) \ y) * z) = ((x \ y) \ y) * ((y / (x \ y)) * z) by Theorem 535. Then x * (((x \ y) \ y) * z) = ((x \ y) \ y) * ((y / (x \ y)) * z) by Proposition 24. Hence we are done by Proposition 24.
Theorem 575 (prov9_403ffa8b9b): T(x \ T(x,z),y) = K(z \ (z / T(x \ e,y)),z)
Proof. We have T((x \ e) * T((x \ e) \ e,z),y) = K(z \ (z / T(x \ e,y)),z) by Theorem 296. Hence we are done by Theorem 145.
Theorem 576 (prov9_60d46bdabd): T(x * T(x \ e,z),y) = T(x,y) * T(T(x,y) \ e,z)
Proof. We have K(z \ (z / T(x,y)),z) = T(x,y) * T(T(x,y) \ e,z) by Theorem 196. Hence we are done by Theorem 296.
Theorem 577 (prov9_c5210a8db9): K(x \ (x / T(x,y)),x) = T(K(x \ e,x),y)
Proof. We have T(x * T(x \ e,x),y) = K(x \ (x / T(x,y)),x) by Theorem 296. Then T((x \ e) * x,y) = K(x \ (x / T(x,y)),x) by Proposition 46. Hence we are done by Proposition 76.
Theorem 578 (prov9_717ded1211): T(z \ e,z \ R(z,x,y)) = z \ e
Proof. We have T(z \ e,(z \ e) * R((z \ e) \ e,x,y)) = z \ e by Theorem 298. Hence we are done by Theorem 531.
Theorem 579 (prov9_e71c1d1148): (R(z,x,y) * (e / z)) / R(z,x,y) = L(e / z,R(z,x,y),e / z)
Proof. We have T(e / z,(e / z) * R((e / z) \ e,x,y)) = e / z by Theorem 298. Then
(L2) T(e / z,(e / z) * R(z,x,y)) = e / z
by Proposition 25. We have (R(z,x,y) * T(e / z,(e / z) * R(z,x,y))) / R(z,x,y) = L(e / z,R(z,x,y),e / z) by Theorem 80. Hence we are done by (L2).
Theorem 580 (prov9_5d828a735c): R(x \ e,R(x,y,z),x \ e) = T(x \ e,R(x,y,z))
Proof. We have T(x \ e,(x \ e) * R((x \ e) \ e,y,z)) = x \ e by Theorem 298. Then
(L2) T(x \ e,R(x,y,z) * (x \ e)) = x \ e
by Proposition 88. We have R(T(x \ e,R(x,y,z) * (x \ e)),R(x,y,z),x \ e) = T(x \ e,R(x,y,z)) by Theorem 113. Hence we are done by (L2).
Theorem 581 (prov9_8bac33ce71): T(y \ T(y,x),y) = T(y \ T(y,x),T(y,y \ e))
Proof. We have T(y \ e,T(y,y \ e)) = T(y,y \ e) \ e by Theorem 463. Then
(L2) T(y \ e,T(y,y \ e)) = T(y \ e,y)
by Theorem 208. We have T(y \ T(y,x),T(y,y \ e)) = K(x \ (x / T(y \ e,T(y,y \ e))),x) by Theorem 575. Then
(L4) T(y \ T(y,x),T(y,y \ e)) = K(x \ (x / T(y \ e,y)),x)
by (L2). We have T(y \ T(y,x),y) = K(x \ (x / T(y \ e,y)),x) by Theorem 575. Hence we are done by (L4).
Theorem 582 (prov9_3b3c459178): K(y \ T(y,x),T(y,z) \ e) = e
Proof. We have T(T(y,z) \ e,y \ T(y,x)) = T(y,z) \ e by Theorem 308. Then T(y \ T(y,x),T(y,z) \ e) = y \ T(y,x) by Proposition 21. Hence we are done by Proposition 22.
Theorem 583 (prov9_ce84994dda): T(x * y,y) / y = y \ T(y * x,y)
Proof. We have T((y \ (y * x)) * y,y) / y = y \ T(y * x,y) by Theorem 311. Hence we are done by Axiom 3.
Theorem 584 (prov9_ddd8debd96): y * (T(x * y,y) / y) = T(y * x,y)
Proof. We have y * (y \ T(y * x,y)) = T(y * x,y) by Axiom 4. Hence we are done by Theorem 583.
Theorem 585 (prov9_3192328a5f): T(K(y,x),T(y,y \ e)) = T(K(y,x),y)
Proof. We have L(T(y \ T(y,x),T(y,y \ e)),y,x) = T((x * y) \ (y * x),T(y,y \ e)) by Theorem 103. Then
(L2) L(T(y \ T(y,x),y),y,x) = T((x * y) \ (y * x),T(y,y \ e))
by Theorem 581. We have L(T(y \ T(y,x),y),y,x) = T((x * y) \ (y * x),y) by Theorem 103. Then T((x * y) \ (y * x),T(y,y \ e)) = T((x * y) \ (y * x),y) by (L2). Then T((x * y) \ (y * x),T(y,y \ e)) = T(K(y,x),y) by Definition 2. Hence we are done by Definition 2.
Theorem 586 (prov9_4b0f0bb6c3): x \ (K(x,x \ e) * y) = K(x,x \ e) * (x \ y)
Proof. We have x * (K(x,x \ e) * (x \ y)) = K(x,x \ e) * y by Theorem 327. Hence we are done by Proposition 2.
Theorem 587 (prov9_84decee727): L(x,y,K(y,y \ e)) = L(x,K(y,y \ e),y)
Proof. We have
(L3) (K(y,y \ e) * y) * L(x,y,K(y,y \ e)) = K(y,y \ e) * (y * x)
by Proposition 52. We have
(L1) y * (K(y,y \ e) * (y \ (y * x))) = K(y,y \ e) * (y * x)
by Theorem 327.
T(y,y \ e) * L(x,y,K(y,y \ e))
=K(y,y \ e) * (y * x) by (L3), Theorem 246
=y * (K(y,y \ e) * x) by (L1), Axiom 3

Then
(L5) T(y,y \ e) * L(x,y,K(y,y \ e)) = y * (K(y,y \ e) * x)
. We have (y * K(y,y \ e)) * L(x,K(y,y \ e),y) = y * (K(y,y \ e) * x) by Proposition 52. Then T(y,y \ e) * L(x,K(y,y \ e),y) = y * (K(y,y \ e) * x) by Theorem 562. Hence we are done by (L5) and Proposition 7.
Theorem 588 (prov9_2e360f9558): L(x,K(y \ e,y),e / y) = L(x,e / y,K(e / y,(e / y) \ e))
Proof. We have L(x,K(e / y,(e / y) \ e),e / y) = L(x,e / y,K(e / y,(e / y) \ e)) by Theorem 587. Hence we are done by Theorem 272.
Theorem 589 (prov9_c2a5e31bab): (x \ ((y * x) / y)) * y = T(y,R(x,x \ e,y))
Proof. We have T(y,R(x,x \ e,y)) = ((y * R(x,x \ e,y)) / y) * (R(x,x \ e,y) \ y) by Theorem 179. Then
(L2) T(y,R(x,x \ e,y)) = ((y * R(x,x \ e,y)) / y) * ((x \ e) * y)
by Theorem 45. We have ((y * R(x,x \ e,y)) / y) * ((x \ e) * y) = (((y * x) / y) * (x \ e)) * y by Theorem 129. Then T(y,R(x,x \ e,y)) = (((y * x) / y) * (x \ e)) * y by (L2). Hence we are done by Theorem 195.
Theorem 590 (prov9_f46b96eae0): T(y,R(x,x \ e,y)) / y = x \ ((y * x) / y)
Proof. We have (x \ ((y * x) / y)) * y = T(y,R(x,x \ e,y)) by Theorem 589. Hence we are done by Proposition 1.
Theorem 591 (prov9_1df445640c): (T(x,y) \ x) * y = T(y,R(T(x,y),T(x,y) \ e,y))
Proof. We have (T(x,y) \ ((y * T(x,y)) / y)) * y = T(y,R(T(x,y),T(x,y) \ e,y)) by Theorem 589. Hence we are done by Proposition 48.
Theorem 592 (prov9_3c13b0153f): K(T(x,x \ e),T(x,x \ e) \ e) = K(x,x \ e)
Proof. We have K((((x \ e) \ e) \ e) \ e,((x \ e) \ e) \ e) = K((x \ e) \ e,e / ((x \ e) \ e)) by Theorem 560. Then K(T((x \ e) \ e,x \ e),((x \ e) \ e) \ e) = K((x \ e) \ e,e / ((x \ e) \ e)) by Theorem 209. Then K(T((x \ e) \ e,x \ e),((x \ e) \ e) \ e) = K((x \ e) \ e,x \ e) by Proposition 24. Then
(L6) K(T((x \ e) \ e,x \ e),T(x \ e,(x \ e) \ e)) = K((x \ e) \ e,x \ e)
by Proposition 49. We have K((x \ e) \ e,x \ e) = K(x,x \ e) by Theorem 561. Then
(L8) K(T((x \ e) \ e,x \ e),T(x \ e,(x \ e) \ e)) = K(x,x \ e)
by (L6). We have T((x \ e) \ e,x \ e) = (((x \ e) \ e) \ e) \ e by Theorem 209. Then T((x \ e) \ e,x \ e) = T(x \ e,x) \ e by Theorem 209. Then K(T(x \ e,x) \ e,T(x \ e,(x \ e) \ e)) = K(x,x \ e) by (L8). Then K(T(x \ e,x) \ e,((x \ e) \ e) \ e) = K(x,x \ e) by Proposition 49. Then
(L11) K(T(x \ e,x) \ e,T(x \ e,x)) = K(x,x \ e)
by Theorem 209. We have e / (T(x,x \ e) \ e) = T(x,x \ e) by Proposition 24. Then
(L13) e / T(x \ e,x) = T(x,x \ e)
by Theorem 208. We have K(e / T(x \ e,x),T(x \ e,x)) = K(T(x \ e,x) \ e,T(x \ e,x)) by Theorem 564. Then K(T(x,x \ e),T(x \ e,x)) = K(T(x \ e,x) \ e,T(x \ e,x)) by (L13). Then K(x,x \ e) = K(T(x,x \ e),T(x \ e,x)) by (L11). Hence we are done by Theorem 208.
Theorem 593 (prov9_b061963be7): T(y,z) \ ((y \ x) \ x) = ((y \ x) \ x) * (T(y,z) \ e)
Proof. We have ((y \ x) \ x) * (T(x / (y \ x),z) \ e) = T(x / (y \ x),z) \ ((y \ x) \ x) by Theorem 573. Then ((y \ x) \ x) * (T(y,z) \ e) = T(x / (y \ x),z) \ ((y \ x) \ x) by Proposition 24. Hence we are done by Proposition 24.
Theorem 594 (prov9_3ddfb500b6): (x * u) * R((x * u) \ (y * u),z,w) = (x * R(x \ y,z,w)) * u
Proof. We have R((y * u) / (x * u),z,w) * (x * u) = (x * u) * R((x * u) \ (y * u),z,w) by Proposition 83. Then (R(y / x,z,w) * x) * u = (x * u) * R((x * u) \ (y * u),z,w) by Theorem 517. Hence we are done by Proposition 83.
Theorem 595 (prov9_aef3be7515): ((x / z) * L((x / z) \ y,w,u)) * z = x * L(x \ (y * z),w,u)
Proof. We have R(L(y / (x / z),w,u),x / z,z) * x = (L(y / (x / z),w,u) * (x / z)) * z by Theorem 23. Then L(R(y / (x / z),x / z,z),w,u) * x = (L(y / (x / z),w,u) * (x / z)) * z by Axiom 10. Then
(L3) L((y * z) / x,w,u) * x = (L(y / (x / z),w,u) * (x / z)) * z
by Theorem 67. We have L((y * z) / x,w,u) * x = x * L(x \ (y * z),w,u) by Theorem 85. Then (L(y / (x / z),w,u) * (x / z)) * z = x * L(x \ (y * z),w,u) by (L3). Hence we are done by Theorem 85.
Theorem 596 (prov9_abaafab195): K(y,(y / (x \ e)) / y) = x \ ((y * x) / y)
Proof. We have (x \ e) * ((y * ((x \ e) \ e)) / y) = x \ ((y * x) / y) by Theorem 546. Hence we are done by Theorem 333.
Theorem 597 (prov9_5a53dde6cf): (x * L(x \ (z * y),w,u)) / y = (x / y) * L((x / y) \ z,w,u)
Proof. We have ((x / y) * L((x / y) \ z,w,u)) * y = x * L(x \ (z * y),w,u) by Theorem 595. Hence we are done by Proposition 1.
Theorem 598 (prov9_659aed49d3): (x * L(x \ y,z,w)) / y = (x / y) * L((x / y) \ e,z,w)
Proof. We have (x * L(x \ (e * y),z,w)) / y = (x / y) * L((x / y) \ e,z,w) by Theorem 597. Hence we are done by Axiom 1.
Theorem 599 (prov9_20e7869699): (x * R(x \ y,z,w)) / y = (x / y) * R((x / y) \ e,z,w)
Proof. We have R(R(e / (x / y),z,w),x / y,y) * x = (R(e / (x / y),z,w) * (x / y)) * y by Theorem 23. Then R(y / ((x / y) * y),z,w) * x = (R(e / (x / y),z,w) * (x / y)) * y by Theorem 40. Then
(L3) (R(e / (x / y),z,w) * (x / y)) * y = R(y / x,z,w) * x
by Axiom 6. We have R(y / x,z,w) * x = x * R(x \ y,z,w) by Proposition 83. Then (R(e / (x / y),z,w) * (x / y)) * y = x * R(x \ y,z,w) by (L3). Then ((x / y) * R((x / y) \ e,z,w)) * y = x * R(x \ y,z,w) by Proposition 83. Hence we are done by Proposition 1.
Theorem 600 (prov9_5df04cb44f): (x \ e) \ (K(x,x \ e) * y) = K(x,x \ e) * ((x \ e) \ y)
Proof. We have (x \ e) * (K((x \ e) \ e,x \ e) * ((x \ e) \ y)) = K((x \ e) \ e,x \ e) * y by Theorem 548. Then (x \ e) * (K(x,x \ e) * ((x \ e) \ y)) = K((x \ e) \ e,x \ e) * y by Theorem 561. Then (x \ e) * (K(x,x \ e) * ((x \ e) \ y)) = K(x,x \ e) * y by Theorem 561. Hence we are done by Proposition 2.
Theorem 601 (prov9_af3f9ae9f5): (x * K(y,y \ e)) * y = (x * y) * K(y,y \ e)
Proof. We have (x * (y * L(y \ e,y,e / y))) * y = (x * y) * (y * L(y \ e,y,e / y)) by Theorem 339. Then (x * (y * (e / y))) * y = (x * y) * (y * L(y \ e,y,e / y)) by Theorem 35. Then (x * y) * (y * L(y \ e,y,e / y)) = (x * K(y,y \ e)) * y by Theorem 250. Then (x * y) * (y * (e / y)) = (x * K(y,y \ e)) * y by Theorem 35. Hence we are done by Theorem 250.
Theorem 602 (prov9_519c86be5b): (x * K(y,y \ e)) \ ((x * y) * K(y,y \ e)) = y
Proof. We have (x * K(y,y \ e)) * y = (x * y) * K(y,y \ e) by Theorem 601. Hence we are done by Proposition 2.
Theorem 603 (prov9_6bf40fa1d3): L(y \ e,y,x \ (x / y)) = e / y
Proof. We have L(y \ e,y,((x / y) * y) \ (x / y)) = e / y by Theorem 342. Hence we are done by Axiom 6.
Theorem 604 (prov9_bc5310bb49): (x \ (x / y)) / (e / y) = (x \ (x / y)) * y
Proof. We have (x \ (x / y)) / L(y \ e,y,x \ (x / y)) = (x \ (x / y)) * y by Theorem 33. Hence we are done by Theorem 603.
Theorem 605 (prov9_a0632ff46e): T(e / (e / (x / y)),y / x) = x / y
Proof. We have T(e / (e / (x / y)),R(e / (x / y),x / y,y)) = x / y by Theorem 359. Hence we are done by Theorem 38.
Theorem 606 (prov9_39a7d0dfaf): ((x \ y) / y) * T(x,x \ e) = x * ((x \ y) / y)
Proof. We have ((x \ y) / y) * T(x,(x \ y) / y) = x * ((x \ y) / y) by Proposition 46. Hence we are done by Theorem 363.
Theorem 607 (prov9_662abfed98): T(y \ e,K(x,y)) = y \ e
Proof. We have T(T(e / y,K(x,y)),y) = T(y \ e,K(x,y)) by Proposition 51. Then
(L2) T(e / y,y) = T(y \ e,K(x,y))
by Theorem 367. We have T(e / y,y) = y \ e by Proposition 47. Hence we are done by (L2).
Theorem 608 (prov9_b489b2c462): (z * w) * ((y * ((z * w) \ (x * w))) / y) = (((y * (x / z)) / y) * z) * w
Proof. We have ((y * R(x / z,z,w)) / y) * (z * w) = (((y * (x / z)) / y) * z) * w by Theorem 129. Hence we are done by Theorem 489.
Theorem 609 (prov9_acc8c2f9d2): L(y,T(y,x) \ y,T(y,x)) = L(y,T(y,x) \ y,y)
Proof. We have T(((T(y,x) \ y) * y) / (T(y,x) \ y),y * (T(y,x) \ y)) = L(y,T(y,x) \ y,y) by Theorem 92. Then T(((T(y,x) \ y) * y) / (T(y,x) \ y),(y * y) / T(y,x)) = L(y,T(y,x) \ y,y) by Theorem 571. Then T(T(y,x) * T(T(y,x) \ y,y),(y * y) / T(y,x)) = L(y,T(y,x) \ y,y) by Theorem 495. Then T(L(y,T(y,x) \ y,T(y,x)),(y * y) / T(y,x)) = L(y,T(y,x) \ y,y) by Theorem 527. Then
(L5) T(L(y,T(y,x) \ y,T(y,x)),y * (T(y,x) \ y)) = L(y,T(y,x) \ y,y)
by Theorem 571. We have L(T(y,y * (T(y,x) \ y)),T(y,x) \ y,T(y,x)) = T(L(y,T(y,x) \ y,T(y,x)),y * (T(y,x) \ y)) by Axiom 8. Then L(T(y,y * (T(y,x) \ y)),T(y,x) \ y,T(y,x)) = L(y,T(y,x) \ y,y) by (L5). Hence we are done by Theorem 569.
Theorem 610 (prov9_564d8e8cd4): ((y \ x) * y) * (e / y) = T(x * (e / y),y)
Proof. We have y * (T(((e / y) * ((e / y) \ (y \ (x * (e / y))))) * y,y) / y) = T(y * ((e / y) * ((e / y) \ (y \ (x * (e / y))))),y) by Theorem 584. Then
(L4) y * ((e / y) * T((e / y) \ (y \ (x * (e / y))),y)) = T(y * ((e / y) * ((e / y) \ (y \ (x * (e / y))))),y)
by Theorem 499. We have y * ((e / y) * T((e / y) \ (y \ (x * (e / y))),y)) = (y * T(y \ x,y)) * (e / y) by Theorem 301. Then
(L6) T(y * ((e / y) * ((e / y) \ (y \ (x * (e / y))))),y) = (y * T(y \ x,y)) * (e / y)
by (L4). We have y * (y \ (x * (e / y))) = x * (e / y) by Axiom 4. Then y * ((e / y) * ((e / y) \ (y \ (x * (e / y))))) = x * (e / y) by Axiom 4. Then T(x * (e / y),y) = (y * T(y \ x,y)) * (e / y) by (L6). Hence we are done by Proposition 46.
Theorem 611 (prov9_48120812c0): (x * y) * (e / y) = T((y * x) * (e / y),y)
Proof. We have ((y \ (y * x)) * y) * (e / y) = T((y * x) * (e / y),y) by Theorem 610. Hence we are done by Axiom 3.
Theorem 612 (prov9_07fcfc83bd): K(x \ y,y \ x) = K(x \ y,(x \ y) \ e)
Proof. We have K(x \ y,(x * (x \ y)) \ x) = K(x \ y,(x \ y) \ e) by Theorem 392. Hence we are done by Axiom 4.
Theorem 613 (prov9_d0cb2ae63e): (x \ y) \ K(x \ y,y \ x) = e / (x \ y)
Proof. We have (x \ y) \ K(x \ y,(x \ y) \ e) = e / (x \ y) by Theorem 251. Hence we are done by Theorem 612.
Theorem 614 (prov9_0766199446): e / (e / (x \ y)) = L(e / (y \ x),y \ x,y)
Proof. We have (y \ x) * T(x \ y,(x \ y) * (y \ x)) = L(x \ y,y \ x,x \ y) * (y \ x) by Theorem 79. Then
(L2) (y \ x) * (x \ y) = L(x \ y,y \ x,x \ y) * (y \ x)
by Theorem 306. We have L(e / (y \ x),y \ x,y) * (y \ x) = (y \ x) * L((y \ x) \ e,y \ x,y) by Theorem 85. Then L(e / (y \ x),y \ x,y) * (y \ x) = (y \ x) * (x \ y) by Theorem 28. Then L(x \ y,y \ x,x \ y) = L(e / (y \ x),y \ x,y) by (L2) and Proposition 8. Hence we are done by Theorem 394.
Theorem 615 (prov9_8dfe6fd141): x \ (y * K(y \ x,x \ y)) = e / (e / (x \ y))
Proof. We have x \ (y * ((y \ x) * (e / (y \ x)))) = L(e / (y \ x),y \ x,y) by Theorem 449. Then x \ (y * K(y \ x,(y \ x) \ e)) = L(e / (y \ x),y \ x,y) by Theorem 250. Then x \ (y * K(y \ x,(y \ x) \ e)) = e / (e / (x \ y)) by Theorem 614. Hence we are done by Theorem 612.
Theorem 616 (prov9_729ac3a2c2): L(x,x \ e,y) = e / (e / ((y * (x \ e)) \ y))
Proof. We have L(e / (x \ e),x \ e,y) = e / (e / ((y * (x \ e)) \ y)) by Theorem 395. Hence we are done by Proposition 24.
Theorem 617 (prov9_22aea79720): L(e / x,x,y) \ e = e / ((y * x) \ y)
Proof. We have (e / (e / ((y * x) \ y))) \ e = e / ((y * x) \ y) by Proposition 25. Hence we are done by Theorem 395.
Theorem 618 (prov9_cda5506640): L(L(y,x,x \ e),K(x \ e,x),e / x) = (e / x) * (x * y)
Proof. We have L(L(y,x,x \ e),K(x \ e,x),e / x) = ((e / x) * K(x \ e,x)) \ ((e / x) * (K(x \ e,x) * L(y,x,x \ e))) by Definition 4. Then
(L2) L(L(y,x,x \ e),K(x \ e,x),e / x) = (x \ e) \ ((e / x) * (K(x \ e,x) * L(y,x,x \ e)))
by Theorem 109. We have (x \ e) \ ((e / x) * (K(x \ e,x) * L(y,x,x \ e))) = (e / x) * ((x \ e) \ (K(x \ e,x) * L(y,x,x \ e))) by Theorem 536. Then L(L(y,x,x \ e),K(x \ e,x),e / x) = (e / x) * ((x \ e) \ (K(x \ e,x) * L(y,x,x \ e))) by (L2). Then (e / x) * ((x \ e) \ ((x \ e) * (x * y))) = L(L(y,x,x \ e),K(x \ e,x),e / x) by Theorem 59. Hence we are done by Axiom 3.
Theorem 619 (prov9_a5f135f27c): T(y \ T(y,z),K(x,y)) = y \ T(y,z)
Proof. We have T(y \ e,K(x,y)) * T(T(y \ e,K(x,y)) \ e,z) = T((y \ e) * T((y \ e) \ e,z),K(x,y)) by Theorem 576. Then (y \ e) * T(T(y \ e,K(x,y)) \ e,z) = T((y \ e) * T((y \ e) \ e,z),K(x,y)) by Theorem 607. Then
(L3) T((y \ e) * T((y \ e) \ e,z),K(x,y)) = (y \ e) * T((y \ e) \ e,z)
by Theorem 607. We have (y \ e) * T((y \ e) \ e,z) = y \ T(y,z) by Theorem 145. Then T((y \ e) * T((y \ e) \ e,z),K(x,y)) = y \ T(y,z) by (L3). Hence we are done by Theorem 145.
Theorem 620 (prov9_bb387a3cd4): K(y,y \ e) * T(K(y,y \ e) \ x,y) = T(x,y)
Proof. We have (T((x / (e / y)) / y,y) * y) * (e / y) = T(x / (y * (e / y)),y) * (y * (e / y)) by Theorem 138. Then (T((x / (e / y)) / y,y) * y) * (e / y) = T(x / (y * (e / y)),y) * K(y,y \ e) by Theorem 250. Then
(L3) (T((x / (e / y)) / y,y) * y) * (e / y) = T(x / K(y,y \ e),y) * K(y,y \ e)
by Theorem 250. We have T(x / K(y,y \ e),y) * K(y,y \ e) = K(y,y \ e) * T(K(y,y \ e) \ x,y) by Proposition 61. Then
(L5) (T((x / (e / y)) / y,y) * y) * (e / y) = K(y,y \ e) * T(K(y,y \ e) \ x,y)
by (L3). We have (T((x / (e / y)) / y,y) * y) * (e / y) = y * ((e / y) * T((e / y) \ (y \ x),y)) by Theorem 293. Then K(y,y \ e) * T(K(y,y \ e) \ x,y) = y * ((e / y) * T((e / y) \ (y \ x),y)) by (L5). Then
(L8) K(y,y \ e) * T(K(y,y \ e) \ x,y) = y * (T((y \ x) * y,y) / y)
by Theorem 498. We have y * (T((y \ x) * y,y) / y) = T(x,y) by Theorem 312. Hence we are done by (L8).
Theorem 621 (prov9_dcd09895ef): T(x * K(y,y \ e),y) = T(x,y) * K(y,y \ e)
Proof. We have K(y,y \ e) * T(K(y,y \ e) \ (x * K(y,y \ e)),y) = T(x,y) * K(y,y \ e) by Theorem 466. Hence we are done by Theorem 620.
Theorem 622 (prov9_2e1f9d2ca7): y * ((y \ x) * K(y,y \ e)) = x * K(y,y \ e)
Proof. We have T((x / y) * K(y,y \ e),y) = y \ (((x / y) * K(y,y \ e)) * y) by Definition 3. Then T((x / y) * K(y,y \ e),y) = y \ (((x / y) * y) * K(y,y \ e)) by Theorem 601. Then
(L5) y \ (x * K(y,y \ e)) = T((x / y) * K(y,y \ e),y)
by Axiom 6. We have T((x / y) * K(y,y \ e),y) = T(x / y,y) * K(y,y \ e) by Theorem 621. Then T((x / y) * K(y,y \ e),y) = (y \ x) * K(y,y \ e) by Proposition 47. Then
(L6) (y \ x) * K(y,y \ e) = y \ (x * K(y,y \ e))
by (L5). We have y * (y \ (x * K(y,y \ e))) = x * K(y,y \ e) by Axiom 4. Hence we are done by (L6).
Theorem 623 (prov9_189cdcf46b): y * (x * K(y,y \ e)) = (y * x) * K(y,y \ e)
Proof. We have y * ((y \ (y * x)) * K(y,y \ e)) = (y * x) * K(y,y \ e) by Theorem 622. Hence we are done by Axiom 3.
Theorem 624 (prov9_71f4c688f3): R(y,x,K(y,y \ e)) = y
Proof. We have (y * x) * K(y,y \ e) = y * (x * K(y,y \ e)) by Theorem 623. Hence we are done by Theorem 64.
Theorem 625 (prov9_cabe555550): L(K(y,y \ e),x,y) = K(y,y \ e)
Proof. We have (y * x) * K(y,y \ e) = y * (x * K(y,y \ e)) by Theorem 623. Hence we are done by Theorem 54.
Theorem 626 (prov9_4368f0ff9c): T(x,y * K(x,x \ e)) = T(x,y)
Proof. We have R(T(x,y * K(x,x \ e)),y,K(x,x \ e)) = T(R(x,y,K(x,x \ e)),y * K(x,x \ e)) by Axiom 9. Then
(L2) R(T(x,y * K(x,x \ e)),y,K(x,x \ e)) = T(x,y * K(x,x \ e))
by Theorem 624. We have T((y * K(x,x \ e)) \ ((y * x) * K(x,x \ e)),y) = R(T(x,y * K(x,x \ e)),y,K(x,x \ e)) by Theorem 519. Then T(x,y) = R(T(x,y * K(x,x \ e)),y,K(x,x \ e)) by Theorem 602. Hence we are done by (L2).
Theorem 627 (prov9_91d1c227a1): T(K(y \ e,y) * x,y) = K(y \ e,y) * T(x,y)
Proof. We have
(L1) T(((e / K(y,y \ e)) * x) * K(y,y \ e),y) = T((e / K(y,y \ e)) * x,y) * K(y,y \ e)
by Theorem 621. We have T(((e / K(y,y \ e)) * x) * K(y,y \ e),y) = ((e / K(y,y \ e)) * T(x,y)) * K(y,y \ e) by Theorem 507. Then ((e / K(y,y \ e)) * T(x,y)) * K(y,y \ e) = T((e / K(y,y \ e)) * x,y) * K(y,y \ e) by (L1). Then (e / K(y,y \ e)) * T(x,y) = T((e / K(y,y \ e)) * x,y) by Proposition 10. Then T(((y \ e) * y) * x,y) = (e / K(y,y \ e)) * T(x,y) by Theorem 462. Then (e / K(y,y \ e)) * T(x,y) = T(K(y \ e,y) * x,y) by Proposition 76. Then ((y \ e) * y) * T(x,y) = T(K(y \ e,y) * x,y) by Theorem 462. Hence we are done by Proposition 76.
Theorem 628 (prov9_1eb9fa21bd): K(y \ e,y) * (x * y) = (K(y \ e,y) * x) * y
Proof. We have y * (K(y \ e,y) * (y \ (x * y))) = K(y \ e,y) * (x * y) by Theorem 548. Then y * (K(y \ e,y) * T(x,y)) = K(y \ e,y) * (x * y) by Definition 3. Then
(L3) y * T(K(y \ e,y) * x,y) = K(y \ e,y) * (x * y)
by Theorem 627. We have y * T(K(y \ e,y) * x,y) = (K(y \ e,y) * x) * y by Proposition 46. Hence we are done by (L3).
Theorem 629 (prov9_76c9634737): L(y,x,K(y \ e,y)) = y
Proof. We have (K(y \ e,y) * x) * y = K(y \ e,y) * (x * y) by Theorem 628. Hence we are done by Theorem 54.
Theorem 630 (prov9_3471963f61): T(x,y) = T(x,K(x \ e,x) * y)
Proof. We have (K(x \ e,x) * y) \ (K(x \ e,x) * (x * y)) = L(T(x,y),y,K(x \ e,x)) by Proposition 62. Then
(L4) (K(x \ e,x) * y) \ (x * (K(x \ e,x) * y)) = L(T(x,y),y,K(x \ e,x))
by Theorem 549. We have T(x,K(x \ e,x) * y) = (K(x \ e,x) * y) \ (x * (K(x \ e,x) * y)) by Definition 3. Then
(L6) T(x,K(x \ e,x) * y) = L(T(x,y),y,K(x \ e,x))
by (L4). We have L(T(x,y),y,K(x \ e,x)) = T(L(x,y,K(x \ e,x)),y) by Axiom 8. Then L(T(x,y),y,K(x \ e,x)) = T(x,y) by Theorem 629. Hence we are done by (L6).
Theorem 631 (prov9_669c76f081): T(K(x \ e,x),K(x \ e,x) \ y) = T(K(x \ e,x),y)
Proof. We have T(x,K(x \ e,x) * (K(x \ e,x) \ y)) = T(x,K(x \ e,x) \ y) by Theorem 630. Then
(L2) T(x,y) = T(x,K(x \ e,x) \ y)
by Axiom 4. We have K(x \ (x / T(x,K(x \ e,x) \ y)),x) = T(K(x \ e,x),K(x \ e,x) \ y) by Theorem 577. Then
(L4) K(x \ (x / T(x,y)),x) = T(K(x \ e,x),K(x \ e,x) \ y)
by (L2). We have K(x \ (x / T(x,y)),x) = T(K(x \ e,x),y) by Theorem 577. Hence we are done by (L4).
Theorem 632 (prov9_69d1baaf9b): T(x,K(y,y \ e)) = (K(y,y \ e) \ x) * K(y,y \ e)
Proof. We have T(K((e / y) \ e,e / y),K((e / y) \ e,e / y) \ x) = (K((e / y) \ e,e / y) \ x) \ x by Proposition 49. Then T(K((e / y) \ e,e / y),x) = (K((e / y) \ e,e / y) \ x) \ x by Theorem 631. Then (K(y,y \ e) \ x) \ x = T(K((e / y) \ e,e / y),x) by Theorem 253. Then
(L4) T(K(y,y \ e),x) = (K(y,y \ e) \ x) \ x
by Theorem 253. We have (K(y,y \ e) \ x) * ((x * ((K(y,y \ e) \ x) \ x)) / x) = T(x,x / (K(y,y \ e) \ x)) by Theorem 515. Then (K(y,y \ e) \ x) * ((x * T(K(y,y \ e),x)) / x) = T(x,x / (K(y,y \ e) \ x)) by (L4). Then (K(y,y \ e) \ x) * K(y,y \ e) = T(x,x / (K(y,y \ e) \ x)) by Proposition 48. Hence we are done by Proposition 24.
Theorem 633 (prov9_d8f9317a21): x * K(y,y \ e) = T(K(y,y \ e) * x,K(y,y \ e))
Proof. We have (K(y,y \ e) \ (K(y,y \ e) * x)) * K(y,y \ e) = T(K(y,y \ e) * x,K(y,y \ e)) by Theorem 632. Hence we are done by Axiom 3.
Theorem 634 (prov9_91d055eda1): (T(y,z) * x) * K(x,y) = x * (K(x,y) * T(K(x,y) \ y,z))
Proof. We have T((x * y) / (x * K(x,y)),z) * (x * K(x,y)) = (x * K(x,y)) * T((x * K(x,y)) \ (x * y),z) by Proposition 61. Then
(L2) T(R(y,x,K(x,y)),z) * (x * K(x,y)) = (x * K(x,y)) * T((x * K(x,y)) \ (x * y),z)
by Theorem 461. We have T(R(y,x,K(x,y)),z) * (x * K(x,y)) = (T(y,z) * x) * K(x,y) by Proposition 65. Then
(L4) (x * K(x,y)) * T((x * K(x,y)) \ (x * y),z) = (T(y,z) * x) * K(x,y)
by (L2). We have (x * K(x,y)) * T((x * K(x,y)) \ (x * y),z) = x * (K(x,y) * T(K(x,y) \ y,z)) by Theorem 486. Hence we are done by (L4).
Theorem 635 (prov9_8ed0c75b70): L(x * T(x \ e,w),y,z) = K(w \ (w / L(x,y,z)),w)
Proof. We have (z * (y * x)) * T((z * (y * x)) \ (z * y),w) = (z * y) * K(w \ (w / ((z * y) \ (z * (y * x)))),w) by Theorem 284. Then
(L2) (z * (y * x)) * T((z * (y * x)) \ (z * y),w) = (z * y) * K(w \ (w / L(x,y,z)),w)
by Definition 4. We have (z * (y * x)) * T((z * (y * x)) \ (z * y),w) = z * ((y * x) * T((y * x) \ y,w)) by Theorem 486. Then (z * y) * K(w \ (w / L(x,y,z)),w) = z * ((y * x) * T((y * x) \ y,w)) by (L2). Then (z * y) * K(w \ (w / L(x,y,z)),w) = z * (y * (x * T(x \ e,w))) by Theorem 98. Hence we are done by Theorem 54.
Theorem 636 (prov9_c1cb332c72): R(x * T(x \ e,w),y,z) = K(w \ (w / R(x,y,z)),w)
Proof. We have K(w \ (w / (((x * y) * z) / (y * z))),w) = (((x * y) * z) * T(((x * y) * z) \ (y * z),w)) / (y * z) by Theorem 287. Then K(w \ (w / R(x,y,z)),w) = (((x * y) * z) * T(((x * y) * z) \ (y * z),w)) / (y * z) by Definition 5. Then
(L3) K(w \ (w / R(x,y,z)),w) = (((x * y) * T((x * y) \ y,w)) * z) / (y * z)
by Theorem 291.
R(x * T(x \ e,w),y,z)
=(((x * T(x \ e,w)) * y) * z) / (y * z) by Definition 5
=K(w \ (w / R(x,y,z)),w) by (L3), Theorem 194

Hence we are done.
Theorem 637 (prov9_bbc9300360): R(x * T(x \ e,R(x,y,z)),y,z) = K(R(x,y,z) \ e,R(x,y,z))
Proof. We have R(x * T(x \ e,R(x,y,z)),y,z) = K(R(x,y,z) \ (R(x,y,z) / R(x,y,z)),R(x,y,z)) by Theorem 636. Hence we are done by Proposition 29.
Theorem 638 (prov9_d56108fc0f): (x \ T(x,y)) \ e = (x \ e) \ (T(x,y) \ e)
Proof. We have T(e / (x \ e),y) * (x \ e) = (x \ e) * T((x \ e) \ e,y) by Proposition 61. Then
(L2) T(e / (x \ e),y) * (x \ e) = x \ T(x,y)
by Theorem 145. We have (T(e / (x \ e),y) * (x \ e)) * ((x \ e) \ ((x \ e) / (x \ T(x,y)))) = ((x \ e) / (x \ T(x,y))) * T(((x \ e) / (x \ T(x,y))) \ ((x \ e) \ ((x \ e) / (x \ T(x,y)))),y) by Theorem 183. Then
(L4) (x \ T(x,y)) * ((x \ e) \ ((x \ e) / (x \ T(x,y)))) = ((x \ e) / (x \ T(x,y))) * T(((x \ e) / (x \ T(x,y))) \ ((x \ e) \ ((x \ e) / (x \ T(x,y)))),y)
by (L2). We have (x \ e) * (((x \ e) / (x \ T(x,y))) * T(((x \ e) / (x \ T(x,y))) \ ((x \ e) \ ((x \ e) / (x \ T(x,y)))),y)) = ((x \ e) * T((x \ e) \ e,y)) * ((x \ e) / (x \ T(x,y))) by Theorem 235. Then (x \ e) * ((x \ T(x,y)) * ((x \ e) \ ((x \ e) / (x \ T(x,y))))) = ((x \ e) * T((x \ e) \ e,y)) * ((x \ e) / (x \ T(x,y))) by (L4). Then
(L7) (x \ e) * ((x \ T(x,y)) * ((x \ e) \ ((x \ e) / (x \ T(x,y))))) = (x \ T(x,y)) * ((x \ e) / (x \ T(x,y)))
by Theorem 145. We have (x \ e) * K(x \ T(x,y),(x \ e) / (x \ T(x,y))) = (x \ T(x,y)) * ((x \ e) / (x \ T(x,y))) by Theorem 459. Then (x \ T(x,y)) * ((x \ e) \ ((x \ e) / (x \ T(x,y)))) = K(x \ T(x,y),(x \ e) / (x \ T(x,y))) by (L7) and Proposition 7. Then
(L10) (x \ T(x,y)) * ((x \ e) \ (T(x,y) \ e)) = K(x \ T(x,y),(x \ e) / (x \ T(x,y)))
by Theorem 374. We have K(x \ T(x,y),T(x,y) \ e) = e by Theorem 582. Then K(x \ T(x,y),(x \ e) / (x \ T(x,y))) = e by Theorem 374. Then (x \ T(x,y)) * ((x \ e) \ (T(x,y) \ e)) = e by (L10). Hence we are done by Proposition 2.
Theorem 639 (prov9_efa69a3b97): K(x,y) = (((y * x) \ y) \ K(x,y)) / (L(x \ e,x,y) \ e)
Proof. We have
(L1) T(L(x \ e,x,y) \ e,y) / (L(x \ e,x,y) \ e) = L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y)
by Theorem 484.
K(x,y)
=L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y) by Theorem 542
=(((y * x) \ y) \ K(x,y)) / (L(x \ e,x,y) \ e) by (L1), Theorem 540

Hence we are done.
Theorem 640 (prov9_73f06ed819): (K(y / x,x / y) * x) / y = e / (e / (x / y))
Proof. We have R(((y / x) * (e / (y / x))) / (y / x),y / x,x) = (((y / x) * (e / (y / x))) * x) / y by Theorem 67. Then ((y / x) * (x / ((y / x) * x))) / (y / x) = (((y / x) * (e / (y / x))) * x) / y by Theorem 485. Then ((y / x) * (x / y)) / (y / x) = (((y / x) * (e / (y / x))) * x) / y by Axiom 6. Then (K(y / x,(y / x) \ e) * x) / y = ((y / x) * (x / y)) / (y / x) by Theorem 250. Then
(L5) (K(y / x,x / y) * x) / y = ((y / x) * (x / y)) / (y / x)
by Theorem 402. We have ((y / x) * T(e / (e / (x / y)),y / x)) / (y / x) = e / (e / (x / y)) by Proposition 48. Then ((y / x) * (x / y)) / (y / x) = e / (e / (x / y)) by Theorem 605. Hence we are done by (L5).
Theorem 641 (prov9_b4f4f476b2): R(x * R(x \ e,w,u),y,z) = R(x,y,z) * R(R(x,y,z) \ e,w,u)
Proof. We have (((x * y) * z) / (y * z)) * R((((x * y) * z) / (y * z)) \ e,w,u) = (((x * y) * z) * R(((x * y) * z) \ (y * z),w,u)) / (y * z) by Theorem 599. Then R(x,y,z) * R((((x * y) * z) / (y * z)) \ e,w,u) = (((x * y) * z) * R(((x * y) * z) \ (y * z),w,u)) / (y * z) by Definition 5. Then (((x * y) * z) * R(((x * y) * z) \ (y * z),w,u)) / (y * z) = R(x,y,z) * R(R(x,y,z) \ e,w,u) by Definition 5. Then (((x * y) * R((x * y) \ y,w,u)) * z) / (y * z) = R(x,y,z) * R(R(x,y,z) \ e,w,u) by Theorem 594. Then
(L5) (((x * R(x \ e,w,u)) * y) * z) / (y * z) = R(x,y,z) * R(R(x,y,z) \ e,w,u)
by Proposition 84. We have R(x * R(x \ e,w,u),y,z) = (((x * R(x \ e,w,u)) * y) * z) / (y * z) by Definition 5. Hence we are done by (L5).
Theorem 642 (prov9_da527ad971): R(x,z,w) * ((y * (R(x,z,w) \ e)) / y) = R(x * ((y * (x \ e)) / y),z,w)
Proof. We have (R(x,z,w) * (z * w)) * ((y * ((R(x,z,w) * (z * w)) \ (z * w))) / y) = (R(x,z,w) * ((y * (R(x,z,w) \ e)) / y)) * (z * w) by Theorem 343. Then ((x * z) * w) * ((y * ((R(x,z,w) * (z * w)) \ (z * w))) / y) = (R(x,z,w) * ((y * (R(x,z,w) \ e)) / y)) * (z * w) by Proposition 54. Then
(L3) ((x * z) * w) * ((y * (((x * z) * w) \ (z * w))) / y) = (R(x,z,w) * ((y * (R(x,z,w) \ e)) / y)) * (z * w)
by Proposition 54. We have ((x * z) * w) * ((y * (((x * z) * w) \ (z * w))) / y) = (((y * (z / (x * z))) / y) * (x * z)) * w by Theorem 608. Then ((x * z) * w) * ((y * (((x * z) * w) \ (z * w))) / y) = ((x * z) * ((y * ((x * z) \ z)) / y)) * w by Theorem 125. Then (R(x,z,w) * ((y * (R(x,z,w) \ e)) / y)) * (z * w) = ((x * z) * ((y * ((x * z) \ z)) / y)) * w by (L3). Then (R(x,z,w) * ((y * (R(x,z,w) \ e)) / y)) * (z * w) = ((x * ((y * (x \ e)) / y)) * z) * w by Theorem 343. Hence we are done by Theorem 64.
Theorem 643 (prov9_59230fbbed): x = L(x,K(x,x \ e),y)
Proof. We have (y * K(x,x \ e)) \ ((y * x) * K(x,x \ e)) = x by Theorem 602. Then
(L2) (y * K(x,x \ e)) \ (y * T(x,x \ e)) = x
by Theorem 406. We have L(x,K(x,x \ e),y) = (y * K(x,x \ e)) \ (y * (K(x,x \ e) * x)) by Definition 4. Then L(x,K(x,x \ e),y) = (y * K(x,x \ e)) \ (y * T(x,x \ e)) by Theorem 246. Hence we are done by (L2).
Theorem 644 (prov9_2cb5457a76): L(e / y,K(y \ e,y),x) = e / y
Proof. We have L(e / y,K(e / y,(e / y) \ e),x) = e / y by Theorem 643. Hence we are done by Theorem 272.
Theorem 645 (prov9_836b2387a3): x / (e / (e / y)) = (x * K(y,y \ e)) / y
Proof. We have R(x / (e / (e / y)),e / (e / y),K(y,y \ e)) = (x * K(y,y \ e)) / ((e / (e / y)) * K(y,y \ e)) by Proposition 55. Then R(x / (e / (e / y)),e / (e / y),K(y,y \ e)) = (x * K(y,y \ e)) / y by Theorem 252. Hence we are done by Theorem 414.
Theorem 646 (prov9_c178d314f0): x / (e / (e / y)) = x * (e / (e / (x \ (x / y))))
Proof. We have x * L(e / y,y,x / y) = (x / y) * (y * (e / y)) by Theorem 53. Then x * (e / (e / (x \ (x / y)))) = (x / y) * (y * (e / y)) by Theorem 396. Then (x / y) * K(y,y \ e) = x * (e / (e / (x \ (x / y)))) by Theorem 250. Hence we are done by Theorem 417.
Theorem 647 (prov9_04c914318a): y * (e / (e / (y \ x))) = y / (e / (e / (x \ y)))
Proof. We have y * (e / (e / (y \ (y / (x \ y))))) = y / (e / (e / (x \ y))) by Theorem 646. Hence we are done by Proposition 24.
Theorem 648 (prov9_8d8578ff3c): x / L(y,y \ e,x) = x * (e / y)
Proof. We have x / (e / (e / ((x * (y \ e)) \ x))) = x * (e / (e / (x \ (x * (y \ e))))) by Theorem 647. Then x / (e / (e / ((x * (y \ e)) \ x))) = x * (e / (e / (y \ e))) by Axiom 3. Then x / L(y,y \ e,x) = x * (e / (e / (y \ e))) by Theorem 616. Hence we are done by Proposition 24.
Theorem 649 (prov9_5f60c30e08): (y * (e / x)) \ y = L(x,x \ e,y)
Proof. We have (y / L(x,x \ e,y)) \ y = L(x,x \ e,y) by Proposition 25. Hence we are done by Theorem 648.
Theorem 650 (prov9_3bac438745): K(x \ (x / L(y,y \ e,x)),z) = K(e / y,z)
Proof. We have
(L3) (z * (x \ (x * (e / y)))) \ ((z \ (z * (x \ (x * (e / y))))) * z) = K(z \ (z * (x \ (x * (e / y)))),z)
by Theorem 2. Then (z * (x \ (x * (e / y)))) \ ((z \ (z * (x \ (x * (e / y))))) * z) = K(x \ (x * (e / y)),z) by Axiom 3. Then
(L5) K(x \ (x * (e / y)),z) = K(z \ (z * (x \ (x * (e / y)))),z)
by (L3). We have x \ (x * (e / y)) = e / y by Axiom 3. Then z \ (z * (x \ (x * (e / y)))) = e / y by Axiom 3. Then K(e / y,z) = K(x \ (x * (e / y)),z) by (L5). Hence we are done by Theorem 648.
Theorem 651 (prov9_6243b173c4): L(x,e / x,y) = L(x,x \ e,y)
Proof. We have L(x,e / x,y) = (y * (e / x)) \ y by Theorem 34. Hence we are done by Theorem 649.
Theorem 652 (prov9_9ea9fe98e0): L(x \ e,x,y) = L(x \ e,(x \ e) \ e,y)
Proof. We have L(x \ e,e / (x \ e),y) = L(x \ e,(x \ e) \ e,y) by Theorem 651. Hence we are done by Proposition 24.
Theorem 653 (prov9_642086269d): L(x \ e,T(x,x \ e),y) = L(x \ e,x,y)
Proof. We have L(x \ e,(x \ e) \ e,y) = L(x \ e,x,y) by Theorem 652. Hence we are done by Proposition 49.
Theorem 654 (prov9_428ef7ef7b): L(x * T(x \ e,y),x \ e,y) = K(e / x,y)
Proof. We have L(x * T(x \ e,y),x \ e,y) = K(y \ (y / L(x,x \ e,y)),y) by Theorem 635. Hence we are done by Theorem 650.
Theorem 655 (prov9_27f924a4c1): x * y = (x * ((y \ e) \ e)) * K(y \ e,y)
Proof. We have ((x * ((y \ e) \ e)) / ((y \ e) \ e)) * y = (x * ((y \ e) \ e)) * K(y \ e,y) by Theorem 424. Hence we are done by Axiom 5.
Theorem 656 (prov9_9b498e750f): x / K(y \ e,y) = x * K(y,y \ e)
Proof. We have ((x * K(y,y \ e)) / K(y,y \ e)) * K(y,y \ e) = x * K(y,y \ e) by Axiom 6. Then ((((x * K(y,y \ e)) / K(y,y \ e)) / K(y \ e,y)) * K(y \ e,y)) * K(y,y \ e) = x * K(y,y \ e) by Axiom 6. Then ((x / K(y \ e,y)) * K(y \ e,y)) * K(y,y \ e) = x * K(y,y \ e) by Axiom 5. Then
(L5) (x * K(y,y \ e)) / K(y,y \ e) = (x / K(y \ e,y)) * K(y \ e,y)
by Proposition 1. We have (x * K(y,y \ e)) / K(y,y \ e) = (x * K(y,y \ e)) * K(y \ e,y) by Theorem 425. Then (x / K(y \ e,y)) * K(y \ e,y) = (x * K(y,y \ e)) * K(y \ e,y) by (L5). Hence we are done by Proposition 10.
Theorem 657 (prov9_b9a17c7e71): (x * K(y,y \ e)) \ x = K(y \ e,y)
Proof. We have (x / K(y \ e,y)) \ x = K(y \ e,y) by Proposition 25. Hence we are done by Theorem 656.
Theorem 658 (prov9_6b7a60c061): T((K(x \ e,x) * y) * K(x,x \ e),K(x \ e,x)) = y
Proof. We have T((K(x \ e,x) * y) / K(x \ e,x),K(x \ e,x)) = y by Theorem 7. Hence we are done by Theorem 656.
Theorem 659 (prov9_4d68ed8bf6): T(y,K(x,x \ e)) * K(x \ e,x) = K(x,x \ e) \ y
Proof. We have
(L1) (T(y,K(x,x \ e)) * K(x \ e,x)) * K(x,x \ e) = T(y,K(x,x \ e))
by Theorem 426. We have (K(x,x \ e) \ y) * K(x,x \ e) = T(y,K(x,x \ e)) by Theorem 632. Hence we are done by (L1) and Proposition 8.
Theorem 660 (prov9_80f3f7df6e): T(y,x) = T(y,x * K(y \ e,y))
Proof. We have T(y,(x * K(y \ e,y)) * K(y,y \ e)) = T(y,x * K(y \ e,y)) by Theorem 626. Hence we are done by Theorem 426.
Theorem 661 (prov9_aa19c283b1): (x * y) \ (x * (e / (e / y))) = K(y \ e,y)
Proof. We have ((x * (e / (e / y))) * K(y,y \ e)) \ (x * (e / (e / y))) = K(y \ e,y) by Theorem 657. Hence we are done by Theorem 415.
Theorem 662 (prov9_0a463fa7a9): (x * (y \ e)) * ((y \ e) \ e) = R(x,y \ e,y)
Proof. We have
(L1) ((x * (y \ e)) * ((y \ e) \ e)) * K(y \ e,y) = (x * (y \ e)) * y
by Theorem 655. We have R(x,y \ e,y) * K(y \ e,y) = (x * (y \ e)) * y by Theorem 469. Hence we are done by (L1) and Proposition 8.
Theorem 663 (prov9_f97e19b031): L(K(y \ e,y),y,x) = K(y \ e,y)
Proof. We have L(y \ (e / (e / y)),y,x) = (x * y) \ (x * (e / (e / y))) by Proposition 53. Then L(K(y \ e,y),y,x) = (x * y) \ (x * (e / (e / y))) by Theorem 257. Hence we are done by Theorem 661.
Theorem 664 (prov9_0d48cfb129): K(y \ e,y) = K((x * y) \ x,x \ (x * y))
Proof. We have x * (x \ ((x * y) * K((x * y) \ x,x \ (x * y)))) = (x * y) * K((x * y) \ x,x \ (x * y)) by Axiom 4. Then
(L3) x * (y * (y \ (x \ ((x * y) * K((x * y) \ x,x \ (x * y)))))) = (x * y) * K((x * y) \ x,x \ (x * y))
by Axiom 4. We have (x * y) * L(y \ (x \ ((x * y) * K((x * y) \ x,x \ (x * y)))),y,x) = x * (y * (y \ (x \ ((x * y) * K((x * y) \ x,x \ (x * y)))))) by Proposition 52. Then L(y \ (x \ ((x * y) * K((x * y) \ x,x \ (x * y)))),y,x) = K((x * y) \ x,x \ (x * y)) by (L3) and Proposition 7. Then L(y \ (e / (e / (x \ (x * y)))),y,x) = K((x * y) \ x,x \ (x * y)) by Theorem 615. Then L(y \ (e / (e / y)),y,x) = K((x * y) \ x,x \ (x * y)) by Axiom 3. Then L(K(y \ e,y),y,x) = K((x * y) \ x,x \ (x * y)) by Theorem 257. Hence we are done by Theorem 663.
Theorem 665 (prov9_f6864f8738): K((x * y) \ x,y) = K(y \ e,y)
Proof. We have K((x * y) \ x,x \ (x * y)) = K(y \ e,y) by Theorem 664. Hence we are done by Axiom 3.
Theorem 666 (prov9_18f6633e12): K(y \ x,x \ y) = K((x \ y) \ e,x \ y)
Proof. We have K((x * (x \ y)) \ x,x \ y) = K((x \ y) \ e,x \ y) by Theorem 665. Hence we are done by Axiom 4.
Theorem 667 (prov9_ca72d6e822): K(x / (y * x),(x / (y * x)) \ e) = K(y \ e,y)
Proof. We have K(x / (y * x),(x / (y * x)) \ e) = K(x / (y * x),(y * x) / x) by Theorem 402. Then K(x / (y * x),(x / (y * x)) \ e) = K(x / (y * x),y) by Axiom 5. Hence we are done by Theorem 428.
Theorem 668 (prov9_98e55f3277): (y * x) * K(y \ e,y) = y * (x * K(y \ e,y))
Proof. We have (y * x) / L(K(y,y \ e),x / K(y,y \ e),y) = y * (x / K(y,y \ e)) by Theorem 472. Then (y * x) / K(y,y \ e) = y * (x / K(y,y \ e)) by Theorem 625. Then (y * x) * K(y \ e,y) = y * (x / K(y,y \ e)) by Theorem 425. Hence we are done by Theorem 425.
Theorem 669 (prov9_e7fd6f41e9): R(y,x,K(y \ e,y)) = y
Proof. We have (y * x) * K(y \ e,y) = y * (x * K(y \ e,y)) by Theorem 668. Hence we are done by Theorem 64.
Theorem 670 (prov9_2adf389667): ((y / (x * y)) \ e) \ e = R(x \ e,x,y)
Proof. We have K(y / (x * y),(y / (x * y)) \ e) * (y / (x * y)) = ((y / (x * y)) \ e) \ e by Theorem 552. Then
(L2) K(x \ e,x) * (y / (x * y)) = ((y / (x * y)) \ e) \ e
by Theorem 667. We have K(x \ e,x) * (y / (x * y)) = (y / (x * y)) * K(x \ e,x) by Theorem 373. Then K(x \ e,x) * (y / (x * y)) = R(x \ e,x,y) by Theorem 427. Hence we are done by (L2).
Theorem 671 (prov9_31add57d56): R(x \ e,x,y) = (R(e / x,x,y) \ e) \ e
Proof. We have R(x \ e,x,y) = ((y / (x * y)) \ e) \ e by Theorem 670. Hence we are done by Proposition 79.
Theorem 672 (prov9_98cb76e388): K(e / (e / y),x) = (y * K(y,x)) / y
Proof. We have L((e / y) * T((e / y) \ e,x),(e / y) \ e,x) = K(e / (e / y),x) by Theorem 654. Then L((e / y) * T((e / y) \ e,x),y,x) = K(e / (e / y),x) by Proposition 25. Then
(L3) L(T(y,x) / y,y,x) = K(e / (e / y),x)
by Theorem 149. We have L((x \ (y * x)) / y,y,x) = (y * ((x * y) \ (y * x))) / y by Proposition 75. Then L((x \ (y * x)) / y,y,x) = (y * K(y,x)) / y by Definition 2. Then L(T(y,x) / y,y,x) = (y * K(y,x)) / y by Definition 3. Hence we are done by (L3).
Theorem 673 (prov9_429e74aece): T(K(e / (e / x),y),x) = K(x,y)
Proof. We have T((x * K(x,y)) / x,x) = K(x,y) by Theorem 7. Hence we are done by Theorem 672.
Theorem 674 (prov9_33810b212c): K((y \ e) \ e,x) = T(K(y,x),y)
Proof. We have T(K(e / (e / T(y,y \ e)),x),T(y,y \ e)) = K(T(y,y \ e),x) by Theorem 673. Then T(K(e / (y \ e),x),T(y,y \ e)) = K(T(y,y \ e),x) by Theorem 15. Then
(L3) T(K(y,x),T(y,y \ e)) = K(T(y,y \ e),x)
by Proposition 24. We have T(K(y,x),T(y,y \ e)) = T(K(y,x),y) by Theorem 585. Then K(T(y,y \ e),x) = T(K(y,x),y) by (L3). Hence we are done by Proposition 49.
Theorem 675 (prov9_eb8faddb45): (((y * x) \ y) \ e) \ (((y * x) \ y) \ K(x,y)) = K((x \ e) \ e,y)
Proof. We have L(e / T(x,x \ e),T(x,x \ e),y) \ e = e / ((y * T(x,x \ e)) \ y) by Theorem 617. Then
(L2) L(e / T(x,x \ e),T(x,x \ e),y) \ e = e / L(T(x,x \ e) \ e,T(x,x \ e),y)
by Proposition 78. We have (e / L(T(x,x \ e) \ e,T(x,x \ e),y)) \ e = L(T(x,x \ e) \ e,T(x,x \ e),y) by Proposition 25. Then (L(e / T(x,x \ e),T(x,x \ e),y) \ e) \ e = L(T(x,x \ e) \ e,T(x,x \ e),y) by (L2). Then (L(x \ e,T(x,x \ e),y) \ e) \ e = L(T(x,x \ e) \ e,T(x,x \ e),y) by Theorem 15. Then L(T(x \ e,x),T(x,x \ e),y) = (L(x \ e,T(x,x \ e),y) \ e) \ e by Theorem 208. Then
(L7) L(T(x \ e,x),T(x,x \ e),y) = (L(x \ e,x,y) \ e) \ e
by Theorem 653. We have y / L(T(x,x \ e) \ e,T(x,x \ e),y) = y * T(x,x \ e) by Theorem 33. Then y / L(T(x \ e,x),T(x,x \ e),y) = y * T(x,x \ e) by Theorem 208. Then
(L10) y / ((L(x \ e,x,y) \ e) \ e) = y * T(x,x \ e)
by (L7). We have T(L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y),L(x \ e,x,y) \ e) = K(y \ (y / T(L(x \ e,x,y),L(x \ e,x,y) \ e)),y) by Theorem 296. Then T(K(x,y),L(x \ e,x,y) \ e) = K(y \ (y / T(L(x \ e,x,y),L(x \ e,x,y) \ e)),y) by Theorem 542. Then K(y \ (y / ((L(x \ e,x,y) \ e) \ e)),y) = T(K(x,y),L(x \ e,x,y) \ e) by Proposition 49. Then T(K(x,y),L(x \ e,x,y) \ e) = K(y \ (y * T(x,x \ e)),y) by (L10). Then T(K(x,y),L(x \ e,x,y) \ e) = K(T(x,x \ e),y) by Axiom 3. Then T(K(x,y),L(x \ e,x,y) \ e) = K((x \ e) \ e,y) by Proposition 49. Then
(L17) T(K(x,y),((y * x) \ y) \ e) = K((x \ e) \ e,y)
by Proposition 78. We have (((y * x) \ y) \ K(x,y)) / (L(x \ e,x,y) \ e) = K(x,y) by Theorem 639. Then
(L19) (((y * x) \ y) \ K(x,y)) / (((y * x) \ y) \ e) = K(x,y)
by Proposition 78. We have T((((y * x) \ y) \ K(x,y)) / (((y * x) \ y) \ e),((y * x) \ y) \ e) = (((y * x) \ y) \ e) \ (((y * x) \ y) \ K(x,y)) by Proposition 47. Then T(K(x,y),((y * x) \ y) \ e) = (((y * x) \ y) \ e) \ (((y * x) \ y) \ K(x,y)) by (L19). Hence we are done by (L17).
Theorem 676 (prov9_d24bb05602): T(x,x \ e) * y = K(x,x \ e) * (x * y)
Proof. We have T(x,x \ e) * (x \ (x * y)) = K(x,x \ e) * (x * y) by Theorem 431. Hence we are done by Axiom 3.
Theorem 677 (prov9_dba33dd287): ((x \ e) \ e) * (x \ y) = K(x,x \ e) * y
Proof. We have T(x,x \ e) * (x \ y) = K(x,x \ e) * y by Theorem 431. Hence we are done by Proposition 49.
Theorem 678 (prov9_b372f76332): L(y,e / x,K(e / x,x)) = y
Proof. We have L(y,e / x,K(e / x,(e / x) \ e)) = y by Theorem 432. Hence we are done by Proposition 25.
Theorem 679 (prov9_b24f76f820): ((x \ e) \ e) * y = K(x,x \ e) * (x * y)
Proof. We have (K(x,x \ e) * x) * L(y,x,K(x,x \ e)) = K(x,x \ e) * (x * y) by Proposition 52. Then ((x \ e) \ e) * L(y,x,K(x,x \ e)) = K(x,x \ e) * (x * y) by Theorem 552. Hence we are done by Theorem 432.
Theorem 680 (prov9_e8792b4a4d): x \ y = K(x,x \ e) * (((x \ e) \ e) \ y)
Proof. We have T(x,x \ e) \ (K(T(x,x \ e),T(x,x \ e) \ e) * y) = K(T(x,x \ e),T(x,x \ e) \ e) * (T(x,x \ e) \ y) by Theorem 586. Then
(L2) T(x,x \ e) \ (K(x,x \ e) * y) = K(T(x,x \ e),T(x,x \ e) \ e) * (T(x,x \ e) \ y)
by Theorem 592. We have T(x,x \ e) \ (K(x,x \ e) * y) = L(x \ y,x,K(x,x \ e)) by Theorem 554. Then K(T(x,x \ e),T(x,x \ e) \ e) * (T(x,x \ e) \ y) = L(x \ y,x,K(x,x \ e)) by (L2). Then K(x,x \ e) * (T(x,x \ e) \ y) = L(x \ y,x,K(x,x \ e)) by Theorem 592. Then K(x,x \ e) * (((x \ e) \ e) \ y) = L(x \ y,x,K(x,x \ e)) by Proposition 49. Hence we are done by Theorem 432.
Theorem 681 (prov9_b31f05ccde): L(y,e / (e / x),K(x,x \ e)) = y
Proof. We have L(y,e / (e / x),K(e / (e / x),e / x)) = y by Theorem 678. Hence we are done by Theorem 566.
Theorem 682 (prov9_c8d61c7422): (x \ e) * y = K(x \ e,x) * ((e / x) * y)
Proof. We have (x \ e) * L(y,e / x,K(x \ e,x)) = K(x \ e,x) * ((e / x) * y) by Theorem 259. Hence we are done by Theorem 433.
Theorem 683 (prov9_827e1ae701): L(x,y,y \ e) = L(x,y,e / y)
Proof. We have (e / y) * (y * x) = L(x,y,e / y) by Proposition 67. Then
(L2) L(L(x,y,y \ e),K(y \ e,y),e / y) = L(x,y,e / y)
by Theorem 618. We have L(L(x,y,y \ e),e / y,K(e / y,(e / y) \ e)) = L(L(x,y,y \ e),K(y \ e,y),e / y) by Theorem 588. Then L(L(x,y,y \ e),e / y,K(y \ e,y)) = L(L(x,y,y \ e),K(y \ e,y),e / y) by Theorem 272. Then L(x,y,e / y) = L(L(x,y,y \ e),e / y,K(y \ e,y)) by (L2). Hence we are done by Theorem 433.
Theorem 684 (prov9_46aae03fb4): x * y = (e / (e / x)) * (K(x,x \ e) * y)
Proof. We have L(y,K(e / (e / x),(e / (e / x)) \ e),e / (e / x)) = L(y,e / (e / x),K(e / (e / x),(e / (e / x)) \ e)) by Theorem 587. Then L(y,K(e / (e / x),e / x),e / (e / x)) = L(y,e / (e / x),K(e / (e / x),(e / (e / x)) \ e)) by Proposition 25. Then L(y,e / (e / x),K(e / (e / x),(e / (e / x)) \ e)) = L(y,K(x,x \ e),e / (e / x)) by Theorem 566. Then
(L4) y = L(y,K(x,x \ e),e / (e / x))
by Theorem 432. We have ((e / (e / x)) * K(x,x \ e)) * L(y,K(x,x \ e),e / (e / x)) = (e / (e / x)) * (K(x,x \ e) * y) by Proposition 52. Then x * L(y,K(x,x \ e),e / (e / x)) = (e / (e / x)) * (K(x,x \ e) * y) by Theorem 252. Hence we are done by (L4).
Theorem 685 (prov9_a21ffdb56f): x * y = K(x,x \ e) * ((e / (e / x)) * y)
Proof. We have (K(x,x \ e) * (e / (e / x))) * L(y,e / (e / x),K(x,x \ e)) = K(x,x \ e) * ((e / (e / x)) * y) by Proposition 52. Then x * L(y,e / (e / x),K(x,x \ e)) = K(x,x \ e) * ((e / (e / x)) * y) by Theorem 557. Hence we are done by Theorem 681.
Theorem 686 (prov9_af8f59c8e4): (e / (e / x)) \ y = x \ (K(x,x \ e) * y)
Proof. We have L((e / (e / x)) \ y,e / (e / x),K(x,x \ e)) = (K(x,x \ e) * (e / (e / x))) \ (K(x,x \ e) * y) by Proposition 53. Then L((e / (e / x)) \ y,e / (e / x),K(x,x \ e)) = x \ (K(x,x \ e) * y) by Theorem 557. Hence we are done by Theorem 681.
Theorem 687 (prov9_eb97b79660): K(x,x \ e) = K((y / (x * y)) \ e,y / (x * y))
Proof. We have y * T(y \ (x * y),y / (x * y)) = T(x,y / (x * y)) * y by Theorem 466. Then K((y / (x * y)) \ e,y / (x * y)) * (x * y) = T(x,y / (x * y)) * y by Theorem 202. Then
(L4) K((y / (x * y)) \ e,y / (x * y)) * (x * y) = T(x,x \ e) * y
by Theorem 362. We have K(x,x \ e) * (x * y) = T(x,x \ e) * y by Theorem 676. Hence we are done by (L4) and Proposition 8.
Theorem 688 (prov9_c1e14280a4): K(y,y \ e) * (T(y,x) \ e) = y * (T(y,x) \ (e / y))
Proof. We have ((y \ e) \ e) \ (((y \ e) \ e) * (T(y,x) \ e)) = T(y,x) \ e by Axiom 3. Then
(L3) ((y \ e) \ e) \ (y * (y \ (((y \ e) \ e) * (T(y,x) \ e)))) = T(y,x) \ e
by Axiom 4. We have ((y \ e) \ e) * (((y \ e) \ e) \ (y * (y \ (((y \ e) \ e) * (T(y,x) \ e))))) = y * (y \ (((y \ e) \ e) * (T(y,x) \ e))) by Axiom 4. Then ((y \ e) \ e) * (y * (y \ (((y \ e) \ e) \ (y * (y \ (((y \ e) \ e) * (T(y,x) \ e))))))) = y * (y \ (((y \ e) \ e) * (T(y,x) \ e))) by Axiom 4. Then
(L6) ((y \ e) \ e) * (y * (y \ (T(y,x) \ e))) = y * (y \ (((y \ e) \ e) * (T(y,x) \ e)))
by (L3). We have ((y \ e) \ e) * (y * (y \ (T(y,x) \ e))) = y * (((y \ e) \ e) * (y \ (T(y,x) \ e))) by Theorem 574. Then y * (y \ (((y \ e) \ e) * (T(y,x) \ e))) = y * (((y \ e) \ e) * (y \ (T(y,x) \ e))) by (L6). Then y \ (((y \ e) \ e) * (T(y,x) \ e)) = ((y \ e) \ e) * (y \ (T(y,x) \ e)) by Proposition 9. Then
(L9) y \ (T(y,x) \ ((y \ e) \ e)) = ((y \ e) \ e) * (y \ (T(y,x) \ e))
by Theorem 593. We have T(y,x) \ (y \ ((y \ e) \ e)) = y \ (T(y,x) \ ((y \ e) \ e)) by Theorem 526. Then
(L11) T(y,x) \ K(y,e / y) = y \ (T(y,x) \ ((y \ e) \ e))
by Theorem 559. We have T(y,x) \ K(y,e / y) = y * (T(y,x) \ (e / y)) by Theorem 523. Then y \ (T(y,x) \ ((y \ e) \ e)) = y * (T(y,x) \ (e / y)) by (L11). Then ((y \ e) \ e) * (y \ (T(y,x) \ e)) = y * (T(y,x) \ (e / y)) by (L9). Hence we are done by Theorem 677.
Theorem 689 (prov9_59f78f3877): K(x,x \ e) \ (x * y) = (e / (e / x)) * y
Proof. We have K(x,x \ e) * ((e / (e / x)) * y) = x * y by Theorem 685. Hence we are done by Proposition 2.
Theorem 690 (prov9_ef46b82306): (e / (e / x)) \ y = K(x,x \ e) * (x \ y)
Proof. We have x \ (K(x,x \ e) * y) = K(x,x \ e) * (x \ y) by Theorem 586. Hence we are done by Theorem 686.
Theorem 691 (prov9_07ff23e70d): (e / (e / y)) * (x * K(y,y \ e)) = T(y * x,K(y,y \ e))
Proof. We have T(y * x,K(y,y \ e)) = K(y,y \ e) \ ((y * x) * K(y,y \ e)) by Definition 3. Then T(y * x,K(y,y \ e)) = K(y,y \ e) \ (y * (x * K(y,y \ e))) by Theorem 623. Hence we are done by Theorem 689.
Theorem 692 (prov9_ba2553c17f): (e / x) \ (K(x,x \ e) * y) = (x \ e) \ y
Proof. We have
(L1) K(x,x \ e) * ((e / x) \ (K(x,x \ e) * y)) = (x \ e) \ (K(x,x \ e) * y)
by Theorem 440. We have K(x,x \ e) * ((x \ e) \ y) = (x \ e) \ (K(x,x \ e) * y) by Theorem 600. Hence we are done by (L1) and Proposition 7.
Theorem 693 (prov9_71e7686756): y = K(x \ e,x) * (K(x,x \ e) * y)
Proof. We have (e / x) \ ((x \ e) * (K(x,x \ e) * y)) = K(x \ e,x) * (K(x,x \ e) * y) by Theorem 438. Hence we are done by Theorem 441.
Theorem 694 (prov9_d89bb51a7d): K(x,x \ e) \ y = K(x \ e,x) * y
Proof. We have K(x,x \ e) * (K(x \ e,x) * y) = y by Theorem 442. Hence we are done by Proposition 2.
Theorem 695 (prov9_ea25f6ee32): K(x \ e,x) * ((e / (e / x)) \ y) = x \ y
Proof. We have
(L1) K(x,x \ e) * (K(x \ e,x) * ((e / (e / x)) \ y)) = (e / (e / x)) \ y
by Theorem 442. We have K(x,x \ e) * (x \ y) = (e / (e / x)) \ y by Theorem 690. Hence we are done by (L1) and Proposition 7.
Theorem 696 (prov9_86e5a2f04b): K(x,x \ e) * y = K(x \ e,x) \ y
Proof. We have K(x,x \ e) * (K(x \ e,x) * (K(x \ e,x) \ y)) = K(x \ e,x) \ y by Theorem 442. Hence we are done by Axiom 4.
Theorem 697 (prov9_d5c3daeef1): K(x,x \ e) * ((x \ e) * y) = (e / x) * y
Proof. We have K(x,x \ e) * (K(x \ e,x) * ((e / x) * y)) = (e / x) * y by Theorem 442. Hence we are done by Theorem 682.
Theorem 698 (prov9_8ddac5b807): T(x,y) = T(x,K(x,x \ e) * y)
Proof. We have T(x,K(x \ e,x) * (K(x,x \ e) * y)) = T(x,K(x,x \ e) * y) by Theorem 630. Hence we are done by Theorem 693.
Theorem 699 (prov9_885c768ddd): T(y,K(x,x \ e)) * K(x \ e,x) = K(x \ e,x) * y
Proof. We have T(y,K(x,x \ e)) * K(x \ e,x) = K(x,x \ e) \ y by Theorem 659. Hence we are done by Theorem 694.
Theorem 700 (prov9_8dcc7b2753): T((e / y) * x,K(y,y \ e)) = ((y \ e) * x) * K(y,y \ e)
Proof. We have T(K(y,y \ e) * ((y \ e) * x),K(y,y \ e)) = ((y \ e) * x) * K(y,y \ e) by Theorem 633. Hence we are done by Theorem 697.
Theorem 701 (prov9_b344f00346): T(T(y,K(x,x \ e)),K(x \ e,x)) = y
Proof. We have K(x \ e,x) * y = T(y,K(x,x \ e)) * K(x \ e,x) by Theorem 699. Hence we are done by Theorem 11.
Theorem 702 (prov9_8ba7cfda25): R(x \ e,x,y) \ y = ((x \ e) \ e) * y
Proof. We have
(L1) K(y / (x * y),(y / (x * y)) \ e) * ((((y / (x * y)) \ e) \ e) \ y) = (y / (x * y)) \ y
by Theorem 680. We have K(y / (x * y),(y / (x * y)) \ e) * (K(y / (x * y),(y / (x * y)) \ e) \ (x * y)) = (y / (x * y)) \ y by Theorem 19. Then (((y / (x * y)) \ e) \ e) \ y = K(y / (x * y),(y / (x * y)) \ e) \ (x * y) by (L1) and Proposition 7. Then K(x \ e,x) \ (x * y) = (((y / (x * y)) \ e) \ e) \ y by Theorem 667. Then
(L5) K(x \ e,x) \ (x * y) = R(x \ e,x,y) \ y
by Theorem 670. We have K(x,x \ e) * (x * y) = ((x \ e) \ e) * y by Theorem 679. Then K(x \ e,x) \ (x * y) = ((x \ e) \ e) * y by Theorem 696. Hence we are done by (L5).
Theorem 703 (prov9_4313b3137c): y / (((x \ e) \ e) * y) = R(x \ e,x,y)
Proof. We have y / (R(x \ e,x,y) \ y) = R(x \ e,x,y) by Proposition 24. Hence we are done by Theorem 702.
Theorem 704 (prov9_b42d7a89da): R(x \ e,(x \ e) \ e,y) = R(x \ e,x,y)
Proof. We have R(x \ e,(x \ e) \ e,y) = y / (((x \ e) \ e) * y) by Proposition 66. Hence we are done by Theorem 703.
Theorem 705 (prov9_c00d58d4c3): y * T(x,(y \ e) \ e) = T(x * y,K(y,y \ e))
Proof. We have (((y \ e) \ e) / y) \ (L(x,y,((y \ e) \ e) / y) * ((y \ e) \ e)) = y * T(x,(y \ e) \ e) by Theorem 491. Then K(y,y \ e) \ (L(x,y,((y \ e) \ e) / y) * ((y \ e) \ e)) = y * T(x,(y \ e) \ e) by Theorem 247. Then K(y,y \ e) \ (L(x,y,K(y,y \ e)) * ((y \ e) \ e)) = y * T(x,(y \ e) \ e) by Theorem 247. Then
(L4) K(y,y \ e) \ (x * ((y \ e) \ e)) = y * T(x,(y \ e) \ e)
by Theorem 432. We have T(x * y,K(y,y \ e)) = K(y,y \ e) \ ((x * y) * K(y,y \ e)) by Definition 3. Then T(x * y,K(y,y \ e)) = K(y,y \ e) \ (x * ((y \ e) \ e)) by Theorem 407. Hence we are done by (L4).
Theorem 706 (prov9_03f3ae16be): T(x,(y \ e) \ e) = T(T(x,y),K(y,y \ e))
Proof. We have (e / (e / y)) * (K(y,y \ e) * T(T(x,y),K(y,y \ e))) = y * T(T(x,y),K(y,y \ e)) by Theorem 684. Then (e / (e / y)) * (T(x,y) * K(y,y \ e)) = y * T(T(x,y),K(y,y \ e)) by Proposition 46. Then
(L3) T(y * T(x,y),K(y,y \ e)) = y * T(T(x,y),K(y,y \ e))
by Theorem 691. We have T(((y * T(x,y)) / y) * y,K(y,y \ e)) = y * T((y * T(x,y)) / y,(y \ e) \ e) by Theorem 705. Then T(y * T(x,y),K(y,y \ e)) = y * T((y * T(x,y)) / y,(y \ e) \ e) by Axiom 6. Then y * T((y * T(x,y)) / y,(y \ e) \ e) = y * T(T(x,y),K(y,y \ e)) by (L3). Then T((y * T(x,y)) / y,(y \ e) \ e) = T(T(x,y),K(y,y \ e)) by Proposition 9. Hence we are done by Proposition 48.
Theorem 707 (prov9_ab0c79e0bc): T(T(x,y),K(y,y \ e)) = T(x,T(y,y \ e))
Proof. We have T(T(x,y),K(y,y \ e)) = T(x,(y \ e) \ e) by Theorem 706. Hence we are done by Proposition 49.
Theorem 708 (prov9_217b55f6cc): T(T(x,(y \ e) \ e),K(y \ e,y)) = T(x,y)
Proof. We have T(T(T(x,y),K(y,y \ e)),K(y \ e,y)) = T(x,y) by Theorem 701. Hence we are done by Theorem 706.
Theorem 709 (prov9_9700fef64a): (x \ e) \ ((x \ y) / y) = x * ((x \ y) / y)
Proof. We have K(x,x \ e) * ((e / x) \ ((x \ y) / y)) = ((x \ y) / y) * T(((x \ y) / y) \ ((e / x) \ ((x \ y) / y)),e / x) by Theorem 563. Then K(x,x \ e) * (((x \ y) / y) * x) = ((x \ y) / y) * T(((x \ y) / y) \ ((e / x) \ ((x \ y) / y)),e / x) by Theorem 543. Then ((x \ y) / y) * T(((x \ y) / y) \ (((x \ y) / y) * x),e / x) = K(x,x \ e) * (((x \ y) / y) * x) by Theorem 543. Then ((x \ y) / y) * T(x,e / x) = K(x,x \ e) * (((x \ y) / y) * x) by Axiom 3. Then
(L7) K(x,x \ e) * (((x \ y) / y) * x) = ((x \ y) / y) * T(x,x \ e)
by Theorem 211. We have ((x \ y) / y) * T(x,x \ e) = x * ((x \ y) / y) by Theorem 606. Then
(L9) K(x,x \ e) * (((x \ y) / y) * x) = x * ((x \ y) / y)
by (L7). We have K(x,x \ e) * ((e / x) \ ((x \ y) / y)) = (x \ e) \ ((x \ y) / y) by Theorem 440. Then K(x,x \ e) * (((x \ y) / y) * x) = (x \ e) \ ((x \ y) / y) by Theorem 543. Hence we are done by (L9).
Theorem 710 (prov9_0540fa15d7): T(e / (e / (y / (x * y))),x * y) = (x * y) \ (K(x,x \ e) * y)
Proof. We have K((y / (x * y)) \ e,y / (x * y)) = K((x * y) / y,y / (x * y)) by Theorem 429. Then
(L2) K(x,x \ e) = K((x * y) / y,y / (x * y))
by Theorem 687. We have (K((x * y) / y,y / (x * y)) * y) / (x * y) = e / (e / (y / (x * y))) by Theorem 640. Then (K((x * y) / y,y / (x * y)) * y) / (x * y) = e / (e / R(e / x,x,y)) by Proposition 79. Then
(L5) (K(x,x \ e) * y) / (x * y) = e / (e / R(e / x,x,y))
by (L2). We have R(e / (e / (e / x)),x,y) = (((e / (e / (e / x))) * x) * y) / (x * y) by Definition 5. Then R(e / (e / (e / x)),x,y) = (K(x,x \ e) * y) / (x * y) by Theorem 555. Then
(L8) e / (e / R(e / x,x,y)) = R(e / (e / (e / x)),x,y)
by (L5). We have
(L9) (((x * y) \ y) / (y / (x * y))) / (e / (y / (x * y))) = T(e / (e / (y / (x * y))),x * y)
by Theorem 502. Then (((x * y) \ y) / (y / (x * y))) / (e / (y / (x * y))) = T(e / (e / R(e / x,x,y)),x * y) by Proposition 79. Then T(e / (e / R(e / x,x,y)),x * y) = T(e / (e / (y / (x * y))),x * y) by (L9). Then
(L12) T(R(e / (e / (e / x)),x,y),x * y) = T(e / (e / (y / (x * y))),x * y)
by (L8). We have R(T(e / (e / (e / x)),x * y),x,y) = T(R(e / (e / (e / x)),x,y),x * y) by Axiom 9. Then
(L14) R(T(e / (e / (e / x)),x * y),x,y) = T(e / (e / (y / (x * y))),x * y)
by (L12). We have R(T((e * K(x,x \ e)) / x,x * y),x,y) = (x * y) \ ((e * K(x,x \ e)) * y) by Theorem 488. Then R(T(e / (e / (e / x)),x * y),x,y) = (x * y) \ ((e * K(x,x \ e)) * y) by Theorem 645. Then T(e / (e / (y / (x * y))),x * y) = (x * y) \ ((e * K(x,x \ e)) * y) by (L14). Hence we are done by Axiom 1.
Theorem 711 (prov9_c119e3c31f): K(T(x,y),x \ e) = K(T(x,y),T(x,y) \ e)
Proof. We have T(x,y) * ((T(x,y) * (T(x,y) \ e)) / T(x,y)) = T(x * ((T(x,y) * (x \ e)) / T(x,y)),y) by Theorem 345. Then
(L2) T(x,y) * (e / T(x,y)) = T(x * ((T(x,y) * (x \ e)) / T(x,y)),y)
by Axiom 4. We have K(T(x,y),T(x,y) \ e) = T(x,y) * (e / T(x,y)) by Theorem 250. Then K(T(x,y),T(x,y) \ e) = T(x * ((T(x,y) * (x \ e)) / T(x,y)),y) by (L2). Then T(x * ((x \ T(x,y)) / T(x,y)),y) = K(T(x,y),T(x,y) \ e) by Theorem 146. Then T(a(T(x,y),e / x,x),y) = K(T(x,y),T(x,y) \ e) by Theorem 399. Hence we are done by Theorem 445.
Theorem 712 (prov9_8f62d3b3b9): T(e / (e / x),y) \ T(x,y) = K(T(x,y),T(x,y) \ e)
Proof. We have T(e / (e / x),y) \ T(x,y) = K(T(x,y),x \ e) by Theorem 444. Hence we are done by Theorem 711.
Theorem 713 (prov9_e6c99ecf71): e / (e / (y \ x)) = T(e / (e / (x / y)),y)
Proof. We have e / (e / T(x / y,y)) = T(e / (e / (x / y)),y) by Theorem 447. Hence we are done by Proposition 47.
Theorem 714 (prov9_3809f782cb): (x * y) * (e / (e / ((x * y) \ y))) = K(x,x \ e) * y
Proof. We have T(e / (e / (y / (x * y))),x * y) = (x * y) \ (K(x,x \ e) * y) by Theorem 710. Then
(L2) e / (e / ((x * y) \ y)) = (x * y) \ (K(x,x \ e) * y)
by Theorem 713. We have (x * y) * ((x * y) \ (K(x,x \ e) * y)) = K(x,x \ e) * y by Axiom 4. Hence we are done by (L2).
Theorem 715 (prov9_796c95a111): K(x \ e,x) * (e / T(x,y)) = x \ (x / T(x,y))
Proof. We have ((y * (e / (e / T(x,y)))) / y) \ ((e / T(x,y)) * T((y * (e / (e / T(x,y)))) / y,e / T(x,y))) = e / T(x,y) by Theorem 454. Then ((y * (e / (e / T(x,y)))) / y) \ ((e / T(x,y)) * ((y * ((e / T(x,y)) \ e)) / y)) = e / T(x,y) by Theorem 94. Then ((y * (e / (e / T(x,y)))) / y) \ ((e / T(x,y)) * ((y * T(x,y)) / y)) = e / T(x,y) by Proposition 25. Then ((y * (e / (e / T(x,y)))) / y) \ ((e / T(x,y)) * x) = e / T(x,y) by Proposition 48. Then ((y * (e / (e / T(x,y)))) / y) \ (x / T(x,y)) = e / T(x,y) by Theorem 550. Then ((y * T(e / (e / x),y)) / y) \ (x / T(x,y)) = e / T(x,y) by Theorem 447. Then
(L7) (e / (e / x)) \ (x / T(x,y)) = e / T(x,y)
by Proposition 48. We have K(x \ e,x) * ((e / (e / x)) \ (x / T(x,y))) = x \ (x / T(x,y)) by Theorem 695. Hence we are done by (L7).
Theorem 716 (prov9_ba3f6e9222): K(x \ e,x) = T(K(x \ e,x),y)
Proof. We have T(e / T(x,y),x) * T(x,y) = T(x,y) * T(T(x,y) \ e,x) by Proposition 61. Then (x \ (x / T(x,y))) * T(x,y) = T(x,y) * T(T(x,y) \ e,x) by Theorem 551. Then T(x * T(x \ e,x),y) = (x \ (x / T(x,y))) * T(x,y) by Theorem 576. Then (x \ (x / T(x,y))) * T(x,y) = T((x \ e) * x,y) by Proposition 46. Then
(L8) (x \ (x / T(x,y))) * T(x,y) = T(K(x \ e,x),y)
by Proposition 76. We have (x \ (x / T(x,y))) / (e / T(x,y)) = (x \ (x / T(x,y))) * T(x,y) by Theorem 604. Then T(e / T(x,y),x) / (e / T(x,y)) = (x \ (x / T(x,y))) * T(x,y) by Theorem 551. Then
(L9) T(e / T(x,y),x) / (e / T(x,y)) = T(K(x \ e,x),y)
by (L8). We have K(x \ (x / (e / (e / T(x,y)))),x) = T(e / T(x,y),x) / (e / T(x,y)) by Theorem 199. Then
(L11) K(x \ (x / (e / (e / T(x,y)))),x) = T(K(x \ e,x),y)
by (L9). We have T(e / T(x,y),x) = K(x \ (x / (e / (e / T(x,y)))),x) * (e / T(x,y)) by Theorem 201. Then x \ (x / T(x,y)) = K(x \ (x / (e / (e / T(x,y)))),x) * (e / T(x,y)) by Theorem 551. Then
(L14) T(K(x \ e,x),y) * (e / T(x,y)) = x \ (x / T(x,y))
by (L11). We have K(x \ e,x) * (e / T(x,y)) = x \ (x / T(x,y)) by Theorem 715. Hence we are done by (L14) and Proposition 8.
Theorem 717 (prov9_65de26e67f): T(y,K(x \ e,x)) = y
Proof. We have T(K(x \ e,x),y) = K(x \ e,x) by Theorem 716. Hence we are done by Proposition 21.
Theorem 718 (prov9_11d99f316a): T(y,K(x,x \ e)) = y
Proof. We have T(y,K((e / x) \ e,e / x)) = y by Theorem 717. Hence we are done by Theorem 253.
Theorem 719 (prov9_6aa2d98774): T(z,K(y \ x,x \ y)) = z
Proof. We have T(z,K((x \ y) \ e,x \ y)) = z by Theorem 717. Hence we are done by Theorem 666.
Theorem 720 (prov9_d0a99fcaaa): (K(x \ e,x) * y) * K(x,x \ e) = y
Proof. We have T((K(x \ e,x) * y) * K(x,x \ e),K(x \ e,x)) = y by Theorem 658. Hence we are done by Theorem 717.
Theorem 721 (prov9_607b311d6d): T(x,(y \ e) \ e) = T(x,y)
Proof. We have T(T(x,(y \ e) \ e),K(y \ e,y)) = T(x,y) by Theorem 708. Hence we are done by Theorem 717.
Theorem 722 (prov9_48a0436e69): T(x,T(y,y \ e)) = T(x,y)
Proof. We have T(T(T(x,y),K(y,y \ e)),K(y \ e,y)) = T(x,y) by Theorem 701. Then T(T(x,T(y,y \ e)),K(y \ e,y)) = T(x,y) by Theorem 707. Hence we are done by Theorem 717.
Theorem 723 (prov9_4dd73de8ac): T(K(y,y \ e),x) = K(y,y \ e)
Proof. We have T(K((e / y) \ e,e / y),x) = K((e / y) \ e,e / y) by Theorem 716. Then K(K((e / y) \ e,e / y),x) = e by Proposition 22. Then K(K(y,y \ e),x) = e by Theorem 253. Hence we are done by Proposition 23.
Theorem 724 (prov9_baba46b798): z * K(y \ x,x \ y) = K(y \ x,x \ y) * z
Proof. We have
(L1) T((K(y \ x,x \ y) * z) * (e / K(y \ x,x \ y)),K(y \ x,x \ y)) = (K(y \ x,x \ y) * z) * (e / K(y \ x,x \ y))
by Theorem 719. We have T((K(y \ x,x \ y) * z) * (e / K(y \ x,x \ y)),K(y \ x,x \ y)) = (z * K(y \ x,x \ y)) * (e / K(y \ x,x \ y)) by Theorem 611. Then (z * K(y \ x,x \ y)) * (e / K(y \ x,x \ y)) = (K(y \ x,x \ y) * z) * (e / K(y \ x,x \ y)) by (L1). Hence we are done by Proposition 10.
Theorem 725 (prov9_6b4e5e1238): T(x,R(y \ e,y,z)) = T(x,R(e / y,y,z))
Proof. We have T(x,(R(e / y,y,z) \ e) \ e) = T(x,R(e / y,y,z)) by Theorem 721. Hence we are done by Theorem 671.
Theorem 726 (prov9_d0596f07fe): T(y,y \ e) * T(x,y) = x * T(y,y \ e)
Proof. We have T(y,y \ e) * T(x,T(y,y \ e)) = x * T(y,y \ e) by Proposition 46. Hence we are done by Theorem 722.
Theorem 727 (prov9_ff5d80ac03): T(x,R(e / y,y,z)) = T(x,R(y \ e,(y \ e) \ e,z))
Proof. We have T(x,R(e / y,y,z)) = T(x,R(y \ e,y,z)) by Theorem 725. Hence we are done by Theorem 704.
Theorem 728 (prov9_49282bddab): T(x \ e,y) = (x \ e) * (x * T(x \ e,y))
Proof. We have R(x \ e,y,K((x \ e) \ e,x \ e)) = x \ e by Theorem 669. Then
(L2) R(x \ e,y,K(x,x \ e)) = x \ e
by Theorem 561. We have R(x \ e,y,K(x,x \ e)) * (y * K(x,x \ e)) = ((x \ e) * y) * K(x,x \ e) by Proposition 54. Then (x \ e) * (y * K(x,x \ e)) = ((x \ e) * y) * K(x,x \ e) by (L2). Then
(L5) T((e / x) * y,K(x,x \ e)) = (x \ e) * (y * K(x,x \ e))
by Theorem 700. We have T(x \ e,y * K(x,x \ e)) = (y * K(x,x \ e)) \ ((x \ e) * (y * K(x,x \ e))) by Definition 3. Then
(L7) T(x \ e,y * K(x,x \ e)) = (y * K(x,x \ e)) \ T((e / x) * y,K(x,x \ e))
by (L5). We have K(x,x \ e) * (K(x \ e,x) * (y * K(x,x \ e))) = y * K(x,x \ e) by Theorem 442. Then
(L9) T(y,K(x,x \ e)) = K(x \ e,x) * (y * K(x,x \ e))
by Theorem 11. We have T(x \ e,K(x \ e,(x \ e) \ e) * (y * K(x,x \ e))) = T(x \ e,y * K(x,x \ e)) by Theorem 698. Then T(x \ e,K(x \ e,x) * (y * K(x,x \ e))) = T(x \ e,y * K(x,x \ e)) by Theorem 249. Then T(x \ e,T(y,K(x,x \ e))) = T(x \ e,y * K(x,x \ e)) by (L9). Then
(L13) T(x \ e,T(y,K(x,x \ e))) = (y * K(x,x \ e)) \ T((e / x) * y,K(x,x \ e))
by (L7). We have T(x \ e,K((x \ e) \ e,x \ e) * T(y,K((x \ e) \ e,x \ e))) = T(x \ e,T(y,K((x \ e) \ e,x \ e))) by Theorem 630. Then T(x \ e,y * K((x \ e) \ e,x \ e)) = T(x \ e,T(y,K((x \ e) \ e,x \ e))) by Proposition 46. Then T(x \ e,T(y,K((x \ e) \ e,x \ e))) = T(x \ e,y) by Theorem 660. Then T(x \ e,T(y,K(x,x \ e))) = T(x \ e,y) by Theorem 561. Then (y * K(x,x \ e)) \ T((e / x) * y,K(x,x \ e)) = T(x \ e,y) by (L13). Then
(L19) (y * K(x,x \ e)) \ ((e / x) * y) = T(x \ e,y)
by Theorem 718. We have (y / K(x \ e,x)) \ (L(e / x,K(x \ e,x),y / K(x \ e,x)) * y) = K(x \ e,x) * T(e / x,y) by Theorem 491. Then (y / K(x \ e,x)) \ ((e / x) * y) = K(x \ e,x) * T(e / x,y) by Theorem 644. Then
(L22) (y * K(x,x \ e)) \ ((e / x) * y) = K(x \ e,x) * T(e / x,y)
by Theorem 656. We have K(x \ e,x) * T(e / x,y) = (x \ e) * (x * T(x \ e,y)) by Theorem 567. Then (y * K(x,x \ e)) \ ((e / x) * y) = (x \ e) * (x * T(x \ e,y)) by (L22). Hence we are done by (L19).
Theorem 729 (prov9_11c192b44e): (x \ e) \ T(x \ e,y) = x * T(x \ e,y)
Proof. We have (x \ e) * (x * T(x \ e,y)) = T(x \ e,y) by Theorem 728. Hence we are done by Proposition 2.
Theorem 730 (prov9_45ca2fa5f2): K(y \ (y / x),y) = (x \ e) \ T(x \ e,y)
Proof. We have K(y \ (y / x),y) = x * T(x \ e,y) by Theorem 196. Hence we are done by Theorem 729.
Theorem 731 (prov9_5237c13b3e): K(y \ x,x \ y) * y = x * (e / (e / (x \ y)))
Proof. We have x * (x \ (y * K(y \ x,x \ y))) = y * K(y \ x,x \ y) by Axiom 4. Then x * (e / (e / (x \ y))) = y * K(y \ x,x \ y) by Theorem 615. Hence we are done by Theorem 724.
Theorem 732 (prov9_17f021d8df): K(x \ e,y) = K(e / x,y)
Proof. We have K(y \ (y / ((x \ e) \ e)),y) = (x \ e) \ T(x \ e,y) by Theorem 198. Then
(L2) K(y \ (y / ((x \ e) \ e)),y) = K(y \ (y / x),y)
by Theorem 730. We have
(L3) L((x \ e) \ T(x \ e,y),x \ e,y) = (y * (x \ e)) \ ((x \ e) * y)
by Proposition 73.
K(x \ e,y)
=(y * (x \ e)) \ ((x \ e) * y) by Definition 2
=L(K(y \ (y / ((x \ e) \ e)),y),x \ e,y) by (L3), Theorem 198

Then K(x \ e,y) = L(K(y \ (y / ((x \ e) \ e)),y),x \ e,y). Then L(K(y \ (y / x),y),x \ e,y) = K(x \ e,y) by (L2). Then
(L8) L(x * T(x \ e,y),x \ e,y) = K(x \ e,y)
by Theorem 196. We have L(x * T(x \ e,y),x \ e,y) = K(e / x,y) by Theorem 654. Hence we are done by (L8).
Theorem 733 (prov9_0f5564de23): K(x,y) = K((x \ e) \ e,y)
Proof. We have K(e / (x \ e),y) = K((x \ e) \ e,y) by Theorem 732. Hence we are done by Proposition 24.
Theorem 734 (prov9_632b51c32e): T(K(x,y),x) = K(x,y)
Proof. We have T(K(x,y),x) = K((x \ e) \ e,y) by Theorem 674. Hence we are done by Theorem 733.
Theorem 735 (prov9_133856ff26): (((y * x) \ y) \ e) \ (((y * x) \ y) \ K(x,y)) = K(x,y)
Proof. We have (((y * x) \ y) \ e) \ (((y * x) \ y) \ K(x,y)) = K((x \ e) \ e,y) by Theorem 675. Hence we are done by Theorem 733.
Theorem 736 (prov9_dbf0d9a7ac): T(y,K(y,x)) = y
Proof. We have T(K(y,x),y) = K(y,x) by Theorem 734. Hence we are done by Proposition 21.
Theorem 737 (prov9_ac6446d09c): K(x \ ((y * x) / y),y) = e
Proof. We have T(K(y,(y / (x \ e)) / y),y) = K(y,(y / (x \ e)) / y) by Theorem 734. Then K(K(y,(y / (x \ e)) / y),y) = e by Proposition 22. Hence we are done by Theorem 596.
Theorem 738 (prov9_816af6fb05): T(T(y,x) \ y,x) = T(y,x) \ y
Proof. We have K(T(y,x) \ ((x * T(y,x)) / x),x) = e by Theorem 737. Then K(T(y,x) \ y,x) = e by Proposition 48. Hence we are done by Proposition 23.
Theorem 739 (prov9_c1ad8eb43b): T(y,T(x,y) \ x) = y
Proof. We have T(y,K(y,(y / (T(x,y) \ e)) / y)) = y by Theorem 736. Then T(y,T(x,y) \ ((y * T(x,y)) / y)) = y by Theorem 596. Hence we are done by Proposition 48.
Theorem 740 (prov9_c6c527b157): (T(y,x) \ y) * x = x * (T(y,x) \ y)
Proof. We have (T(y,x) \ y) * T(x,T(y,x) \ y) = x * (T(y,x) \ y) by Proposition 46. Hence we are done by Theorem 739.
Theorem 741 (prov9_88741b4b5d): K(x \ T(x,y),T(z,y) \ z) = e
Proof. We have T(y \ T(y,R(T(z,y),T(z,y) \ e,y)),K(y \ (y / (x \ e)),y)) = y \ T(y,R(T(z,y),T(z,y) \ e,y)) by Theorem 619. Then T(K(y \ (y / (x \ e)),y),y \ T(y,R(T(z,y),T(z,y) \ e,y))) = K(y \ (y / (x \ e)),y) by Proposition 21. Then K(K(y \ (y / (x \ e)),y),y \ T(y,R(T(z,y),T(z,y) \ e,y))) = e by Proposition 22. Then K(K(y \ (y / (x \ e)),y),y \ ((T(z,y) \ z) * y)) = e by Theorem 591. Then K(K(y \ (y / (x \ e)),y),T(T(z,y) \ z,y)) = e by Definition 3. Then K(K(y \ (y / (x \ e)),y),T(z,y) \ z) = e by Theorem 738. Hence we are done by Theorem 198.
Theorem 742 (prov9_9231fd31e1): T(x,R(T(y,x),T(y,x) \ e,x)) = x * (T(y,x) \ y)
Proof. We have T(x,R(T(y,x),T(y,x) \ e,x)) = (T(y,x) \ y) * x by Theorem 591. Hence we are done by Theorem 740.
Theorem 743 (prov9_c52a3e41d6): K(T(y,x),(y * x) \ x) = K(y,y \ e)
Proof. We have
(L1) K(x \ (y * x),(y * x) \ x) * x = (y * x) * (e / (e / ((y * x) \ x)))
by Theorem 731. We have K(y,y \ e) * x = (y * x) * (e / (e / ((y * x) \ x))) by Theorem 714. Then K(x \ (y * x),(y * x) \ x) = K(y,y \ e) by (L1) and Proposition 8. Hence we are done by Definition 3.
Theorem 744 (prov9_1c1158d7eb): K(T(y,x),T(y,x) \ e) = K(y,y \ e)
Proof. We have K(T(y,x),(x * T(y,x)) \ x) = K(T(y,x),T(y,x) \ e) by Theorem 392. Then K(T(y,x),(y * x) \ x) = K(T(y,x),T(y,x) \ e) by Proposition 46. Hence we are done by Theorem 743.
Theorem 745 (prov9_ded6e67136): T(x,y) \ K(x,x \ e) = e / T(x,y)
Proof. We have T(x,y) \ K(T(x,y),T(x,y) \ e) = e / T(x,y) by Theorem 251. Hence we are done by Theorem 744.
Theorem 746 (prov9_79b082cca2): (T(x,y) \ e) * K(x,x \ e) = e / T(x,y)
Proof. We have (T(x,y) \ e) * K(T(x,y),T(x,y) \ e) = e / T(x,y) by Theorem 553. Hence we are done by Theorem 744.
Theorem 747 (prov9_35fd57c29a): e / T(y,x) = (e / y) * (T(y,x) \ y)
Proof. We have (e / y) * (y * (T(y,x) \ e)) = L(T(y,x) \ e,y,e / y) by Proposition 67. Then (e / y) * (T(y,x) \ y) = L(T(y,x) \ e,y,e / y) by Theorem 175. Then
(L3) L(T(y,x) \ e,y,y \ e) = (e / y) * (T(y,x) \ y)
by Theorem 683. We have T(y,x) * L(T(y,x) \ e,y,y \ e) = T(y * L(y \ e,y,y \ e),x) by Theorem 351. Then T(y,x) * L(T(y,x) \ e,y,y \ e) = T(y * ((e / y) * e),x) by Theorem 434. Then T(y,x) * ((e / y) * (T(y,x) \ y)) = T(y * ((e / y) * e),x) by (L3). Then T(y,x) * ((e / y) * (T(y,x) \ y)) = T(y * (e / y),x) by Axiom 2. Then T(y,x) * ((e / y) * (T(y,x) \ y)) = T(K(y,y \ e),x) by Theorem 250. Then T(y,x) \ T(K(y,y \ e),x) = (e / y) * (T(y,x) \ y) by Proposition 2. Then T(y,x) \ K(y,y \ e) = (e / y) * (T(y,x) \ y) by Theorem 723. Hence we are done by Theorem 745.
Theorem 748 (prov9_6d3fd7baa8): K(x \ e,x) * (e / T(x,y)) = T(x,y) \ e
Proof.
(K(x \ e,x) * (e / T(x,y))) * K(x,x \ e)
=e / T(x,y) by Theorem 720
=(T(x,y) \ e) * K(x,x \ e) by Theorem 746

Then (K(x \ e,x) * (e / T(x,y))) * K(x,x \ e) = (T(x,y) \ e) * K(x,x \ e). Hence we are done by Proposition 10.
Theorem 749 (prov9_db0c1a1b92): (y \ T(y,x)) \ e = T(y,x) \ y
Proof. We have T(y,x) \ K(y,e / y) = y * (T(y,x) \ (e / y)) by Theorem 523. Then T(y,x) \ K(y,y \ e) = y * (T(y,x) \ (e / y)) by Theorem 248. Then
(L3) e / T(y,x) = y * (T(y,x) \ (e / y))
by Theorem 745. We have (y \ e) \ (T(y,x) \ e) = (y \ T(y,x)) \ e by Theorem 638. Then (e / y) \ (K(y,y \ e) * (T(y,x) \ e)) = (y \ T(y,x)) \ e by Theorem 692. Then (e / y) \ (y * (T(y,x) \ (e / y))) = (y \ T(y,x)) \ e by Theorem 688. Then
(L7) (e / y) \ (e / T(y,x)) = (y \ T(y,x)) \ e
by (L3). We have (e / y) * (T(y,x) \ y) = e / T(y,x) by Theorem 747. Then (e / y) \ (e / T(y,x)) = T(y,x) \ y by Proposition 2. Hence we are done by (L7).
Theorem 750 (prov9_4a6642a5c7): (y * T(y \ e,x)) \ e = T(y \ e,x) \ (y \ e)
Proof. We have ((y \ e) \ T(y \ e,x)) \ e = T(y \ e,x) \ (y \ e) by Theorem 749. Hence we are done by Theorem 729.
Theorem 751 (prov9_9b7c9ef815): e / (y \ T(y,x)) = T(y,x) \ y
Proof. We have (y \ T(y,x)) \ K(y \ T(y,x),T(y,x) \ y) = e / (y \ T(y,x)) by Theorem 613. Then (y \ T(y,x)) \ e = e / (y \ T(y,x)) by Theorem 741. Hence we are done by Theorem 749.
Theorem 752 (prov9_07870b53a9): T(x,y) \ e = x \ (x / T(x,y))
Proof. We have K(x \ e,x) * (e / T(x,y)) = x \ (x / T(x,y)) by Theorem 715. Hence we are done by Theorem 748.
Theorem 753 (prov9_eb03855ce4): y / T(y,x) = T(y,x) \ y
Proof. We have y * (y \ (y / T(y,x))) = y / T(y,x) by Axiom 4. Then
(L2) y * (T(y,x) \ e) = y / T(y,x)
by Theorem 752. We have y * (T(y,x) \ e) = T(y,x) \ y by Theorem 175. Hence we are done by (L2).
Theorem 754 (prov9_eacb5a0258): K(x,y) = T(K(x,y),y)
Proof. We have y * (T(y,x) \ y) = (y / T(y,x)) * y by Theorem 572. Then
(L6) T(y / T(y,x),y) = T(y,x) \ y
by Theorem 11. We have T(y / T(y,x),y) * T(y,x) = T(y,x) * T(T(y,x) \ y,y) by Proposition 61. Then
(L8) (T(y,x) \ y) * T(y,x) = T(y,x) * T(T(y,x) \ y,y)
by (L6). We have T(y,x) * T(T(y,x) \ y,y) = L(y,T(y,x) \ y,T(y,x)) by Theorem 527. Then T(y,x) * T(T(y,x) \ y,y) = L(y,T(y,x) \ y,y) by Theorem 609. Then
(L9) L(y,T(y,x) \ y,y) = (T(y,x) \ y) * T(y,x)
by (L8). We have (y / T(y,x)) * T(y,x) = y by Axiom 6. Then (T(y,x) \ y) * T(y,x) = y by Theorem 753. Then
(L10) L(y,T(y,x) \ y,y) = y
by (L9). We have (y * x) * L(T(T(y,x) \ y,y),T(y,x),x) = x * (T(y,x) * T(T(y,x) \ y,y)) by Theorem 58. Then (y * x) * L(T(T(y,x) \ y,y),T(y,x),x) = x * L(y,T(y,x) \ y,T(y,x)) by Theorem 527. Then (y * x) * T((y * x) \ (x * y),y) = x * L(y,T(y,x) \ y,T(y,x)) by Theorem 487. Then (y * x) * T(K(x,y),y) = x * L(y,T(y,x) \ y,T(y,x)) by Definition 2. Then x * L(y,T(y,x) \ y,y) = (y * x) * T(K(x,y),y) by Theorem 609. Then x * y = (y * x) * T(K(x,y),y) by (L10). Hence we are done by Theorem 460.
Theorem 755 (prov9_860a038556): T(y,K(x,y)) = y
Proof. We have T(K(x,y),y) = K(x,y) by Theorem 754. Hence we are done by Proposition 21.
Theorem 756 (prov9_d0d9d38aba): T(x,y) * K(x \ e,x) = T(e / (e / x),y)
Proof. We have T(e / (e / x),y) \ T(x,y) = K(T(x,y),T(x,y) \ e) by Theorem 712. Then
(L2) T(e / (e / x),y) \ T(x,y) = K(x,x \ e)
by Theorem 744. We have T(e / (e / x),y) * (T(e / (e / x),y) \ T(x,y)) = T(x,y) by Axiom 4. Then
(L4) T(e / (e / x),y) * K(x,x \ e) = T(x,y)
by (L2). We have (T(x,y) * K(x \ e,x)) * K(x,x \ e) = T(x,y) by Theorem 426. Then (T(x,y) * K(x \ e,x)) * K(x,x \ e) = T(e / (e / x),y) * K(x,x \ e) by (L4). Hence we are done by Proposition 10.
Theorem 757 (prov9_f06aaef1ca): T(y,x) = y * (T(y,x) / y)
Proof. We have L(T(y,x),e / y,y) = (y * (e / y)) \ (y * ((e / y) * T(y,x))) by Definition 4. Then L(T(y,x),e / y,y) = K(y,e / y) \ (y * ((e / y) * T(y,x))) by Proposition 77. Then
(L6) K(y,e / y) \ (y * (T(y,x) / y)) = L(T(y,x),e / y,y)
by Theorem 150. We have L(T(y,x),e / y,y) = T(L(y,e / y,y),x) by Axiom 8. Then L(T(y,x),e / y,y) = T(e / (e / y),x) by Theorem 266. Then
(L7) K(y,e / y) \ (y * (T(y,x) / y)) = T(e / (e / y),x)
by (L6). We have K(y,e / y) * (K(y,e / y) \ (y * (T(y,x) / y))) = y * (T(y,x) / y) by Axiom 4. Then K(y,e / y) * T(e / (e / y),x) = y * (T(y,x) / y) by (L7). Then K(y,y \ e) * T(e / (e / y),x) = y * (T(y,x) / y) by Theorem 248. Then
(L11) K(y,y \ e) * T(K(y,y \ e) \ y,x) = y * (T(y,x) / y)
by Theorem 558. We have (K(y,y \ e) * T(K(y,y \ e) \ y,x)) * (K(y,y \ e) \ e) = T(y * (K(y,y \ e) \ e),x) by Theorem 506. Then (y * (T(y,x) / y)) * (K(y,y \ e) \ e) = T(y * (K(y,y \ e) \ e),x) by (L11). Then (y * (T(y,x) / y)) * K(y \ e,y) = T(y * (K(y,y \ e) \ e),x) by Theorem 565. Then (y * (T(y,x) / y)) * K(y \ e,y) = T(y * K(y \ e,y),x) by Theorem 565. Then
(L16) (y * (T(y,x) / y)) * K(y \ e,y) = T(e / (e / y),x)
by Theorem 262. We have T(y,x) * K(y \ e,y) = T(e / (e / y),x) by Theorem 756. Hence we are done by (L16) and Proposition 8.
Theorem 758 (prov9_799a000eb7): y \ T(y,x) = T(y,x) / y
Proof. We have y * (T(y,x) / y) = T(y,x) by Theorem 757. Hence we are done by Proposition 2.
Theorem 759 (prov9_8e93d4c502): (x \ T(x,y)) * x = T(x,y)
Proof. We have (T(x,y) / x) * x = T(x,y) by Axiom 6. Then (x \ (x * (T(x,y) / x))) * x = T(x,y) by Axiom 3. Hence we are done by Theorem 757.
Theorem 760 (prov9_536e9f6685): y \ T(y,R(x,x \ e,y)) = x \ ((y * x) / y)
Proof. We have T(y,R(x,x \ e,y)) / y = x \ ((y * x) / y) by Theorem 590. Hence we are done by Theorem 758.
Theorem 761 (prov9_bb0c7517b4): T(y,y \ T(y,x)) = y
Proof. We have (y \ T(y,x)) * y = T(y,x) by Theorem 759. Hence we are done by Theorem 20.
Theorem 762 (prov9_eccd7787f9): x \ e = T(e / x,T(x,y))
Proof. We have T(x,x \ T(x,y)) = x by Theorem 761. Then K(x,x \ T(x,y)) = e by Proposition 22. Then
(L3) K(x,T(x,y) / x) = e
by Theorem 758. We have x * T(e / x,T(x,y)) = K(x,T(x,y) / x) by Theorem 529. Then x \ K(x,T(x,y) / x) = T(e / x,T(x,y)) by Proposition 2. Hence we are done by (L3).
Theorem 763 (prov9_e54647eeba): K(x,y) \ e = T(L(x \ e,x,y) \ e,y) \ (L(x \ e,x,y) \ e)
Proof. We have (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y)) \ e = T(L(x \ e,x,y) \ e,y) \ (L(x \ e,x,y) \ e) by Theorem 750. Hence we are done by Theorem 542.
Theorem 764 (prov9_9d6628f043): (y \ (z * T(z \ y,x))) \ e = T((y \ z) \ e,x) \ ((y \ z) \ e)
Proof. We have ((y \ z) * T((y \ z) \ e,x)) \ e = T((y \ z) \ e,x) \ ((y \ z) \ e) by Theorem 750. Hence we are done by Theorem 568.
Theorem 765 (prov9_d16a73ad24): (K(x,y) \ y) * K(x,y) = y
Proof. We have x * (K(x,y) * T(K(x,y) \ y,K(x,y))) = (T(y,K(x,y)) * x) * K(x,y) by Theorem 634. Then x * ((K(x,y) \ y) * K(x,y)) = (T(y,K(x,y)) * x) * K(x,y) by Proposition 46. Then
(L3) (y * x) * K(x,y) = x * ((K(x,y) \ y) * K(x,y))
by Theorem 755. We have x * y = (y * x) * K(x,y) by Proposition 82. Hence we are done by (L3) and Proposition 7.
Theorem 766 (prov9_5f04c9c295): y / K(x,y) = K(x,y) \ y
Proof. We have (K(x,y) \ y) * K(x,y) = y by Theorem 765. Hence we are done by Proposition 1.
Theorem 767 (prov9_8a003da801): y \ ((x * y) / x) = (((x * y) / x) \ y) \ e
Proof. We have (((x * y) / x) \ T((x * y) / x,x)) \ e = T((x * y) / x,x) \ ((x * y) / x) by Theorem 749. Then (((x * y) / x) \ (x \ (x * y))) \ e = T((x * y) / x,x) \ ((x * y) / x) by Proposition 47. Then (x \ (x * y)) \ ((x * y) / x) = (((x * y) / x) \ (x \ (x * y))) \ e by Proposition 47. Then (((x * y) / x) \ y) \ e = (x \ (x * y)) \ ((x * y) / x) by Axiom 3. Hence we are done by Axiom 3.
Theorem 768 (prov9_4ec972b840): e / (y \ ((x * y) / x)) = ((x * y) / x) \ y
Proof. We have e / ((((x * y) / x) \ y) \ e) = ((x * y) / x) \ y by Proposition 24. Hence we are done by Theorem 767.
Theorem 769 (prov9_c247770242): T(x,R(y,y \ e,x)) \ x = ((x * y) / x) \ y
Proof. We have e / (x \ T(x,R(y,y \ e,x))) = T(x,R(y,y \ e,x)) \ x by Theorem 751. Then e / (y \ ((x * y) / x)) = T(x,R(y,y \ e,x)) \ x by Theorem 760. Hence we are done by Theorem 768.
Theorem 770 (prov9_9491bcb5a2): T(y,R(T(x,y),T(x,y) \ e,y)) \ y = x \ T(x,y)
Proof. We have T(y,R(T(x,y),T(x,y) \ e,y)) \ y = ((y * T(x,y)) / y) \ T(x,y) by Theorem 769. Hence we are done by Proposition 48.
Theorem 771 (prov9_66759c8aff): (y * (T(x,y) \ x)) \ y = x \ T(x,y)
Proof. We have T(y,R(T(x,y),T(x,y) \ e,y)) \ y = x \ T(x,y) by Theorem 770. Hence we are done by Theorem 742.
Theorem 772 (prov9_7dec43c66c): T(y,K(x,y) \ e) = y
Proof. We have T(y,T(L(x \ e,x,y) \ e,y) \ (L(x \ e,x,y) \ e)) = y by Theorem 739. Hence we are done by Theorem 763.
Theorem 773 (prov9_9f15cbf6e4): (x \ R(x,y,z)) \ (x \ e) = R(x,y,z) \ e
Proof. We have R(T(x,x \ e) * T(T(x,x \ e) \ e,x \ R(x,y,z)),y,z) = K((x \ R(x,y,z)) \ ((x \ R(x,y,z)) / R(T(x,x \ e),y,z)),x \ R(x,y,z)) by Theorem 636. Then R(T(x,x \ e) * T(T(x,x \ e) \ e,x \ R(x,y,z)),y,z) = K((x \ R(x,y,z)) \ (x \ e),x \ R(x,y,z)) by Theorem 532. Then
(L5) R(T(x,x \ e) * T(T(x \ e,x),x \ R(x,y,z)),y,z) = K((x \ R(x,y,z)) \ (x \ e),x \ R(x,y,z))
by Theorem 208. We have T(T(x \ e,x \ R(x,y,z)),x) = T(T(x \ e,x),x \ R(x,y,z)) by Axiom 7. Then T(x \ e,x) = T(T(x \ e,x),x \ R(x,y,z)) by Theorem 578. Then R(T(x,x \ e) * T(x \ e,x),y,z) = K((x \ R(x,y,z)) \ (x \ e),x \ R(x,y,z)) by (L5). Then K((x \ R(x,y,z)) \ (x \ e),x \ R(x,y,z)) = R((x \ e) * T(x,x \ e),y,z) by Theorem 726. Then
(L8) K((x \ R(x,y,z)) \ (x \ e),x \ R(x,y,z)) = R(e,y,z)
by Theorem 8. We have R(e,y,z) = ((e * y) * z) / (y * z) by Definition 5. Then
(L10) R(e,y,z) = (y * z) / (y * z)
by Axiom 1. We have (y * z) / (y * z) = e by Proposition 29. Then R(e,y,z) = e by (L10). Then
(L13) K((x \ R(x,y,z)) \ (x \ e),x \ R(x,y,z)) = e
by (L8). We have (x \ e) * K((x \ R(x,y,z)) \ (x \ e),x \ R(x,y,z)) = ((x \ R(x,y,z)) \ (x \ e)) * (x \ R(x,y,z)) by Theorem 17. Then
(L15) (x \ e) * e = ((x \ R(x,y,z)) \ (x \ e)) * (x \ R(x,y,z))
by (L13). We have (((x \ R(x,y,z)) \ (x \ e)) * (x \ R(x,y,z))) * e = ((x \ R(x,y,z)) \ (x \ e)) * (x \ R(x,y,z)) by Axiom 2. Then (x \ e) * e = (((x \ R(x,y,z)) \ (x \ e)) * (x \ R(x,y,z))) * e by (L15). Then x \ e = ((x \ R(x,y,z)) \ (x \ e)) * (x \ R(x,y,z)) by Proposition 10. Then
(L19) (x \ e) / (x \ R(x,y,z)) = (x \ R(x,y,z)) \ (x \ e)
by Proposition 1. We have ((x \ e) / (R(x,y,z) * (x \ e))) * R(x,y,z) = R(x,y,z) * R(R(x,y,z) \ e,R(x,y,z),x \ e) by Theorem 90. Then ((x \ e) / (x \ R(x,y,z))) * R(x,y,z) = R(x,y,z) * R(R(x,y,z) \ e,R(x,y,z),x \ e) by Proposition 89. Then R(x * R(x \ e,R(x,y,z),x \ e),y,z) = ((x \ e) / (x \ R(x,y,z))) * R(x,y,z) by Theorem 641. Then
(L23) R(x * T(x \ e,R(x,y,z)),y,z) = ((x \ e) / (x \ R(x,y,z))) * R(x,y,z)
by Theorem 580. We have R(x * T(x \ e,R(x,y,z)),y,z) = K(R(x,y,z) \ e,R(x,y,z)) by Theorem 637. Then
(L25) ((x \ e) / (x \ R(x,y,z))) * R(x,y,z) = K(R(x,y,z) \ e,R(x,y,z))
by (L23). We have (R(x,y,z) \ e) * R(x,y,z) = K(R(x,y,z) \ e,R(x,y,z)) by Proposition 76. Then (x \ e) / (x \ R(x,y,z)) = R(x,y,z) \ e by (L25) and Proposition 8. Hence we are done by (L19).
Theorem 774 (prov9_d9fdabed9f): a(R(x,y,z),e / x,x) = x * ((x \ R(x,y,z)) / R(x,y,z))
Proof. We have R(x,y,z) \ T(R(x,y,z),R(x \ e,(x \ e) \ e,R(x,y,z))) = (x \ e) \ ((R(x,y,z) * (x \ e)) / R(x,y,z)) by Theorem 760. Then R(x,y,z) \ T(R(x,y,z),R(x \ e,(x \ e) \ e,R(x,y,z))) = (x \ e) \ ((x \ R(x,y,z)) / R(x,y,z)) by Proposition 89. Then R(x,y,z) \ T(R(x,y,z),R(e / x,x,R(x,y,z))) = (x \ e) \ ((x \ R(x,y,z)) / R(x,y,z)) by Theorem 727. Then R(x,y,z) \ R(T(x,R(e / x,x,R(x,y,z))),y,z) = (x \ e) \ ((x \ R(x,y,z)) / R(x,y,z)) by Axiom 9. Then
(L5) R(x,y,z) \ R(T(x,x \ e),y,z) = (x \ e) \ ((x \ R(x,y,z)) / R(x,y,z))
by Theorem 361. We have R(x,y,z) \ R(T(x,x \ e),y,z) = a(R(x,y,z),e / x,x) by Theorem 545. Then
(L7) (x \ e) \ ((x \ R(x,y,z)) / R(x,y,z)) = a(R(x,y,z),e / x,x)
by (L5). We have (x \ e) \ ((x \ R(x,y,z)) / R(x,y,z)) = x * ((x \ R(x,y,z)) / R(x,y,z)) by Theorem 709. Hence we are done by (L7).
Theorem 775 (prov9_598d906826): z * L(z \ e,R(z,x,y),e / z) = a(R(z,x,y),e / z,z)
Proof. We have T((R(z,x,y) * (e / z)) / R(z,x,y),z) = (R(z,x,y) * (z \ e)) / R(z,x,y) by Theorem 94. Then T((R(z,x,y) * (e / z)) / R(z,x,y),z) = (z \ R(z,x,y)) / R(z,x,y) by Proposition 89. Then
(L3) T(L(e / z,R(z,x,y),e / z),z) = (z \ R(z,x,y)) / R(z,x,y)
by Theorem 579. We have T(L(e / z,R(z,x,y),e / z),z) = L(z \ e,R(z,x,y),e / z) by Proposition 72. Then
(L5) (z \ R(z,x,y)) / R(z,x,y) = L(z \ e,R(z,x,y),e / z)
by (L3). We have z * ((z \ R(z,x,y)) / R(z,x,y)) = a(R(z,x,y),e / z,z) by Theorem 774. Hence we are done by (L5).
Theorem 776 (prov9_5e4e970c6f): y / K(x,y) = y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e))
Proof. We have y / ((y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e))) \ y) = y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e)) by Proposition 24. Then y / ((((y * x) \ y) \ e) \ T(((y * x) \ y) \ e,y)) = y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e)) by Theorem 771. Then y / ((((y * x) \ y) \ e) \ (((y * x) \ y) \ K(x,y))) = y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e)) by Theorem 538. Hence we are done by Theorem 735.
Theorem 777 (prov9_16c10f5b3f): y * (K(x,y) \ e) = K(x,y) \ y
Proof. We have y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e)) = y / K(x,y) by Theorem 776. Then y * (((y * x) \ (y * T(y \ (y * x),y))) \ e) = y / K(x,y) by Theorem 764. Then y * (((y * x) \ (y * T(x,y))) \ e) = y / K(x,y) by Axiom 3. Then y * (((y * x) \ (x * y)) \ e) = y / K(x,y) by Proposition 46. Then
(L5) y * (K(x,y) \ e) = y / K(x,y)
by Definition 2. We have y / K(x,y) = K(x,y) \ y by Theorem 766. Hence we are done by (L5).
Theorem 778 (prov9_b51b848792): (K(x,y) \ e) * T(y,z) = K(x,y) \ T(y,z)
Proof. We have ((K(x,y) \ e) * T((K(x,y) \ e) \ (K(x,y) \ y),K(x,y) \ e)) / (K(x,y) \ e) = (K(x,y) \ e) \ (K(x,y) \ y) by Proposition 48. Then (K(x,y) \ T(y,K(x,y) \ e)) / (K(x,y) \ e) = (K(x,y) \ e) \ (K(x,y) \ y) by Theorem 496. Then
(L5) (K(x,y) \ y) / (K(x,y) \ e) = (K(x,y) \ e) \ (K(x,y) \ y)
by Theorem 772. We have y * (K(x,y) \ e) = K(x,y) \ y by Theorem 777. Then (K(x,y) \ y) / (K(x,y) \ e) = y by Proposition 1. Then
(L6) y = (K(x,y) \ e) \ (K(x,y) \ y)
by (L5). We have (K(x,y) \ e) * T((K(x,y) \ e) \ (K(x,y) \ y),z) = K(x,y) \ T(y,z) by Theorem 496. Hence we are done by (L6).
Theorem 779 (prov9_2c57734c99): K(x,y) = K(x,(y \ e) \ e)
Proof. We have L(y \ e,y,T(y,R(T(L(x \ e,x,y) \ e,y),T(L(x \ e,x,y) \ e,y) \ e,y)) / y) = T(e / y,T(y,R(T(L(x \ e,x,y) \ e,y),T(L(x \ e,x,y) \ e,y) \ e,y))) by Theorem 503. Then L(y \ e,y,T(L(x \ e,x,y) \ e,y) \ ((y * T(L(x \ e,x,y) \ e,y)) / y)) = T(e / y,T(y,R(T(L(x \ e,x,y) \ e,y),T(L(x \ e,x,y) \ e,y) \ e,y))) by Theorem 590. Then y \ e = L(y \ e,y,T(L(x \ e,x,y) \ e,y) \ ((y * T(L(x \ e,x,y) \ e,y)) / y)) by Theorem 762. Then L(y \ e,y,T(L(x \ e,x,y) \ e,y) \ (L(x \ e,x,y) \ e)) = y \ e by Proposition 48. Then
(L9) L(y \ e,y,K(x,y) \ e) = y \ e
by Theorem 763. We have (K(x,y) \ e) * T(y,K(x,y) \ e) = y * (K(x,y) \ e) by Proposition 46. Then
(L2) (K(x,y) \ e) * y = y * (K(x,y) \ e)
by Theorem 772. We have L(y \ e,y,K(x,y) \ e) = ((K(x,y) \ e) * y) \ (K(x,y) \ e) by Proposition 78. Then L(y \ e,y,K(x,y) \ e) = (y * (K(x,y) \ e)) \ (K(x,y) \ e) by (L2). Then (y * (K(x,y) \ e)) \ (K(x,y) \ e) = y \ e by (L9). Then
(L11) (K(x,y) \ y) \ (K(x,y) \ e) = y \ e
by Theorem 777. We have (K(x,y) \ y) * ((K(x,y) \ y) \ (K(x,y) \ e)) = K(x,y) \ e by Axiom 4. Then
(L13) (K(x,y) \ y) * (y \ e) = K(x,y) \ e
by (L11). We have ((K(x,y) \ y) * (y \ e)) * ((y \ e) \ e) = R(K(x,y) \ y,y \ e,y) by Theorem 662. Then
(L15) (K(x,y) \ e) * ((y \ e) \ e) = R(K(x,y) \ y,y \ e,y)
by (L13). We have (K(x,y) \ e) * T(y,y \ e) = K(x,y) \ T(y,y \ e) by Theorem 778. Then (K(x,y) \ e) * ((y \ e) \ e) = K(x,y) \ T(y,y \ e) by Proposition 49. Then K(x,y) \ ((y \ e) \ e) = (K(x,y) \ e) * ((y \ e) \ e) by Proposition 49. Then
(L19) R(K(x,y) \ y,y \ e,y) = K(x,y) \ ((y \ e) \ e)
by (L15). We have R((x * y) / (x * K(x,y)),y \ e,y) * (x * K(x,y)) = (x * K(x,y)) * R((x * K(x,y)) \ (x * y),y \ e,y) by Proposition 83. Then
(L21) R(R(y,x,K(x,y)),y \ e,y) * (x * K(x,y)) = (x * K(x,y)) * R((x * K(x,y)) \ (x * y),y \ e,y)
by Theorem 461. We have R(R(y,x,K(x,y)),y \ e,y) * (x * K(x,y)) = (R(y,y \ e,y) * x) * K(x,y) by Theorem 468. Then
(L23) (x * K(x,y)) * R((x * K(x,y)) \ (x * y),y \ e,y) = (R(y,y \ e,y) * x) * K(x,y)
by (L21). We have (x * K(x,y)) * R((x * K(x,y)) \ (x * y),y \ e,y) = x * (K(x,y) * R(K(x,y) \ y,y \ e,y)) by Theorem 516. Then (R(y,y \ e,y) * x) * K(x,y) = x * (K(x,y) * R(K(x,y) \ y,y \ e,y)) by (L23). Then x * (K(x,y) * R(K(x,y) \ y,y \ e,y)) = (T(y,y \ e) * x) * K(x,y) by Theorem 207. Then
(L27) x * (K(x,y) * (K(x,y) \ ((y \ e) \ e))) = (T(y,y \ e) * x) * K(x,y)
by (L19). We have x \ (x * ((y \ e) \ e)) = (y \ e) \ e by Axiom 3. Then
(L29) x \ ((T(y,y \ e) * x) * ((T(y,y \ e) * x) \ (x * ((y \ e) \ e)))) = (y \ e) \ e
by Axiom 4. We have x * (x \ ((T(y,y \ e) * x) * ((T(y,y \ e) * x) \ (x * ((y \ e) \ e))))) = (T(y,y \ e) * x) * ((T(y,y \ e) * x) \ (x * ((y \ e) \ e))) by Axiom 4. Then x * (K(x,y) * (K(x,y) \ (x \ ((T(y,y \ e) * x) * ((T(y,y \ e) * x) \ (x * ((y \ e) \ e))))))) = (T(y,y \ e) * x) * ((T(y,y \ e) * x) \ (x * ((y \ e) \ e))) by Axiom 4. Then x * (K(x,y) * (K(x,y) \ ((y \ e) \ e))) = (T(y,y \ e) * x) * ((T(y,y \ e) * x) \ (x * ((y \ e) \ e))) by (L29). Then (T(y,y \ e) * x) * ((T(y,y \ e) * x) \ (x * ((y \ e) \ e))) = (T(y,y \ e) * x) * K(x,y) by (L27). Then
(L34) (T(y,y \ e) * x) \ (x * ((y \ e) \ e)) = K(x,y)
by Proposition 9. We have (T(y,y \ e) * x) \ (x * ((y \ e) \ e)) = K(x,(y \ e) \ e) by Theorem 478. Hence we are done by (L34).
Theorem 780 (prov9_ec8d3736e7): K(x,y \ e) = K(x,e / y)
Proof. We have K(x,((e / y) \ e) \ e) = K(x,e / y) by Theorem 779. Hence we are done by Proposition 25.
Theorem 781 (prov9_824a06f439): z * ((z \ R(z,x,y)) / R(z,x,y)) = K(R(z,x,y),z \ e)
Proof. We have K(R(z,x,y),(R(z,x,y) / z) / R(z,x,y)) = z * ((R(z,x,y) * (z \ e)) / R(z,x,y)) by Theorem 333. Then K(R(z,x,y),e / z) = z * ((R(z,x,y) * (z \ e)) / R(z,x,y)) by Theorem 533. Then z * ((z \ R(z,x,y)) / R(z,x,y)) = K(R(z,x,y),e / z) by Proposition 89. Hence we are done by Theorem 780.
Theorem 782 (prov9_8e311b756b): K(R(z,x,y),z \ e) = a(R(z,x,y),e / z,z)
Proof. We have z * ((z \ R(z,x,y)) / R(z,x,y)) = a(R(z,x,y),e / z,z) by Theorem 774. Hence we are done by Theorem 781.
Theorem 783 (prov9_5f7239ea5d): R(e / (e / x),y,z) \ e = e / R(x,y,z)
Proof. We have (((x * y) * z) / (y * z)) * L((((x * y) * z) / (y * z)) \ e,R(x,y,z),e / x) = (((x * y) * z) * L(((x * y) * z) \ (y * z),R(x,y,z),e / x)) / (y * z) by Theorem 598. Then R(x,y,z) * L((((x * y) * z) / (y * z)) \ e,R(x,y,z),e / x) = (((x * y) * z) * L(((x * y) * z) \ (y * z),R(x,y,z),e / x)) / (y * z) by Definition 5. Then (((x * y) * z) * L(((x * y) * z) \ (y * z),R(x,y,z),e / x)) / (y * z) = R(x,y,z) * L(R(x,y,z) \ e,R(x,y,z),e / x) by Definition 5. Then (((x * y) * L((x * y) \ y,R(x,y,z),e / x)) * z) / (y * z) = R(x,y,z) * L(R(x,y,z) \ e,R(x,y,z),e / x) by Theorem 332. Then
(L5) (((x * L(x \ e,R(x,y,z),e / x)) * y) * z) / (y * z) = R(x,y,z) * L(R(x,y,z) \ e,R(x,y,z),e / x)
by Theorem 294. We have R(x * L(x \ e,R(x,y,z),e / x),y,z) = (((x * L(x \ e,R(x,y,z),e / x)) * y) * z) / (y * z) by Definition 5. Then R(x * L(x \ e,R(x,y,z),e / x),y,z) = R(x,y,z) * L(R(x,y,z) \ e,R(x,y,z),e / x) by (L5). Then R(x,y,z) * ((R(x,y,z) / x) \ ((e / x) * e)) = R(x * L(x \ e,R(x,y,z),e / x),y,z) by Theorem 534. Then R(x,y,z) * ((R(x,y,z) / x) \ (e / x)) = R(x * L(x \ e,R(x,y,z),e / x),y,z) by Axiom 2. Then R(a(R(x,y,z),e / x,x),y,z) = R(x,y,z) * ((R(x,y,z) / x) \ (e / x)) by Theorem 775. Then
(L11) R(x,y,z) * ((R(x,y,z) / x) \ (e / x)) = R(K(R(x,y,z),x \ e),y,z)
by Theorem 782. We have R(x,y,z) * ((R(x,y,z) * (R(x,y,z) \ e)) / R(x,y,z)) = R(x * ((R(x,y,z) * (x \ e)) / R(x,y,z)),y,z) by Theorem 642. Then
(L13) R(x,y,z) * (e / R(x,y,z)) = R(x * ((R(x,y,z) * (x \ e)) / R(x,y,z)),y,z)
by Axiom 4. We have
(L14) K(R(x,y,z),R(x,y,z) \ e) = R(x,y,z) * (e / R(x,y,z))
by Theorem 250. Then K(R(x,y,z),R(x,y,z) \ e) = R(x * ((R(x,y,z) * (x \ e)) / R(x,y,z)),y,z) by (L13). Then R(x * ((x \ R(x,y,z)) / R(x,y,z)),y,z) = K(R(x,y,z),R(x,y,z) \ e) by Proposition 89. Then R(K(R(x,y,z),x \ e),y,z) = K(R(x,y,z),R(x,y,z) \ e) by Theorem 781. Then R(x,y,z) * ((R(x,y,z) / x) \ (e / x)) = K(R(x,y,z),R(x,y,z) \ e) by (L11). Then
(L19) (R(x,y,z) / x) \ (e / x) = e / R(x,y,z)
by (L14) and Proposition 7. We have ((e / (e / x)) \ R(e / (e / x),y,z)) \ ((e / (e / x)) \ e) = R(e / (e / x),y,z) \ e by Theorem 773. Then ((e / (e / x)) \ R(e / (e / x),y,z)) \ (e / x) = R(e / (e / x),y,z) \ e by Proposition 25. Then
(L24) ((e / x) * R((e / x) \ e,y,z)) \ (e / x) = R(e / (e / x),y,z) \ e
by Theorem 187. We have R(e / (e / x),y,z) * (e / x) = (e / x) * R((e / x) \ e,y,z) by Proposition 83. Then R(x,y,z) / x = (e / x) * R((e / x) \ e,y,z) by Theorem 191. Then (R(x,y,z) / x) \ (e / x) = R(e / (e / x),y,z) \ e by (L24). Hence we are done by (L19).
Theorem 784 (R_prop): e / (e / R(x,y,z)) = R(e / (e / x),y,z)
Proof. We have e / (R(e / (e / x),y,z) \ e) = R(e / (e / x),y,z) by Proposition 24. Hence we are done by Theorem 783.
Theorem 785 (prov9_cbe4af6c5b): R(x,x \ y,z) = (y * z) / ((x \ y) * z)
Proof. We have R(x,x \ y,z) = ((x * (x \ y)) * z) / ((x \ y) * z) by Definition 5. Hence we are done by Axiom 4.
Theorem 786 (prov9_845f6e70af): R(x,y,(x * y) \ z) = z / (y * ((x * y) \ z))
Proof. We have R(x,y,(x * y) \ z) = ((x * y) * ((x * y) \ z)) / (y * ((x * y) \ z)) by Definition 5. Hence we are done by Axiom 4.
Theorem 787 (prov9_1bc67e7d4e): R(x,z,T(y,z)) = ((x * z) * T(y,z)) / (y * z)
Proof. We have R(x,z,T(y,z)) = ((x * z) * T(y,z)) / (z * T(y,z)) by Definition 5. Hence we are done by Proposition 46.
Theorem 788 (prov9_83e8f9153e): T(y,T(y,x)) = y
Proof. We have T(T(y,x),y) = T(y,x) by Theorem 451. Hence we are done by Proposition 21.
Theorem 789 (prov9_040b9fbaa2): T(y,x) = T(T(y,x),T(y,z))
Proof. We have T(T(y,T(y,z)),x) = T(T(y,x),T(y,z)) by Axiom 7. Hence we are done by Theorem 788.
Theorem 790 (prov9_f695d3986a): x \ ((x * y) * L(z,y,x)) = y * z
Proof. We have x \ (x * (y * z)) = y * z by Axiom 3. Then x \ ((x * y) * ((x * y) \ (x * (y * z)))) = y * z by Axiom 4. Hence we are done by Definition 4.
Theorem 791 (prov9_4a8a8adb4f): R(x \ e,x,y) = (K(x \ e,x) * y) / (x * y)
Proof. We have R(x \ e,x,y) = (((x \ e) * x) * y) / (x * y) by Definition 5. Hence we are done by Proposition 76.
Theorem 792 (prov9_9c93896ffc): T(x \ e,x) = x \ K(x \ e,x)
Proof. We have T(x \ e,x) = x \ ((x \ e) * x) by Definition 3. Hence we are done by Proposition 76.
Theorem 793 (prov9_8bb915e5ca): T(y,z) * T(y,x) = T(y,x) * T(y,z)
Proof. We have T(y,z) * T(T(y,x),T(y,z)) = T(y,x) * T(y,z) by Proposition 46. Hence we are done by Theorem 789.
Theorem 794 (prov9_3a753e04fe): R(y / x,x,y \ z) = z / (x * (y \ z))
Proof. We have (z / (y \ z)) * (y \ z) = z by Axiom 6. Then (((z / (y \ z)) / x) * x) * (y \ z) = z by Axiom 6. Then
(L3) ((y / x) * x) * (y \ z) = z
by Proposition 24. We have R(y / x,x,y \ z) = (((y / x) * x) * (y \ z)) / (x * (y \ z)) by Definition 5. Hence we are done by (L3).
Theorem 795 (prov9_2d7407fe98): (y \ z) \ z = x * (x \ T(y,y \ z))
Proof. We have (z / T(y,y \ z)) \ z = x * (x \ T(y,y \ z)) by Theorem 19. Hence we are done by Theorem 15.
Theorem 796 (prov9_b7edf9b8fd): x / L(y,e / y,x) = x * (e / y)
Proof. We have x / L((e / y) \ e,e / y,x) = x * (e / y) by Theorem 33. Hence we are done by Proposition 25.
Theorem 797 (prov9_2d8d30a905): R(z * y,L(x,y,z),w) = ((z * (y * x)) * w) / (L(x,y,z) * w)
Proof. We have R(z * y,L(x,y,z),w) = (((z * y) * L(x,y,z)) * w) / (L(x,y,z) * w) by Definition 5. Hence we are done by Proposition 52.
Theorem 798 (prov9_3ad3065ea0): L(T(x,y) \ z,T(x,y),y) = (x * y) \ (y * z)
Proof. We have L(T(x,y) \ z,T(x,y),y) = (y * T(x,y)) \ (y * z) by Proposition 53. Hence we are done by Proposition 46.
Theorem 799 (prov9_7d841233b6): L(x \ ((y * x) / y),x,y) = K(y,(y * x) / y)
Proof. We have (y * x) \ (y * ((y * x) / y)) = K(y,(y * x) / y) by Theorem 3. Hence we are done by Proposition 53.
Theorem 800 (prov9_e90b3ee71c): R(x / y,y,z) \ (x * z) = y * z
Proof. We have ((x * z) / (y * z)) \ (x * z) = y * z by Proposition 25. Hence we are done by Proposition 55.
Theorem 801 (prov9_b9d05e5d6b): R(z / x,x,T(y,z)) \ (y * z) = x * T(y,z)
Proof. We have R(z / x,x,T(y,z)) \ (z * T(y,z)) = x * T(y,z) by Theorem 800. Hence we are done by Proposition 46.
Theorem 802 (prov9_a397c88599): R(x * (w \ e),y,z) = R(R(x / w,y,z),w,w \ e)
Proof. We have R(R(x / w,w,w \ e),y,z) = R(R(x / w,y,z),w,w \ e) by Axiom 12. Hence we are done by Proposition 71.
Theorem 803 (prov9_6964b4abb8): x \ L(L(y,x \ e,x),z,w) = (x \ e) * L(y,z,w)
Proof. We have x \ L(L(y,z,w),x \ e,x) = (x \ e) * L(y,z,w) by Proposition 57. Hence we are done by Axiom 11.
Theorem 804 (prov9_b74af8d548): (z \ w) * L((z \ w) \ w,x,y) = L(z,x,y) * (z \ w)
Proof. We have (z \ w) * L(T(z,z \ w),x,y) = L(z,x,y) * (z \ w) by Proposition 58. Hence we are done by Proposition 49.
Theorem 805 (prov9_686ff032d1): y \ (L(x,z,y) * (y * z)) = z * T(x,y * z)
Proof. We have y \ ((y * z) * L(T(x,y * z),z,y)) = z * T(x,y * z) by Theorem 790. Hence we are done by Proposition 58.
Theorem 806 (prov9_553e0855a7): x * ((y \ z) \ z) = x * T(y,y \ z)
Proof. We have x * (x \ (x * T(y,y \ z))) = x * T(y,y \ z) by Axiom 4. Then x * (e * (e \ (x \ (x * T(y,y \ z))))) = x * T(y,y \ z) by Axiom 4. Then x * (e * (e \ T(y,y \ z))) = x * T(y,y \ z) by Axiom 3. Hence we are done by Theorem 795.
Theorem 807 (prov9_5ffd5c79dd): (z * T(x,x \ y)) / ((x \ y) \ y) = z
Proof. We have
(L1) ((z * T(x,x \ y)) / ((x \ y) \ y)) * T(x,x \ y) = ((z * T(x,x \ y)) / ((x \ y) \ y)) * ((x \ y) \ y)
by Theorem 806. We have z * T(x,x \ y) = ((z * T(x,x \ y)) / ((x \ y) \ y)) * ((x \ y) \ y) by Axiom 6. Hence we are done by (L1) and Proposition 8.
Theorem 808 (prov9_7b1aa21d1c): R(x,x \ (y / z),z) \ y = (x \ (y / z)) * z
Proof. We have e * y = y by Axiom 1. Then
(L2) y * (y \ (e * y)) = y
by Axiom 4. We have y * (y \ (e * y)) = e * y by Axiom 4. Then ((y * (y \ (e * y))) / z) * z = e * y by Axiom 6. Then
(L5) (y / z) * z = e * y
by (L2). We have e \ (e * y) = y by Axiom 3. Then
(L7) e \ (((e * y) / ((x \ (y / z)) * z)) * ((x \ (y / z)) * z)) = y
by Axiom 6. We have ((e * y) / ((x \ (y / z)) * z)) * ((x \ (y / z)) * z) = e \ (((e * y) / ((x \ (y / z)) * z)) * ((x \ (y / z)) * z)) by Proposition 26. Then (x \ (y / z)) * z = ((e * y) / ((x \ (y / z)) * z)) \ (e \ (((e * y) / ((x \ (y / z)) * z)) * ((x \ (y / z)) * z))) by Proposition 2. Then ((e * y) / ((x \ (y / z)) * z)) \ y = (x \ (y / z)) * z by (L7). Then (((y / z) * z) / ((x \ (y / z)) * z)) \ y = (x \ (y / z)) * z by (L5). Hence we are done by Theorem 785.
Theorem 809 (prov9_078b26e255): R(T(y,x),x / y,y) = T(y,x / y)
Proof. We have R(T(y,(x / y) * y),x / y,y) = T(y,x / y) by Theorem 113. Hence we are done by Axiom 6.
Theorem 810 (prov9_094233280c): K(x \ e,x) \ x = T(x,x \ e)
Proof. We have e * x = x by Axiom 1. Then
(L2) (x * (x \ e)) * x = x
by Axiom 4. We have ((x \ e) * x) * T(x,x \ e) = (x * (x \ e)) * x by Theorem 114. Then ((x \ e) * x) \ ((x * (x \ e)) * x) = T(x,x \ e) by Proposition 2. Then T(x,x \ e) = ((x \ e) * x) \ x by (L2). Hence we are done by Proposition 76.
Theorem 811 (prov9_263c90dc0e): R(R(y / x,z,w),x,x \ e) / (x \ e) = x * R(x \ y,z,w)
Proof. We have R(R(y / x,z,w),x,x \ e) / (x \ e) = R(y / x,z,w) * x by Theorem 73. Hence we are done by Proposition 83.
Theorem 812 (prov9_555f37a177): z * R(x,y,y \ z) = (((z * x) / z) * y) * (y \ z)
Proof. We have R((z * x) / z,y,y \ z) * z = (((z * x) / z) * y) * (y \ z) by Theorem 25. Hence we are done by Theorem 87.
Theorem 813 (prov9_d8f48a9cc1): ((x / y) * R(y,z,w)) / (x / y) = R(x / (x / y),z,w)
Proof. We have ((x / y) * R((x / y) \ x,z,w)) / (x / y) = R(x / (x / y),z,w) by Theorem 88. Hence we are done by Proposition 25.
Theorem 814 (prov9_270a7a6ecd): K(x,e / x) \ (e / x) = x \ e
Proof. We have T(e / x,(e / x) \ e) = ((e / x) \ e) \ e by Proposition 49. Then K((e / x) \ e,e / x) \ (e / x) = ((e / x) \ e) \ e by Theorem 810. Then K(x,e / x) \ (e / x) = ((e / x) \ e) \ e by Proposition 25. Hence we are done by Proposition 25.
Theorem 815 (prov9_c3e00aa628): (x * y) * ((w * L(z,y,x)) / w) = x * (y * ((w * z) / w))
Proof. We have (x * y) * L((w * z) / w,y,x) = x * (y * ((w * z) / w)) by Proposition 52. Hence we are done by Theorem 122.
Theorem 816 (prov9_da33de7593): (x * y) * L(L(z,T(x,y),y),w,u) = y * (T(x,y) * L(z,w,u))
Proof. We have (x * y) * L(L(z,w,u),T(x,y),y) = y * (T(x,y) * L(z,w,u)) by Theorem 58. Hence we are done by Axiom 11.
Theorem 817 (prov9_d6dfe46a45): R((w * u) \ (u * x),y,z) = L(R(T(w,u) \ x,y,z),T(w,u),u)
Proof. We have R(L(T(w,u) \ x,T(w,u),u),y,z) = L(R(T(w,u) \ x,y,z),T(w,u),u) by Axiom 10. Hence we are done by Theorem 798.
Theorem 818 (prov9_3582eddc34): (x * y) / x = (y / x) * T(x,y)
Proof. We have (x * T((y / x) * T(x,y),x)) / x = (y / x) * T(x,y) by Proposition 48. Hence we are done by Theorem 493.
Theorem 819 (prov9_0b3df48f91): T(x,y) / (z \ e) = T((x / (z \ e)) / z,y) * z
Proof. We have T((x / (z \ e)) * (z \ e),y) / (z \ e) = T((x / (z \ e)) / z,y) * z by Theorem 490. Hence we are done by Axiom 6.
Theorem 820 (prov9_a82875fd46): y \ T(x,y \ e) = ((y \ e) \ (y \ x)) * (y \ e)
Proof. We have (y \ e) * T((y \ e) \ (y \ x),y \ e) = ((y \ e) \ (y \ x)) * (y \ e) by Proposition 46. Hence we are done by Theorem 496.
Theorem 821 (prov9_495d31d1d0): (y \ (x / (y \ e))) * y = T(x,y) / (y \ e)
Proof. We have T((x / (y \ e)) / y,y) * y = T(x,y) / (y \ e) by Theorem 819. Hence we are done by Proposition 47.
Theorem 822 (prov9_4ef55507a4): R(y,x,T(y,x)) = x \ T(y * x,y)
Proof. We have ((y * x) * T(y,x)) / (y * x) = T(((y * x) * y) / (y * x),x) by Theorem 50. Then R(y,x,T(y,x)) = T(((y * x) * y) / (y * x),x) by Theorem 787. Then
(L3) T(T(y * x,y) / x,x) = R(y,x,T(y,x))
by Theorem 508. We have T(T(y * x,y) / x,x) = x \ T(y * x,y) by Proposition 47. Hence we are done by (L3).
Theorem 823 (prov9_725a97fd92): L(R(y,x,T(y,x)),x,y) = y
Proof. We have
(L1) (y * x) * L(x \ T(y * x,y),x,y) = y * (x * (x \ T(y * x,y)))
by Proposition 52. We have (y * x) * y = y * (x * (x \ T(y * x,y))) by Theorem 9. Then L(x \ T(y * x,y),x,y) = y by (L1) and Proposition 7. Hence we are done by Theorem 822.
Theorem 824 (prov9_9f70398aa7): (x / y) * ((x * y) / x) = T(x,x / (x / y))
Proof. We have (x / y) * ((x * ((x / y) \ x)) / x) = T(x,x / (x / y)) by Theorem 515. Hence we are done by Proposition 25.
Theorem 825 (prov9_0c25b44a0e): (y / T(x,y)) * x = T(y,y / (y / T(x,y)))
Proof. We have (y / T(x,y)) * ((y * T(x,y)) / y) = T(y,y / (y / T(x,y))) by Theorem 824. Hence we are done by Proposition 48.
Theorem 826 (prov9_55d9c487b4): x \ R(x * y,z,w) = (x \ e) * R((x \ e) \ y,z,w)
Proof. We have x \ R(L((x \ e) \ y,x \ e,x),z,w) = (x \ e) * R((x \ e) \ y,z,w) by Theorem 474. Hence we are done by Theorem 62.
Theorem 827 (prov9_673d201758): x \ L(x * y,z,w) = (x \ e) * L((x \ e) \ y,z,w)
Proof. We have x \ L(L((x \ e) \ y,x \ e,x),z,w) = (x \ e) * L((x \ e) \ y,z,w) by Theorem 803. Hence we are done by Theorem 62.
Theorem 828 (prov9_88f5918a54): (y * R(x,z,z \ y)) / (z \ y) = ((y * x) / y) * z
Proof. We have (((y * x) / y) * z) * (z \ y) = y * R(x,z,z \ y) by Theorem 812. Hence we are done by Proposition 1.
Theorem 829 (prov9_85ec3cb8d8): (x * y) * L((x * (y \ e)) / x,y,x) = T(x,x / (x * y))
Proof. We have (x * y) * ((x * ((x * y) \ x)) / x) = T(x,x / (x * y)) by Theorem 515. Hence we are done by Theorem 127.
Theorem 830 (prov9_80b8334467): T(y,x) * z = y * (T(y,x) * (y \ z))
Proof. We have T(y,x) * (y * (y \ z)) = y * (T(y,x) * (y \ z)) by Theorem 172. Hence we are done by Axiom 4.
Theorem 831 (prov9_9f0b784152): (T(z,x) * y) / (T(z,x) * (z \ y)) = z
Proof. We have (T(z,x) * y) / (z \ (T(z,x) * y)) = z by Proposition 24. Hence we are done by Theorem 173.
Theorem 832 (prov9_6a4d0068c4): z * (T(z,y) \ ((z \ T(z,y)) * x)) = L(x,z \ T(z,y),z)
Proof. We have T(z,y) \ (z * ((z \ T(z,y)) * x)) = L(x,z \ T(z,y),z) by Theorem 449. Hence we are done by Theorem 174.
Theorem 833 (prov9_30a54ee612): T(y,z) * (y \ T(y,x)) = T(y,x) * (y \ T(y,z))
Proof. We have y \ (T(y,z) * T(y,x)) = T(y,z) * (y \ T(y,x)) by Theorem 173. Then
(L2) y \ (T(y,x) * T(y,z)) = T(y,z) * (y \ T(y,x))
by Theorem 793. We have y \ (T(y,x) * T(y,z)) = T(y,x) * (y \ T(y,z)) by Theorem 173. Hence we are done by (L2).
Theorem 834 (prov9_b75293408d): (x \ R(x,y,z)) / (x \ e) = R(x,y,z)
Proof. We have R(x,y,z) * (x \ e) = x \ R(x,y,z) by Proposition 89. Hence we are done by Proposition 1.
Theorem 835 (prov9_c9ea83aa3f): R(z,x,y) \ (z \ R(z,x,y)) = z \ e
Proof. We have R(z,x,y) * (z \ e) = z \ R(z,x,y) by Proposition 89. Hence we are done by Proposition 2.
Theorem 836 (prov9_a5d57961d6): (x \ e) * L((x \ e) \ e,y,z) = x \ L(x,y,z)
Proof. We have (x \ e) * L((x \ e) \ e,y,z) = x \ L(x * e,y,z) by Theorem 827. Hence we are done by Axiom 2.
Theorem 837 (prov9_9c638c4e39): z \ L(z,x,y) = L(z,x,y) * (z \ e)
Proof. We have (z \ e) * L((z \ e) \ e,x,y) = L(z,x,y) * (z \ e) by Theorem 804. Hence we are done by Theorem 836.
Theorem 838 (prov9_bb08aa8c2c): ((z \ e) / (z \ L(z,x,y))) * L(z,x,y) = L(z,x,y) * R(L(z,x,y) \ e,L(z,x,y),z \ e)
Proof. We have ((z \ e) / (L(z,x,y) * (z \ e))) * L(z,x,y) = L(z,x,y) * R(L(z,x,y) \ e,L(z,x,y),z \ e) by Theorem 90. Hence we are done by Theorem 837.
Theorem 839 (prov9_b677e9dba9): T(e / z,R(z,x,y)) = R(z,x,y) \ (R(z,x,y) / z)
Proof. We have T(e / z,R(z,x,y)) = R(z,x,y) \ ((e / z) * R(z,x,y)) by Definition 3. Hence we are done by Theorem 192.
Theorem 840 (prov9_a8258529f2): ((y \ (y / x)) * y) * x = y * (x * T(x \ e,y))
Proof. We have (y * x) * T((y * x) \ y,y) = y * (x * T(x \ e,y)) by Theorem 98. Hence we are done by Theorem 530.
Theorem 841 (prov9_22b72f9b9c): (x * T(x \ e,z)) * (x \ y) = y * T(y \ (x \ y),z)
Proof. We have (T(e / x,z) * x) * (x \ y) = y * T(y \ (x \ y),z) by Theorem 183. Hence we are done by Proposition 61.
Theorem 842 (prov9_0e9e37d178): K(x \ e,x) * x = x * K(x \ e,x)
Proof. We have K(x \ e,x) * T(x,K(x \ e,x)) = x * K(x \ e,x) by Proposition 46. Hence we are done by Theorem 755.
Theorem 843 (prov9_92db09f064): (x * T(x \ (y \ x),z)) / (y \ x) = y * T(y \ e,z)
Proof. We have (y * T(y \ e,z)) * (y \ x) = x * T(x \ (y \ x),z) by Theorem 841. Hence we are done by Proposition 1.
Theorem 844 (prov9_3367889222): (y * x) * ((w * ((y * x) \ z)) / w) = y * (x * ((w * (x \ (y \ z))) / w))
Proof. We have (y * x) * ((w * L(x \ (y \ z),x,y)) / w) = y * (x * ((w * (x \ (y \ z))) / w)) by Theorem 815. Hence we are done by Proposition 70.
Theorem 845 (prov9_bf2c4a4563): T(y,y / (y * x)) = y * (x * ((y * (x \ e)) / y))
Proof. We have (y * x) * L((y * (x \ e)) / y,x,y) = y * (x * ((y * (x \ e)) / y)) by Proposition 52. Hence we are done by Theorem 829.
Theorem 846 (prov9_7da9e210d7): y \ T(y,y / (y * x)) = x * ((y * (x \ e)) / y)
Proof. We have y \ ((y * x) * L((y * (x \ e)) / y,x,y)) = x * ((y * (x \ e)) / y) by Theorem 790. Hence we are done by Theorem 829.
Theorem 847 (prov9_42dae10454): x * y = ((x / T(y,z)) * y) * T(y,z)
Proof. We have ((x / T(y,z)) * T(y,z)) * y = ((x / T(y,z)) * y) * T(y,z) by Theorem 238. Hence we are done by Axiom 6.
Theorem 848 (prov9_01eebde8d8): y * (y * x) = T(y,y / (y / T(x,y))) * (x * y)
Proof. We have ((y / T(x,y)) * x) * T(x,y) = y * x by Theorem 847. Then
(L2) T(y,y / (y / T(x,y))) * T(x,y) = y * x
by Theorem 825. We have y * (T(y,y / (y / T(x,y))) * T(x,y)) = T(y,y / (y / T(x,y))) * (x * y) by Theorem 520. Hence we are done by (L2).
Theorem 849 (prov9_87177dc353): K(x,x \ e) * (x \ e) = e / x
Proof. We have K(x,e / x) * (K(x,e / x) \ (e / x)) = e / x by Axiom 4. Then K(x,e / x) * (x \ e) = e / x by Theorem 814. Hence we are done by Theorem 780.
Theorem 850 (prov9_17821100a3): (e / (e / x)) / (x * x) = R(x \ e,x,x)
Proof. We have (K(x \ e,x) * x) / (x * x) = R(x \ e,x,x) by Theorem 791. Then (x * K(x \ e,x)) / (x * x) = R(x \ e,x,x) by Theorem 842. Hence we are done by Theorem 262.
Theorem 851 (prov9_6a60610f18): R(e / (e / z),x,y) = L(R(z,x,y),z \ e,z)
Proof. We have R(L(z,z \ e,z),x,y) = L(R(z,x,y),z \ e,z) by Axiom 10. Hence we are done by Theorem 263.
Theorem 852 (prov9_941ef09a58): K(z \ (z / (x \ (x / y))),z) = x \ ((x / y) * T(y,z))
Proof. We have K(z \ (z / (x \ (x / y))),z) = x \ ((x / y) * T((x / y) \ x,z)) by Theorem 285. Hence we are done by Proposition 25.
Theorem 853 (prov9_5964faec58): K(x \ (x / (x \ (x / y))),x) = x \ ((y * x) / y)
Proof. We have K(x \ (x / (x \ (x / y))),x) = x \ ((x / y) * T(y,x)) by Theorem 852. Hence we are done by Theorem 818.
Theorem 854 (prov9_0e52f0e656): y * (((x * y) / x) * z) = ((x * y) / x) * (y * z)
Proof. We have ((x * y) / x) * (T((x * y) / x,x) * z) = (x \ (x * y)) * (((x * y) / x) * z) by Theorem 521. Then ((x * y) / x) * (y * z) = (x \ (x * y)) * (((x * y) / x) * z) by Theorem 7. Hence we are done by Axiom 3.
Theorem 855 (prov9_e0d51dc6ce): y * (((x * y) / x) \ z) = ((x * y) / x) \ (y * z)
Proof. We have y \ (y * z) = z by Axiom 3. Then
(L3) y \ (((x * y) / x) * (((x * y) / x) \ (y * z))) = z
by Axiom 4. We have y * (y \ (((x * y) / x) * (((x * y) / x) \ (y * z)))) = ((x * y) / x) * (((x * y) / x) \ (y * z)) by Axiom 4. Then y * (((x * y) / x) * (((x * y) / x) \ (y \ (((x * y) / x) * (((x * y) / x) \ (y * z)))))) = ((x * y) / x) * (((x * y) / x) \ (y * z)) by Axiom 4. Then
(L6) y * (((x * y) / x) * (((x * y) / x) \ z)) = ((x * y) / x) * (((x * y) / x) \ (y * z))
by (L3). We have ((x * y) / x) * (y * (((x * y) / x) \ z)) = y * (((x * y) / x) * (((x * y) / x) \ z)) by Theorem 854. Hence we are done by (L6) and Proposition 7.
Theorem 856 (prov9_20b97c2068): y * ((x / (x / y)) * z) = (x / (x / y)) * (y * z)
Proof. We have (x / (x / y)) * (T(x / (x / y),x / y) * z) = ((x / y) \ x) * ((x / (x / y)) * z) by Theorem 521. Then (x / (x / y)) * (y * z) = ((x / y) \ x) * ((x / (x / y)) * z) by Theorem 14. Hence we are done by Proposition 25.
Theorem 857 (prov9_7611b0fe35): y * ((x / (x / y)) \ z) = (x / (x / y)) \ (y * z)
Proof. We have y \ (y * z) = z by Axiom 3. Then
(L3) y \ ((x / (x / y)) * ((x / (x / y)) \ (y * z))) = z
by Axiom 4. We have y * (y \ ((x / (x / y)) * ((x / (x / y)) \ (y * z)))) = (x / (x / y)) * ((x / (x / y)) \ (y * z)) by Axiom 4. Then y * ((x / (x / y)) * ((x / (x / y)) \ (y \ ((x / (x / y)) * ((x / (x / y)) \ (y * z)))))) = (x / (x / y)) * ((x / (x / y)) \ (y * z)) by Axiom 4. Then
(L6) y * ((x / (x / y)) * ((x / (x / y)) \ z)) = (x / (x / y)) * ((x / (x / y)) \ (y * z))
by (L3). We have (x / (x / y)) * (y * ((x / (x / y)) \ z)) = y * ((x / (x / y)) * ((x / (x / y)) \ z)) by Theorem 856. Hence we are done by (L6) and Proposition 7.
Theorem 858 (prov9_cab9a4497e): (e / y) * ((x * y) / x) = ((x * y) / x) / y
Proof. We have (e / y) * ((x * ((e / y) \ e)) / x) = ((x * y) / x) / y by Theorem 290. Hence we are done by Proposition 25.
Theorem 859 (prov9_bbf024ddd8): y * ((x * (y \ e)) / x) = ((x * (y \ e)) / x) / (y \ e)
Proof. We have (e / (y \ e)) * ((x * (y \ e)) / x) = ((x * (y \ e)) / x) / (y \ e) by Theorem 858. Hence we are done by Proposition 24.
Theorem 860 (prov9_2d0ca6619a): (x * z) * ((y * z) / y) = (x * ((y * z) / y)) * z
Proof. We have (x * ((y * z) / y)) * T((y * z) / y,y) = (x * (y \ (y * z))) * ((y * z) / y) by Theorem 240. Then (x * ((y * z) / y)) * z = (x * (y \ (y * z))) * ((y * z) / y) by Theorem 7. Hence we are done by Axiom 3.
Theorem 861 (prov9_3023c0e538): T(y,y / (y / x)) * (y \ x) = (y * x) / y
Proof. We have ((y / x) * ((y * x) / y)) * x = ((y / x) * x) * ((y * x) / y) by Theorem 860. Then
(L2) T(y,y / (y / x)) * x = ((y / x) * x) * ((y * x) / y)
by Theorem 824.
y * (T(y,y / (y / x)) * (y \ x))
=T(y,y / (y / x)) * x by Theorem 830
=y * ((y * x) / y) by (L2), Axiom 6

Then y * (T(y,y / (y / x)) * (y \ x)) = y * ((y * x) / y). Hence we are done by Proposition 9.
Theorem 862 (prov9_30e3757fbd): R(x \ e,L(x,y,z),x \ e) = T(x \ e,L(x,y,z))
Proof. We have T(x \ e,(x \ e) * L((x \ e) \ e,y,z)) = x \ e by Theorem 304. Then
(L2) T(x \ e,L(x,y,z) * (x \ e)) = x \ e
by Theorem 804. We have R(T(x \ e,L(x,y,z) * (x \ e)),L(x,y,z),x \ e) = T(x \ e,L(x,y,z)) by Theorem 113. Hence we are done by (L2).
Theorem 863 (prov9_743b82cd03): R(y * (x \ e),z,w) / (x \ e) = x * R(x \ y,z,w)
Proof. We have R(R(y / x,z,w),x,x \ e) / (x \ e) = x * R(x \ y,z,w) by Theorem 811. Hence we are done by Theorem 802.
Theorem 864 (prov9_57a371ed6b): z * R(e / z,x,y) = R(e / z,x,y) / (z \ e)
Proof. We have R(K(z,z \ e) * (z \ e),x,y) / (z \ e) = z * R(z \ K(z,z \ e),x,y) by Theorem 863. Then R(e / z,x,y) / (z \ e) = z * R(z \ K(z,z \ e),x,y) by Theorem 849. Hence we are done by Theorem 251.
Theorem 865 (prov9_0784cc30d7): T(y,T(e / y,x)) = R(y,T(e / y,x),y)
Proof. We have R(T(y,y * T(y \ e,x)),(y * T(y \ e,x)) / y,y) = T(y,(y * T(y \ e,x)) / y) by Theorem 809. Then R(y,(y * T(y \ e,x)) / y,y) = T(y,(y * T(y \ e,x)) / y) by Theorem 203. Then R(y,T(e / y,x),y) = T(y,(y * T(y \ e,x)) / y) by Theorem 49. Hence we are done by Theorem 49.
Theorem 866 (prov9_9a4c925a42): (x / y) * (y / x) = (y / x) / ((x / y) \ e)
Proof. We have R(e / (x / y),x / y,y) / ((x / y) \ e) = (x / y) * R(e / (x / y),x / y,y) by Theorem 864. Then (y / x) / ((x / y) \ e) = (x / y) * R(e / (x / y),x / y,y) by Theorem 38. Hence we are done by Theorem 38.
Theorem 867 (prov9_bb461b28bf): ((x \ y) / y) * x = x / (((x \ y) / y) \ e)
Proof. We have (y / (x \ y)) / (((x \ y) / y) \ e) = ((x \ y) / y) * (y / (x \ y)) by Theorem 866. Then x / (((x \ y) / y) \ e) = ((x \ y) / y) * (y / (x \ y)) by Proposition 24. Hence we are done by Proposition 24.
Theorem 868 (prov9_60dc56642a): (x * L(y,z,w)) * y = (x * y) * L(y,z,w)
Proof. We have L(((x * y) * y) / (x * y),z,w) * (x * y) = (x * y) * L(y,z,w) by Theorem 86. Then (L((x * y) / x,z,w) * x) * y = (x * y) * L(y,z,w) by Theorem 157. Hence we are done by Theorem 86.
Theorem 869 (prov9_441cf8765d): z / L(z,x,y) = (e / L(z,x,y)) * z
Proof. We have e * z = z by Axiom 1. Then
(L3) z * (z \ (e * z)) = z
by Axiom 4. We have z * (z \ (e * z)) = e * z by Axiom 4. Then ((z * (z \ (e * z))) / L(z,x,y)) * L(z,x,y) = e * z by Axiom 6. Then (z / L(z,x,y)) * L(z,x,y) = e * z by (L3). Then
(L7) (z / L(z,x,y)) * L(z,x,y) = ((e / L(z,x,y)) * L(z,x,y)) * z
by Axiom 6. We have ((e / L(z,x,y)) * L(z,x,y)) * z = ((e / L(z,x,y)) * z) * L(z,x,y) by Theorem 868. Then (z / L(z,x,y)) * L(z,x,y) = ((e / L(z,x,y)) * z) * L(z,x,y) by (L7). Hence we are done by Proposition 10.
Theorem 870 (prov9_3aabf03e8b): x * K(x,(x / y) / x) = T(x,x / (x * y))
Proof. We have x * (y * ((x * (y \ e)) / x)) = T(x,x / (x * y)) by Theorem 845. Hence we are done by Theorem 333.
Theorem 871 (prov9_342c8e6511): T(x,R(y,x,(y * x) \ x)) = x * K(x,y)
Proof. We have x * K(x,(x / ((y * x) \ x)) / x) = T(x,x / (x * ((y * x) \ x))) by Theorem 870. Then x * K(x,(y * x) / x) = T(x,x / (x * ((y * x) \ x))) by Proposition 24. Then T(x,R(y,x,(y * x) \ x)) = x * K(x,(y * x) / x) by Theorem 786. Hence we are done by Axiom 5.
Theorem 872 (prov9_84e8484d27): (y * L(y \ (z / x),w,u)) * x = y * (x * L(x \ (y \ z),w,u))
Proof. We have L(R((z / x) / y,y,x),w,u) * (y * x) = (L((z / x) / y,w,u) * y) * x by Theorem 65. Then
(L2) L(z / (y * x),w,u) * (y * x) = (L((z / x) / y,w,u) * y) * x
by Proposition 74. We have L(z / (y * x),w,u) * (y * x) = (y * x) * L((y * x) \ z,w,u) by Theorem 85. Then (L((z / x) / y,w,u) * y) * x = (y * x) * L((y * x) \ z,w,u) by (L2). Then
(L5) (y * L(y \ (z / x),w,u)) * x = (y * x) * L((y * x) \ z,w,u)
by Theorem 85. We have (y * x) * L((y * x) \ z,w,u) = y * (x * L(x \ (y \ z),w,u)) by Theorem 155. Hence we are done by (L5).
Theorem 873 (prov9_7f135db2ee): (x * R(x \ y,w,u)) * z = x * (z * R(z \ (x \ (y * z)),w,u))
Proof. We have (x * z) * R((x * z) \ (y * z),w,u) = x * (z * R(z \ (x \ (y * z)),w,u)) by Proposition 87. Hence we are done by Theorem 594.
Theorem 874 (prov9_bff27a8904): T(y * L(y \ (z / x),w,u),x) = x \ (y * (x * L(x \ (y \ z),w,u)))
Proof. We have T(y * L(y \ (z / x),w,u),x) = x \ ((y * L(y \ (z / x),w,u)) * x) by Definition 3. Hence we are done by Theorem 872.
Theorem 875 (prov9_2c4477413e): x \ (y * ((w * (y \ (x * z))) / w)) = (x \ y) * ((w * ((x \ y) \ z)) / w)
Proof. We have x * ((x \ y) * ((w * ((x \ y) \ z)) / w)) = y * ((w * (y \ (x * z))) / w) by Theorem 318. Hence we are done by Proposition 2.
Theorem 876 (prov9_f8a82689f2): x \ (y * ((z * (y \ x)) / z)) = (x \ y) * ((z * ((x \ y) \ e)) / z)
Proof. We have x \ (y * ((z * (y \ (x * e))) / z)) = (x \ y) * ((z * ((x \ y) \ e)) / z) by Theorem 875. Hence we are done by Axiom 2.
Theorem 877 (prov9_743ea6251b): (L(x \ e,x,y) * y) / L(x \ e,x,y) = y * (x * T(x \ e,y))
Proof. We have (y / L(x \ e,x,y)) * T(L(x \ e,x,y),y) = (L(x \ e,x,y) * y) / L(x \ e,x,y) by Theorem 818. Then
(L2) (y * x) * T(L(x \ e,x,y),y) = (L(x \ e,x,y) * y) / L(x \ e,x,y)
by Theorem 33. We have (y * x) * T(L(x \ e,x,y),y) = y * (x * T(x \ e,y)) by Theorem 55. Hence we are done by (L2).
Theorem 878 (prov9_4957b31c78): y * (T(x,y) * L(T(x,y) \ z,w,u)) = x * (y * L(y \ (x \ (y * z)),w,u))
Proof. We have (x * y) * L(L(T(x,y) \ z,T(x,y),y),w,u) = y * (T(x,y) * L(T(x,y) \ z,w,u)) by Theorem 816. Then
(L2) (x * y) * L((x * y) \ (y * z),w,u) = y * (T(x,y) * L(T(x,y) \ z,w,u))
by Theorem 798. We have (x * y) * L((x * y) \ (y * z),w,u) = x * (y * L(y \ (x \ (y * z)),w,u)) by Theorem 155. Hence we are done by (L2).
Theorem 879 (prov9_26d82c9212): T(x,u) * L(T(x,u) \ y,z,w) = T(x * L(x \ ((u * y) / u),z,w),u)
Proof. We have u * (T(x,u) * L(T(x,u) \ y,z,w)) = x * (u * L(u \ (x \ (u * y)),z,w)) by Theorem 878. Then
(L2) u \ (x * (u * L(u \ (x \ (u * y)),z,w))) = T(x,u) * L(T(x,u) \ y,z,w)
by Proposition 2. We have u \ (x * (u * L(u \ (x \ (u * y)),z,w))) = T(x * L(x \ ((u * y) / u),z,w),u) by Theorem 874. Hence we are done by (L2).
Theorem 880 (prov9_71bc4865f9): (z * ((y / x) / z)) * x = (z * x) * ((z * ((z * x) \ y)) / z)
Proof. We have (z * x) * ((z * ((z * x) \ ((y / x) * x))) / z) = (((z * ((y / x) / z)) / z) * z) * x by Theorem 608. Then (z * x) * ((z * ((z * x) \ y)) / z) = (((z * ((y / x) / z)) / z) * z) * x by Axiom 6. Hence we are done by Axiom 6.
Theorem 881 (prov9_fc14da0361): (x * ((x / y) / x)) * y = T(x,x / (x * y))
Proof. We have (x * y) * ((x * ((x * y) \ x)) / x) = T(x,x / (x * y)) by Theorem 515. Hence we are done by Theorem 880.
Theorem 882 (prov9_5596b61626): (z * ((y / x) / z)) * x = z * (x * ((z * (x \ (z \ y))) / z))
Proof. We have (z * x) * ((z * ((z * x) \ y)) / z) = z * (x * ((z * (x \ (z \ y))) / z)) by Theorem 844. Hence we are done by Theorem 880.
Theorem 883 (prov9_d19b8b8e2a): (L(y,e / y,x) * x) / L(y,e / y,x) = x * (T(y,x) / y)
Proof. We have (x / L(y,e / y,x)) * T(L(y,e / y,x),x) = (L(y,e / y,x) * x) / L(y,e / y,x) by Theorem 818. Then
(L2) (x * (e / y)) * T(L(y,e / y,x),x) = (L(y,e / y,x) * x) / L(y,e / y,x)
by Theorem 796. We have (x * (e / y)) * T(L(y,e / y,x),x) = x * ((e / y) * T(y,x)) by Theorem 55. Then (L(y,e / y,x) * x) / L(y,e / y,x) = x * ((e / y) * T(y,x)) by (L2). Hence we are done by Theorem 150.
Theorem 884 (prov9_35d51b349e): (z * (y * T(y \ e,w))) * x = (z * y) * (x * T(x \ L(y \ x,y,z),w))
Proof. We have (z * y) * (x * T(x \ ((z * y) \ (z * x)),w)) = ((z * y) * T((z * y) \ z,w)) * x by Theorem 301. Then (z * y) * (x * T(x \ L(y \ x,y,z),w)) = ((z * y) * T((z * y) \ z,w)) * x by Proposition 53. Hence we are done by Theorem 98.
Theorem 885 (prov9_444bf981cd): (z \ e) * R(z,x,y) = T((z \ e) * R(z,x,y),z \ e)
Proof. We have T(z \ e,z \ R(z,x,y)) = z \ e by Theorem 578. Then
(L2) T(z \ R(z,x,y),z \ e) = z \ R(z,x,y)
by Proposition 21. We have (z \ e) * (T(R(z,x,y) * (z \ e),z \ e) / (z \ e)) = T((z \ e) * R(z,x,y),z \ e) by Theorem 584. Then (z \ e) * (T(z \ R(z,x,y),z \ e) / (z \ e)) = T((z \ e) * R(z,x,y),z \ e) by Proposition 89. Then (z \ e) * ((z \ R(z,x,y)) / (z \ e)) = T((z \ e) * R(z,x,y),z \ e) by (L2). Hence we are done by Theorem 834.
Theorem 886 (prov9_2c13594536): R(T(x,T(e / x,R(x,y,z))),y,z) = R(T(x,x \ e),y,z)
Proof. We have L((e / (e / (R(x,y,z) \ (R(x,y,z) / x)))) / ((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * ((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) = ((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) * L(((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) \ (e / (e / (R(x,y,z) \ (R(x,y,z) / x)))),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) by Theorem 85. Then
(L5) L(R((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z) \ (R(x,y,z) / x)),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * ((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) = ((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) * L(((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) \ (e / (e / (R(x,y,z) \ (R(x,y,z) / x)))),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z))
by Theorem 850. We have L(R((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z) \ (R(x,y,z) / x)),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * ((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) = (L((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * (R(x,y,z) \ (R(x,y,z) / x))) * (R(x,y,z) \ (R(x,y,z) / x)) by Theorem 65. Then
(L7) ((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) * L(((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) \ (e / (e / (R(x,y,z) \ (R(x,y,z) / x)))),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) = (L((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * (R(x,y,z) \ (R(x,y,z) / x))) * (R(x,y,z) \ (R(x,y,z) / x))
by (L5). We have ((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) * L(((R(x,y,z) \ (R(x,y,z) / x)) * (R(x,y,z) \ (R(x,y,z) / x))) \ (e / (e / (R(x,y,z) \ (R(x,y,z) / x)))),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) = (R(x,y,z) \ (R(x,y,z) / x)) * ((R(x,y,z) \ (R(x,y,z) / x)) * L((R(x,y,z) \ (R(x,y,z) / x)) \ ((R(x,y,z) \ (R(x,y,z) / x)) \ (e / (e / (R(x,y,z) \ (R(x,y,z) / x))))),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z))) by Theorem 155. Then (L((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * (R(x,y,z) \ (R(x,y,z) / x))) * (R(x,y,z) \ (R(x,y,z) / x)) = (R(x,y,z) \ (R(x,y,z) / x)) * ((R(x,y,z) \ (R(x,y,z) / x)) * L((R(x,y,z) \ (R(x,y,z) / x)) \ ((R(x,y,z) \ (R(x,y,z) / x)) \ (e / (e / (R(x,y,z) \ (R(x,y,z) / x))))),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z))) by (L7). Then (R(x,y,z) \ (R(x,y,z) / x)) * ((R(x,y,z) \ (R(x,y,z) / x)) * L((R(x,y,z) \ (R(x,y,z) / x)) \ K((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x)),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z))) = (L((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * (R(x,y,z) \ (R(x,y,z) / x))) * (R(x,y,z) \ (R(x,y,z) / x)) by Theorem 257. Then (R(x,y,z) \ (R(x,y,z) / x)) * ((R(x,y,z) \ (R(x,y,z) / x)) * L(T((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x)),R(x,y,z) \ (R(x,y,z) / x),R(x,y,z))) = (L((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * (R(x,y,z) \ (R(x,y,z) / x))) * (R(x,y,z) \ (R(x,y,z) / x)) by Theorem 792. Then (R(x,y,z) \ (R(x,y,z) / x)) * (L((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * (R(x,y,z) \ (R(x,y,z) / x))) = (L((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * (R(x,y,z) \ (R(x,y,z) / x))) * (R(x,y,z) \ (R(x,y,z) / x)) by Proposition 58. Then T(R(x,y,z) \ (R(x,y,z) / x),L((R(x,y,z) \ (R(x,y,z) / x)) \ e,R(x,y,z) \ (R(x,y,z) / x),R(x,y,z)) * (R(x,y,z) \ (R(x,y,z) / x))) = R(x,y,z) \ (R(x,y,z) / x) by Theorem 11. Then T(R(x,y,z) \ (R(x,y,z) / x),((R(x,y,z) / x) \ R(x,y,z)) * (R(x,y,z) \ (R(x,y,z) / x))) = R(x,y,z) \ (R(x,y,z) / x) by Theorem 28. Then
(L15) T(R(x,y,z) \ (R(x,y,z) / x),x * (R(x,y,z) \ (R(x,y,z) / x))) = R(x,y,z) \ (R(x,y,z) / x)
by Proposition 25. We have R(T(R(x,y,z) \ (R(x,y,z) / x),x * (R(x,y,z) \ (R(x,y,z) / x))),x,R(x,y,z) \ (R(x,y,z) / x)) = T(R(x,y,z) \ (R(x,y,z) / x),x) by Theorem 113. Then
(L17) R(R(x,y,z) \ (R(x,y,z) / x),x,R(x,y,z) \ (R(x,y,z) / x)) = T(R(x,y,z) \ (R(x,y,z) / x),x)
by (L15). We have R(R(x,y,z) \ (R(x,y,z) / x),x,R(x,y,z) \ (R(x,y,z) / x)) \ T((R(x,y,z) \ (R(x,y,z) / x)) * x,R(x,y,z) \ (R(x,y,z) / x)) = T(x,R(x,y,z) \ (R(x,y,z) / x)) by Theorem 231. Then
(L19) T(R(x,y,z) \ (R(x,y,z) / x),x) \ T((R(x,y,z) \ (R(x,y,z) / x)) * x,R(x,y,z) \ (R(x,y,z) / x)) = T(x,R(x,y,z) \ (R(x,y,z) / x))
by (L17). We have T(R(x,y,z) \ (R(x,y,z) / x),(R(x,y,z) \ (R(x,y,z) / x)) * ((R(x,y,z) / x) \ R(x,y,z))) = R(x,y,z) \ (R(x,y,z) / x) by Theorem 306. Then T(R(x,y,z) \ (R(x,y,z) / x),(R(x,y,z) \ (R(x,y,z) / x)) * x) = R(x,y,z) \ (R(x,y,z) / x) by Proposition 25. Then T((R(x,y,z) \ (R(x,y,z) / x)) * x,R(x,y,z) \ (R(x,y,z) / x)) = (R(x,y,z) \ (R(x,y,z) / x)) * x by Proposition 21. Then
(L20) T(R(x,y,z) \ (R(x,y,z) / x),x) \ ((R(x,y,z) \ (R(x,y,z) / x)) * x) = T(x,R(x,y,z) \ (R(x,y,z) / x))
by (L19). We have T(R(x,y,z) \ (R(x,y,z) / x),x) \ ((R(x,y,z) \ (R(x,y,z) / x)) * x) = T(x,T(R(x,y,z) \ (R(x,y,z) / x),x)) by Theorem 13. Then T(x,R(x,y,z) \ (R(x,y,z) / x)) = T(x,T(R(x,y,z) \ (R(x,y,z) / x),x)) by (L20). Then T(x,T(T(e / x,R(x,y,z)),x)) = T(x,R(x,y,z) \ (R(x,y,z) / x)) by Theorem 839. Then T(x,R(x,y,z) \ (R(x,y,z) / x)) = T(x,T(x \ e,R(x,y,z))) by Proposition 51. Then
(L25) T(x,T(e / x,R(x,y,z))) = T(x,T(x \ e,R(x,y,z)))
by Theorem 839. We have R(x \ e,R(x,y,z),x \ e) \ T((x \ e) * R(x,y,z),x \ e) = T(R(x,y,z),x \ e) by Theorem 231. Then T(x \ e,R(x,y,z)) \ T((x \ e) * R(x,y,z),x \ e) = T(R(x,y,z),x \ e) by Theorem 580. Then
(L28) T(x \ e,R(x,y,z)) \ ((x \ e) * R(x,y,z)) = T(R(x,y,z),x \ e)
by Theorem 885. We have T(x \ e,R(x,y,z)) \ ((x \ e) * R(x,y,z)) = T(R(x,y,z),T(x \ e,R(x,y,z))) by Theorem 13. Then
(L30) T(R(x,y,z),x \ e) = T(R(x,y,z),T(x \ e,R(x,y,z)))
by (L28). We have R(T(x,T(x \ e,R(x,y,z))),y,z) = T(R(x,y,z),T(x \ e,R(x,y,z))) by Axiom 9. Then
(L32) R(T(x,T(x \ e,R(x,y,z))),y,z) = T(R(x,y,z),x \ e)
by (L30). We have R(T(x,x \ e),y,z) = T(R(x,y,z),x \ e) by Axiom 9. Then R(T(x,x \ e),y,z) = R(T(x,T(x \ e,R(x,y,z))),y,z) by (L32). Hence we are done by (L25).
Theorem 887 (prov9_a97cb9758b): T(x * y,y \ e) = (((y \ e) \ x) * (y \ e)) * y
Proof. We have (((y \ e) \ x) * (y \ e)) * (e / (y \ e)) = T(x * (e / (y \ e)),y \ e) by Theorem 610. Then (((y \ e) \ x) * (y \ e)) * y = T(x * (e / (y \ e)),y \ e) by Proposition 24. Hence we are done by Proposition 24.
Theorem 888 (prov9_3c6e90a5e0): (y \ T(x,y \ e)) * y = T((y \ x) * y,y \ e)
Proof. We have (((y \ e) \ (y \ x)) * (y \ e)) * y = T((y \ x) * y,y \ e) by Theorem 887. Hence we are done by Theorem 820.
Theorem 889 (prov9_e9016dfa8d): (y \ ((y \ e) \ x)) * y = (y \ e) \ T(x,y)
Proof. We have T((y \ (x / (y \ e))) * y,y \ e) = (y \ T(x / (y \ e),y \ e)) * y by Theorem 888. Then T(T(x,y) / (y \ e),y \ e) = (y \ T(x / (y \ e),y \ e)) * y by Theorem 821. Then
(L3) (y \ ((y \ e) \ x)) * y = T(T(x,y) / (y \ e),y \ e)
by Proposition 47. We have T(T(x,y) / (y \ e),y \ e) = (y \ e) \ T(x,y) by Proposition 47. Hence we are done by (L3).
Theorem 890 (prov9_467705de4b): (y \ x) * y = (y \ e) \ T((y \ e) * x,y)
Proof. We have (y \ ((y \ e) \ ((y \ e) * x))) * y = (y \ e) \ T((y \ e) * x,y) by Theorem 889. Hence we are done by Axiom 3.
Theorem 891 (prov9_4b892e9207): (y \ e) * ((y \ x) * y) = T((y \ e) * x,y)
Proof. We have (y \ e) * ((y \ e) \ T((y \ e) * x,y)) = T((y \ e) * x,y) by Axiom 4. Hence we are done by Theorem 890.
Theorem 892 (prov9_f91eb79733): (y \ e) * (x * y) = T((y \ e) * (y * x),y)
Proof. We have (y \ e) * ((y \ (y * x)) * y) = T((y \ e) * (y * x),y) by Theorem 891. Hence we are done by Axiom 3.
Theorem 893 (prov9_d40fbea793): ((y \ e) * (y * x)) * y = L(x * y,y \ e,y)
Proof. We have y * T((y \ e) * (y * x),y) = ((y \ e) * (y * x)) * y by Proposition 46. Then
(L2) y * ((y \ e) * (x * y)) = ((y \ e) * (y * x)) * y
by Theorem 892. We have y * ((y \ e) * (x * y)) = L(x * y,y \ e,y) by Proposition 56. Hence we are done by (L2).
Theorem 894 (prov9_8069b13dc5): L(y * x,x \ e,x) / x = (x \ e) * (x * y)
Proof. We have ((x \ e) * (x * y)) * x = L(y * x,x \ e,x) by Theorem 893. Hence we are done by Proposition 1.
Theorem 895 (prov9_2048a027c6): K(y,x \ (x / y)) = K(y,y \ e)
Proof. We have K(y,((x / y) * y) \ (x / y)) = K(y,y \ e) by Theorem 392. Hence we are done by Axiom 6.
Theorem 896 (prov9_559b3bffb1): (y * K(x,y)) \ K(x,y) = y \ e
Proof. We have ((L(L(x \ e,x,y) \ e,L(x \ e,x,y),y) * y) / L(L(x \ e,x,y) \ e,L(x \ e,x,y),y)) * (y \ e) = y \ ((L(L(x \ e,x,y) \ e,L(x \ e,x,y),y) * y) / L(L(x \ e,x,y) \ e,L(x \ e,x,y),y)) by Theorem 195. Then (y * (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y))) * (y \ e) = y \ ((L(L(x \ e,x,y) \ e,L(x \ e,x,y),y) * y) / L(L(x \ e,x,y) \ e,L(x \ e,x,y),y)) by Theorem 877. Then
(L3) y \ (y * (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y))) = (y * (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y))) * (y \ e)
by Theorem 877. We have y \ (y * (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y))) = L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y) by Axiom 3. Then (y * (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y))) * (y \ e) = L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y) by (L3). Then (y * (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y))) \ (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y)) = y \ e by Proposition 2. Then (y * K(x,y)) \ (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y)) = y \ e by Theorem 542. Hence we are done by Theorem 542.
Theorem 897 (prov9_fee03c3305): (y * K(x,y)) * (y \ e) = K(x,y)
Proof. We have (y * K(x,y)) * ((y * K(x,y)) \ K(x,y)) = K(x,y) by Axiom 4. Hence we are done by Theorem 896.
Theorem 898 (prov9_28fc3e37a6): K(x,y) / (y \ e) = y * K(x,y)
Proof. We have K(x,y) / ((y * K(x,y)) \ K(x,y)) = y * K(x,y) by Proposition 24. Hence we are done by Theorem 896.
Theorem 899 (prov9_b86d39a868): T(K(y,z),K(x,y)) = K(y,z)
Proof. We have L(T(y \ T(y,z),K(x,y)),y,z) = T((z * y) \ (y * z),K(x,y)) by Theorem 103. Then
(L2) L(y \ T(y,z),y,z) = T((z * y) \ (y * z),K(x,y))
by Theorem 619. We have L(y \ T(y,z),y,z) = (z * y) \ (y * z) by Proposition 73. Then
(L4) T((z * y) \ (y * z),K(x,y)) = (z * y) \ (y * z)
by (L2). We have K(y,z) = (z * y) \ (y * z) by Definition 2. Then K(y,z) = T((z * y) \ (y * z),K(x,y)) by (L4). Hence we are done by Definition 2.
Theorem 900 (prov9_e2bba7fdeb): K(x,y) = (K(x,y) \ e) \ e
Proof. We have T(K(x,y),(x * y) \ ((x * y) / K(x,y))) = T(K(x,y),K(x,y) \ e) by Theorem 383. Then T(K(x,y),(x * y) \ (y * x)) = T(K(x,y),K(x,y) \ e) by Theorem 452. Then
(L3) T(K(x,y),K(y,x)) = T(K(x,y),K(x,y) \ e)
by Definition 2. We have T(K(x,y),K(x,y) \ e) = (K(x,y) \ e) \ e by Proposition 49. Then T(K(x,y),K(y,x)) = (K(x,y) \ e) \ e by (L3). Hence we are done by Theorem 899.
Theorem 901 (prov9_b773c3cd41): e / K(x,y) = K(x,y) \ e
Proof. We have e / ((K(x,y) \ e) \ e) = K(x,y) \ e by Proposition 24. Hence we are done by Theorem 900.
Theorem 902 (prov9_dd2b7eaeb7): L(K(y \ e,y),x,K(y \ e,y)) = K(y \ e,y)
Proof. We have L(y * T(y \ e,y),x,K(y \ e,y)) = K(y \ (y / L(y,x,K(y \ e,y))),y) by Theorem 635. Then L((y \ e) * y,x,K(y \ e,y)) = K(y \ (y / L(y,x,K(y \ e,y))),y) by Proposition 46. Then
(L5) K(y \ (y / L(y,x,K(y \ e,y))),y) = L(K(y \ e,y),x,K(y \ e,y))
by Proposition 76. We have T(e / L(y,x,K(y \ e,y)),y) = y \ ((e / L(y,x,K(y \ e,y))) * y) by Definition 3. Then T(e / L(y,x,K(y \ e,y)),y) = y \ (y / L(y,x,K(y \ e,y))) by Theorem 869. Then K(T(e / L(y,x,K(y \ e,y)),y),y) = L(K(y \ e,y),x,K(y \ e,y)) by (L5). Then K(T(e / y,y),y) = L(K(y \ e,y),x,K(y \ e,y)) by Theorem 629. Hence we are done by Proposition 47.
Theorem 903 (prov9_050565f522): T(K(y \ e,y) * x,K(y \ e,y)) = x * K(y \ e,y)
Proof. We have (K(y \ e,y) * x) * L(K(y \ e,y),x,K(y \ e,y)) = K(y \ e,y) * (x * K(y \ e,y)) by Proposition 52. Then (K(y \ e,y) * x) * K(y \ e,y) = K(y \ e,y) * (x * K(y \ e,y)) by Theorem 902. Hence we are done by Theorem 11.
Theorem 904 (prov9_edcf7888f9): (x * y) * (L(z,y,x) * R(L(z,y,x) \ e,w,u)) = x * (y * (z * R(z \ e,w,u)))
Proof. We have ((x * y) * L(z,y,x)) * R(((x * y) * L(z,y,x)) \ (x * y),w,u) = (x * y) * (L(z,y,x) * R(L(z,y,x) \ e,w,u)) by Theorem 133. Then (x * (y * z)) * R(((x * y) * L(z,y,x)) \ (x * y),w,u) = (x * y) * (L(z,y,x) * R(L(z,y,x) \ e,w,u)) by Proposition 52. Then
(L3) (x * (y * z)) * R((x * (y * z)) \ (x * y),w,u) = (x * y) * (L(z,y,x) * R(L(z,y,x) \ e,w,u))
by Proposition 52. We have (x * (y * z)) * R((x * (y * z)) \ (x * y),w,u) = x * ((y * z) * R((y * z) \ y,w,u)) by Theorem 516. Then (x * y) * (L(z,y,x) * R(L(z,y,x) \ e,w,u)) = x * ((y * z) * R((y * z) \ y,w,u)) by (L3). Hence we are done by Theorem 133.
Theorem 905 (prov9_8bcc2ad8b7): L(x * R(x \ e,w,u),y,z) = L(x,y,z) * R(L(x,y,z) \ e,w,u)
Proof. We have (z * y) * (L(x,y,z) * R(L(x,y,z) \ e,w,u)) = z * (y * (x * R(x \ e,w,u))) by Theorem 904. Hence we are done by Theorem 54.
Theorem 906 (prov9_3b1fb05fb5): (x \ e) / (x \ L(x,y,z)) = L(x,y,z) \ e
Proof. We have L(x,y,z) * R(L(x,y,z) \ e,L(x,y,z),x \ e) = ((x \ e) / (x \ L(x,y,z))) * L(x,y,z) by Theorem 838. Then L(x * R(x \ e,L(x,y,z),x \ e),y,z) = ((x \ e) / (x \ L(x,y,z))) * L(x,y,z) by Theorem 905. Then
(L3) L(x * T(x \ e,L(x,y,z)),y,z) = ((x \ e) / (x \ L(x,y,z))) * L(x,y,z)
by Theorem 862. We have L(x * T(x \ e,L(x,y,z)),y,z) = K(L(x,y,z) \ (L(x,y,z) / L(x,y,z)),L(x,y,z)) by Theorem 635. Then L(x * T(x \ e,L(x,y,z)),y,z) = K(L(x,y,z) \ e,L(x,y,z)) by Proposition 29. Then
(L6) ((x \ e) / (x \ L(x,y,z))) * L(x,y,z) = K(L(x,y,z) \ e,L(x,y,z))
by (L3). We have (L(x,y,z) \ e) * L(x,y,z) = K(L(x,y,z) \ e,L(x,y,z)) by Proposition 76. Hence we are done by (L6) and Proposition 8.
Theorem 907 (prov9_a39085ef5e): x * K(y \ e,y) = (x * y) / ((y \ e) \ e)
Proof. We have ((x * y) / y) * K(y \ e,y) = (x * y) / ((y \ e) \ e) by Theorem 423. Hence we are done by Axiom 5.
Theorem 908 (prov9_08a1ec0c7c): (x * K(y \ e,y)) * ((y \ e) \ e) = x * y
Proof. We have ((x * y) / ((y \ e) \ e)) * ((y \ e) \ e) = x * y by Axiom 6. Hence we are done by Theorem 907.
Theorem 909 (prov9_1fb580ca98): x * y = (x * T(y,y \ e)) * K(y \ e,y)
Proof. We have ((x * T(y,y \ e)) / ((y \ e) \ e)) * y = (x * T(y,y \ e)) * K(y \ e,y) by Theorem 424. Hence we are done by Theorem 807.
Theorem 910 (prov9_3731c71c77): y * K(x \ y,y \ x) = x * (((x \ y) \ e) \ e)
Proof. We have x * T(x \ y,y \ x) = y * K((y \ x) \ ((y \ x) / (y \ x)),y \ x) by Theorem 284. Then x * T(x \ y,y \ x) = y * K((y \ x) \ e,y \ x) by Proposition 29. Then x * (((x \ y) \ e) \ e) = y * K((y \ x) \ e,y \ x) by Theorem 387. Hence we are done by Theorem 666.
Theorem 911 (prov9_998a5847cc): (x * (e / y)) * y = R(x,y \ e,y)
Proof. We have ((x * (e / y)) * K(y \ e,y)) * ((y \ e) \ e) = (x * (e / y)) * y by Theorem 908. Then
(L2) (x * (y \ e)) * ((y \ e) \ e) = (x * (e / y)) * y
by Theorem 412. We have (x * (y \ e)) * ((y \ e) \ e) = R(x,y \ e,y) by Theorem 662. Hence we are done by (L2).
Theorem 912 (prov9_fb84f22559): x * y = R(x / (e / y),y \ e,y)
Proof. We have ((x / (e / y)) * (e / y)) * y = R(x / (e / y),y \ e,y) by Theorem 911. Hence we are done by Axiom 6.
Theorem 913 (prov9_4454738885): (e / x) \ L(y,x,x \ e) = x * y
Proof. We have (e / x) \ L(y,x,e / x) = x * y by Theorem 71. Hence we are done by Theorem 683.
Theorem 914 (prov9_820ec9c9c8): K(x \ e,x) * (x * y) = (e / (e / x)) * y
Proof. We have
(L1) K(x,x \ e) * (K(x \ e,x) * (x * y)) = x * y
by Theorem 442. We have K(x,x \ e) * ((e / (e / x)) * y) = x * y by Theorem 685. Hence we are done by (L1) and Proposition 7.
Theorem 915 (prov9_631fb29e50): K(x \ e,x) * (x \ y) = T(x,x \ e) \ y
Proof. We have T(x,x \ e) * (x \ (T(x,x \ e) \ y)) = x \ y by Theorem 522. Then
(L3) K(x,x \ e) * (T(x,x \ e) \ y) = x \ y
by Theorem 431. We have K(x,x \ e) * (K(x \ e,x) * (x \ y)) = x \ y by Theorem 442. Hence we are done by (L3) and Proposition 7.
Theorem 916 (prov9_420ed44658): T((e / (e / y)) * x,K(y \ e,y)) = (y * x) * K(y \ e,y)
Proof. We have T(K(y \ e,y) * (y * x),K(y \ e,y)) = (y * x) * K(y \ e,y) by Theorem 903. Hence we are done by Theorem 914.
Theorem 917 (prov9_9082519005): T(e / x,y) \ e = e / T(x \ e,y)
Proof. We have T(e / (e / (x \ e)),y) \ e = e / T(x \ e,y) by Theorem 446. Hence we are done by Proposition 24.
Theorem 918 (prov9_b17f112c60): e / T((x \ e) \ e,y) = T(x,y) \ e
Proof. We have e / T((x \ e) \ e,y) = T(e / (x \ e),y) \ e by Theorem 917. Hence we are done by Proposition 24.
Theorem 919 (prov9_b0fe4ff9c2): (T(x,y) \ e) \ e = T((x \ e) \ e,y)
Proof. We have (e / T((x \ e) \ e,y)) \ e = T((x \ e) \ e,y) by Proposition 25. Hence we are done by Theorem 918.
Theorem 920 (prov9_4b8aa50617): ((y \ x) \ e) \ e = T(((x / y) \ e) \ e,y)
Proof. We have (T(x / y,y) \ e) \ e = T(((x / y) \ e) \ e,y) by Theorem 919. Hence we are done by Proposition 47.
Theorem 921 (prov9_ecbaae7d4b): y * (((y \ x) \ e) \ e) = (((x / y) \ e) \ e) * y
Proof. We have y * T(((x / y) \ e) \ e,y) = (((x / y) \ e) \ e) * y by Proposition 46. Hence we are done by Theorem 920.
Theorem 922 (prov9_77fbda22f1): (((x / y) \ e) \ e) \ x = x * (((x \ y) \ e) \ e)
Proof. We have R((y / x) \ e,y / x,x) \ x = (((y / x) \ e) \ e) * x by Theorem 702. Then R((y / x) \ e,y / x,x) \ x = T(y / x,(y / x) \ e) * x by Proposition 49. Then (((x / ((y / x) * x)) \ e) \ e) \ x = T(y / x,(y / x) \ e) * x by Theorem 670. Then (((x / y) \ e) \ e) \ x = T(y / x,(y / x) \ e) * x by Axiom 6. Then (((y / x) \ e) \ e) * x = (((x / y) \ e) \ e) \ x by Proposition 49. Hence we are done by Theorem 921.
Theorem 923 (prov9_337c77cbff): y * K(x \ e,x) = K(x \ e,x) * y
Proof. We have y * T(K(x \ e,x),y) = K(x \ e,x) * y by Proposition 46. Hence we are done by Theorem 716.
Theorem 924 (prov9_b6884251b3): (K(x,x \ e) * y) * K(x \ e,x) = y
Proof. We have (K(x \ e,x) \ y) * T(K(x \ e,x),K(x \ e,x) \ y) = y by Theorem 8. Then (K(x \ e,x) \ y) * T(K(x \ e,x),y) = y by Theorem 631. Then (K(x,x \ e) * y) * T(K(x \ e,x),y) = y by Theorem 696. Hence we are done by Theorem 716.
Theorem 925 (prov9_0c618d0a6e): x * T(y,y \ e) = K(y,y \ e) * (x * y)
Proof.
(x * T(y,y \ e)) * K(y \ e,y)
=x * y by Theorem 909
=(K(y,y \ e) * (x * y)) * K(y \ e,y) by Theorem 924

Then (x * T(y,y \ e)) * K(y \ e,y) = (K(y,y \ e) * (x * y)) * K(y \ e,y). Hence we are done by Proposition 10.
Theorem 926 (prov9_3fbba21a48): (x * ((y * (x \ e)) / y)) * y = T(y,R(x \ e,x,y))
Proof. We have (x * ((y * (x \ e)) / y)) * y = T(y,R(e / x,x,y)) by Theorem 289. Hence we are done by Theorem 725.
Theorem 927 (prov9_028642573b): K(w \ e,w) * R(x * T(w,w \ e),y,z) = (w \ R(w * x,y,z)) * w
Proof. We have K(w \ e,w) * L(R(w \ ((w \ e) \ (x * w)),y,z),w,w \ e) = (w \ e) * (w * R(w \ ((w \ e) \ (x * w)),y,z)) by Theorem 59. Then K(w \ e,w) * R(((w \ e) * w) \ (x * w),y,z) = (w \ e) * (w * R(w \ ((w \ e) \ (x * w)),y,z)) by Proposition 85. Then
(L3) K(w \ e,w) * R(K(w \ e,w) \ (x * w),y,z) = (w \ e) * (w * R(w \ ((w \ e) \ (x * w)),y,z))
by Proposition 76. We have (w \ e) * (w * R(w \ ((w \ e) \ (x * w)),y,z)) = ((w \ e) * R((w \ e) \ x,y,z)) * w by Theorem 873. Then K(w \ e,w) * R(K(w \ e,w) \ (x * w),y,z) = ((w \ e) * R((w \ e) \ x,y,z)) * w by (L3). Then K(w \ e,w) * R(K(w \ e,w) \ (x * w),y,z) = (w \ R(w * x,y,z)) * w by Theorem 826. Then K(w \ e,w) * R(K(w,w \ e) * (x * w),y,z) = (w \ R(w * x,y,z)) * w by Theorem 696. Hence we are done by Theorem 925.
Theorem 928 (prov9_645df71613): y * K(y,x) = K(y,x) * y
Proof. We have y * T(K(y,x),y) = K(y,x) * y by Proposition 46. Then y * K((y \ e) \ e,x) = K(y,x) * y by Theorem 674. Hence we are done by Theorem 733.
Theorem 929 (prov9_49d60bba91): K(T(y,x) \ e,T(y,x)) = K(y \ e,y)
Proof.
K(T(y,x) \ e,T(y,x)) * (e / T(y,x))
=T(y,x) \ e by Theorem 255
=K(y \ e,y) * (e / T(y,x)) by Theorem 748

Then K(T(y,x) \ e,T(y,x)) * (e / T(y,x)) = K(y \ e,y) * (e / T(y,x)). Hence we are done by Proposition 10.
Theorem 930 (prov9_ead35f8d68): (T(x,y) \ x) \ x = T(x,y)
Proof. We have (x / T(x,y)) \ x = T(x,y) by Proposition 25. Hence we are done by Theorem 753.
Theorem 931 (prov9_56ed1a8c2e): (T(y,x) \ y) * y = y * (T(y,x) \ y)
Proof. We have (y / T(y,x)) * y = y * (T(y,x) \ y) by Theorem 572. Hence we are done by Theorem 753.
Theorem 932 (prov9_1af8f7ad3b): K(y \ (y / ((x \ y) / T(y,x))),y) = T(y,x) \ y
Proof. We have K(y \ (y / ((x \ y) / T(y,x))),y) = ((x \ y) * T((x \ y) \ T(y,x),y)) / T(y,x) by Theorem 287. Then K(y \ (y / ((x \ y) / T(y,x))),y) = ((x \ y) * ((x \ y) \ y)) / T(y,x) by Theorem 514. Then K(y \ (y / ((x \ y) / T(y,x))),y) = y / T(y,x) by Axiom 4. Hence we are done by Theorem 753.
Theorem 933 (prov9_452e908697): y * K(x,y) = K(x,y) * y
Proof. We have y * T(K(x,y),y) = K(x,y) * y by Proposition 46. Hence we are done by Theorem 754.
Theorem 934 (prov9_960bb661e4): (y * K(x,y)) / y = K(x,y)
Proof. We have (y * T(K(x,y),y)) / y = K(x,y) by Proposition 48. Hence we are done by Theorem 754.
Theorem 935 (prov9_6fbf5f2a1b): T(z,x) = T(T(z,x),K(y,z))
Proof. We have T(T(z,K(y,z)),x) = T(T(z,x),K(y,z)) by Axiom 7. Hence we are done by Theorem 755.
Theorem 936 (prov9_e8ca577368): T(y,x \ T(x,y)) = y
Proof. We have T(y,K(y \ (y / (x \ e)),y)) = y by Theorem 755. Hence we are done by Theorem 198.
Theorem 937 (prov9_140550a2fc): (x \ T(x,y)) * y = y * (x \ T(x,y))
Proof. We have (x \ T(x,y)) * T(y,x \ T(x,y)) = y * (x \ T(x,y)) by Proposition 46. Hence we are done by Theorem 936.
Theorem 938 (prov9_3e0bd1691f): R(x,K(y,z),z) * (z * K(y,z)) = (x * K(y,z)) * z
Proof. We have R(x,K(y,z),z) * (K(y,z) * z) = (x * K(y,z)) * z by Proposition 54. Hence we are done by Theorem 933.
Theorem 939 (prov9_ff5cb89ba5): T(K(y,z),T(z,x)) = K(y,z)
Proof. We have T(T(z,x),K(y,z)) = T(z,x) by Theorem 935. Hence we are done by Proposition 21.
Theorem 940 (prov9_eb4ef335ba): ((y * x) / y) / x = x \ ((y * x) / y)
Proof. We have T(x,K(x \ (x / (x \ (x / y))),x)) = x by Theorem 755. Then
(L2) T(x,x \ ((y * x) / y)) = x
by Theorem 853. We have ((y * x) / y) / T(x,x \ ((y * x) / y)) = x \ ((y * x) / y) by Theorem 15. Hence we are done by (L2).
Theorem 941 (prov9_d2d0e40a28): K(x \ e,x) = K((x * y) \ y,T(x,y))
Proof. We have K((y * T(x,y)) \ y,T(x,y)) = K(T(x,y) \ e,T(x,y)) by Theorem 665. Then K((x * y) \ y,T(x,y)) = K(T(x,y) \ e,T(x,y)) by Proposition 46. Hence we are done by Theorem 929.
Theorem 942 (prov9_f7fd9397ee): x * (((y * x) / y) / x) = (y * x) / y
Proof. We have x * (x \ ((y * x) / y)) = (y * x) / y by Axiom 4. Hence we are done by Theorem 940.
Theorem 943 (prov9_640429af57): R(y \ T(y,x),y,z) = (T(y,x) * z) / (y * z)
Proof. We have R(T(y,x) / y,y,z) = (T(y,x) * z) / (y * z) by Proposition 55. Hence we are done by Theorem 758.
Theorem 944 (prov9_c23b321e2e): T(x,R(e / y,y,x)) = T(x,x / (x * y))
Proof. We have (y * ((x * (y \ e)) / x)) * x = T(x,R(e / y,y,x)) by Theorem 289. Then (x \ T(x,x / (x * y))) * x = T(x,R(e / y,y,x)) by Theorem 846. Hence we are done by Theorem 759.
Theorem 945 (prov9_10438141ac): T(z / (y * z),T(y,x)) = R(y \ e,y,z)
Proof. We have T(z / (y * z),T(y,x)) = R(T(e / y,T(y,x)),y,z) by Theorem 39. Hence we are done by Theorem 762.
Theorem 946 (prov9_53e729199b): (x * y) / x = (((x * y) / x) \ y) \ y
Proof. We have T((x * y) / x,((x * y) / x) \ T((x * y) / x,x)) = (x * y) / x by Theorem 761. Then
(L2) T((x * y) / x,K(x \ (x / (((x * y) / x) \ e)),x)) = (x * y) / x
by Theorem 198. We have T((x * y) / x,((x * y) / x) \ y) = (((x * y) / x) \ y) \ y by Proposition 49. Then T((x * y) / x,K(x \ (x / (((x * y) / x) \ e)),x)) = (((x * y) / x) \ y) \ y by Theorem 200. Hence we are done by (L2).
Theorem 947 (prov9_30ed8a747a): K(x \ e,x) * y = (x * y) * ((((x * y) \ y) \ e) \ e)
Proof. We have K((x * y) \ y,T(x,y)) = K(x \ e,x) by Theorem 941. Then
(L2) K((x * y) \ y,y \ (x * y)) = K(x \ e,x)
by Definition 3. We have y * K((x * y) \ y,y \ (x * y)) = (x * y) * ((((x * y) \ y) \ e) \ e) by Theorem 910. Then K((x * y) \ y,y \ (x * y)) * y = (x * y) * ((((x * y) \ y) \ e) \ e) by Theorem 724. Hence we are done by (L2).
Theorem 948 (prov9_bb65563e05): (x * ((y * (x \ e)) / y)) * y = T(y,R(x \ e,(x \ e) \ e,y))
Proof. We have (x * ((y * (x \ e)) / y)) * y = T(y,R(x \ e,x,y)) by Theorem 926. Hence we are done by Theorem 704.
Theorem 949 (prov9_17c7d62f80): T(x,R(y,x,(y * x) \ x)) \ x = K(x,y) \ e
Proof. We have (x \ T(x,R(y,x,(y * x) \ x))) \ e = T(x,R(y,x,(y * x) \ x)) \ x by Theorem 749. Then (x \ (x * K(x,y))) \ e = T(x,R(y,x,(y * x) \ x)) \ x by Theorem 871. Hence we are done by Axiom 3.
Theorem 950 (prov9_49a53211cc): z * L(x,z,z \ T(z,y)) = L(z * x,z \ T(z,y),z)
Proof. We have L(x,z,z \ T(z,y)) = ((z \ T(z,y)) * z) \ ((z \ T(z,y)) * (z * x)) by Definition 4. Then
(L2) L(x,z,z \ T(z,y)) = T(z,y) \ ((z \ T(z,y)) * (z * x))
by Theorem 759. We have z * (T(z,y) \ ((z \ T(z,y)) * (z * x))) = L(z * x,z \ T(z,y),z) by Theorem 832. Hence we are done by (L2).
Theorem 951 (prov9_db3780fdee): L(x,y \ T(y,z),y) = y * L(y \ x,y,y \ T(y,z))
Proof. We have L(y * (y \ x),y \ T(y,z),y) = y * L(y \ x,y,y \ T(y,z)) by Theorem 950. Hence we are done by Axiom 4.
Theorem 952 (prov9_44d2d155fc): T(x,y) \ x = (y * (x \ T(x,y))) \ y
Proof. We have (y * (T(y,R(T(x,y),T(x,y) \ e,y)) \ y)) \ (y * y) = T(y,R(T(x,y),T(x,y) \ e,y)) by Theorem 570. Then
(L7) (y * (((T(x,y) \ x) * y) \ y)) \ (y * y) = T(y,R(T(x,y),T(x,y) \ e,y))
by Theorem 591. We have T(y,R(T(x,y),T(x,y) \ e,y)) = (T(x,y) \ x) * y by Theorem 591. Then (y * (((T(x,y) \ x) * y) \ y)) \ (y * y) = (T(x,y) \ x) * y by (L7). Then
(L10) (y * ((y * (T(x,y) \ x)) \ y)) \ (y * y) = (T(x,y) \ x) * y
by Theorem 740. We have (T(x,y) \ x) * y = y * (T(x,y) \ x) by Theorem 740. Then (y * ((y * (T(x,y) \ x)) \ y)) \ (y * y) = y * (T(x,y) \ x) by (L10). Then
(L13) (y * (x \ T(x,y))) \ (y * y) = y * (T(x,y) \ x)
by Theorem 771. We have ((L(x,e / x,y) * y) / L(x,e / x,y)) \ (y * y) = y * (((L(x,e / x,y) * y) / L(x,e / x,y)) \ y) by Theorem 855. Then (y * (T(x,y) / x)) \ (y * y) = y * (((L(x,e / x,y) * y) / L(x,e / x,y)) \ y) by Theorem 883. Then y * ((y * (T(x,y) / x)) \ y) = (y * (T(x,y) / x)) \ (y * y) by Theorem 883. Then (y * (x \ T(x,y))) \ (y * y) = y * ((y * (T(x,y) / x)) \ y) by Theorem 758. Then (y * (x \ T(x,y))) \ (y * y) = y * ((y * (x \ T(x,y))) \ y) by Theorem 758. Then y * (T(x,y) \ x) = y * ((y * (x \ T(x,y))) \ y) by (L13). Hence we are done by Proposition 9.
Theorem 953 (prov9_b6bf8a346f): T(z,R((y \ x) \ e,((y \ x) \ e) \ e,z)) = (y \ (x * ((z * (x \ y)) / z))) * z
Proof. We have T(z,R((y \ x) \ e,((y \ x) \ e) \ e,z)) = ((y \ x) * ((z * ((y \ x) \ e)) / z)) * z by Theorem 948. Hence we are done by Theorem 876.
Theorem 954 (prov9_e3a9ba5c89): T(y,R((x \ y) \ e,((x \ y) \ e) \ e,y)) = y * K(y,x / y)
Proof. We have T(y,R((x \ y) \ e,((x \ y) \ e) \ e,y)) = (x \ (y * ((y * (y \ x)) / y))) * y by Theorem 953. Then T(y,R((x \ y) \ e,((x \ y) \ e) \ e,y)) = (x \ (y * (x / y))) * y by Axiom 4. Then
(L3) T(y,R((x \ y) \ e,((x \ y) \ e) \ e,y)) = K(y,x / y) * y
by Theorem 3. We have K(y,x / y) * y = y * K(y,x / y) by Theorem 928. Hence we are done by (L3).
Theorem 955 (prov9_78b50b8f42): L(z,x \ T(x,y),x) = (x \ T(x,y)) * (x * (T(x,y) \ z))
Proof. We have ((x \ T(x,y)) * x) * L(T(x,y) \ z,x,x \ T(x,y)) = (x \ T(x,y)) * (x * (T(x,y) \ z)) by Proposition 52. Then
(L2) T(x,y) * L(T(x,y) \ z,x,x \ T(x,y)) = (x \ T(x,y)) * (x * (T(x,y) \ z))
by Theorem 759. We have T(x * L(x \ ((y * z) / y),x,x \ T(x,y)),y) = T(x,y) * L(T(x,y) \ z,x,x \ T(x,y)) by Theorem 879. Then T(x * L(x \ ((y * z) / y),x,x \ T(x,y)),y) = (x \ T(x,y)) * (x * (T(x,y) \ z)) by (L2). Then
(L5) T(L((y * z) / y,x \ T(x,y),x),y) = (x \ T(x,y)) * (x * (T(x,y) \ z))
by Theorem 951. We have T(L((y * z) / y,x \ T(x,y),x),y) = L(y \ (y * z),x \ T(x,y),x) by Proposition 72. Then (x \ T(x,y)) * (x * (T(x,y) \ z)) = L(y \ (y * z),x \ T(x,y),x) by (L5). Hence we are done by Axiom 3.
Theorem 956 (prov9_cb097a09aa): (y \ T(y,x)) \ y = y * (T(y,x) \ y)
Proof. We have (y \ T(y,x)) * T(T(y,y \ T(y,x)),y \ T(y,x)) = T(y,y \ T(y,x)) * (y \ T(y,x)) by Proposition 46. Then
(L2) (y \ T(y,x)) * T(y,y \ T(y,x)) = T(y,y \ T(y,x)) * (y \ T(y,x))
by Theorem 761. We have T(y,y \ T(y,x)) * (y \ T(y,x)) = T(y,x) * (y \ T(y,y \ T(y,x))) by Theorem 833. Then
(L4) (y \ T(y,x)) * T(y,y \ T(y,x)) = T(y,x) * (y \ T(y,y \ T(y,x)))
by (L2). We have (T(y,x) * T(y,y \ T(y,x))) / (T(y,x) * (y \ T(y,y \ T(y,x)))) = y by Theorem 831. Then
(L6) (T(y,x) * T(y,y \ T(y,x))) / ((y \ T(y,x)) * T(y,y \ T(y,x))) = y
by (L4). We have R(y,y \ T(y,x),T(y,y \ T(y,x))) = (T(y,x) * T(y,y \ T(y,x))) / ((y \ T(y,x)) * T(y,y \ T(y,x))) by Theorem 785. Then
(L8) R(y,y \ T(y,x),T(y,y \ T(y,x))) = y
by (L6). We have L(R(y,y \ T(y,x),T(y,y \ T(y,x))),y \ T(y,x),y) = y by Theorem 823. Then L(y,y \ T(y,x),y) = y by (L8). Then (y \ T(y,x)) * (y * (T(y,x) \ y)) = y by Theorem 955. Hence we are done by Proposition 2.
Theorem 957 (prov9_39181eba79): x \ (K(x,y) \ x) = K(x,y) \ e
Proof. We have (x \ T(x,R(y,x,(y * x) \ x))) \ x = x * (T(x,R(y,x,(y * x) \ x)) \ x) by Theorem 956. Then (x \ (x * K(x,y))) \ x = x * (T(x,R(y,x,(y * x) \ x)) \ x) by Theorem 871. Then x * (T(x,R(y,x,(y * x) \ x)) \ x) = K(x,y) \ x by Axiom 3. Then x * (K(x,y) \ e) = K(x,y) \ x by Theorem 949. Hence we are done by Proposition 2.
Theorem 958 (prov9_a527441c25): y \ (K(x,y) \ y) = K(x,y) \ e
Proof. We have y * (K(x,y) \ e) = K(x,y) \ y by Theorem 777. Hence we are done by Proposition 2.
Theorem 959 (prov9_87fe4376ba): y * (T(L(x \ e,x,y) \ e,y) \ (L(x \ e,x,y) \ e)) = K(x,y) \ y
Proof. We have y * (K(x,y) \ e) = K(x,y) \ y by Theorem 777. Hence we are done by Theorem 763.
Theorem 960 (prov9_cb16af3e36): R(T(x,x \ e),y,z) \ R(x,y,z) = K(x \ e,R(x,y,z))
Proof. We have R((x / (((x \ R(x,y,z)) / R(x,y,z)) \ e)) * (((x \ R(x,y,z)) / R(x,y,z)) \ e),y,z) / (((x \ R(x,y,z)) / R(x,y,z)) \ e) = ((x \ R(x,y,z)) / R(x,y,z)) * R(((x \ R(x,y,z)) / R(x,y,z)) \ (x / (((x \ R(x,y,z)) / R(x,y,z)) \ e)),y,z) by Theorem 863. Then R(x,y,z) / (((x \ R(x,y,z)) / R(x,y,z)) \ e) = ((x \ R(x,y,z)) / R(x,y,z)) * R(((x \ R(x,y,z)) / R(x,y,z)) \ (x / (((x \ R(x,y,z)) / R(x,y,z)) \ e)),y,z) by Axiom 6. Then ((x \ R(x,y,z)) / R(x,y,z)) * R(((x \ R(x,y,z)) / R(x,y,z)) \ (((x \ R(x,y,z)) / R(x,y,z)) * x),y,z) = R(x,y,z) / (((x \ R(x,y,z)) / R(x,y,z)) \ e) by Theorem 867. Then
(L4) ((x \ R(x,y,z)) / R(x,y,z)) * R(x,y,z) = R(x,y,z) / (((x \ R(x,y,z)) / R(x,y,z)) \ e)
by Axiom 3. We have K(R(x,y,z) \ (R(x,y,z) / (((x \ R(x,y,z)) / R(x,y,z)) \ e)),R(x,y,z)) = ((x \ R(x,y,z)) / R(x,y,z)) \ T((x \ R(x,y,z)) / R(x,y,z),R(x,y,z)) by Theorem 198. Then K(R(x,y,z) \ (R(x,y,z) / (((x \ R(x,y,z)) / R(x,y,z)) \ e)),R(x,y,z)) = ((x \ R(x,y,z)) / R(x,y,z)) \ (R(x,y,z) \ (x \ R(x,y,z))) by Proposition 47. Then K(R(x,y,z) \ (R(x,y,z) / (((x \ R(x,y,z)) / R(x,y,z)) \ e)),R(x,y,z)) = ((x \ R(x,y,z)) / R(x,y,z)) \ (x \ e) by Theorem 835. Then K(R(x,y,z) \ (((x \ R(x,y,z)) / R(x,y,z)) * R(x,y,z)),R(x,y,z)) = ((x \ R(x,y,z)) / R(x,y,z)) \ (x \ e) by (L4). Then K(R(x,y,z) \ (x \ R(x,y,z)),R(x,y,z)) = ((x \ R(x,y,z)) / R(x,y,z)) \ (x \ e) by Axiom 6. Then
(L10) K(x \ e,R(x,y,z)) = ((x \ R(x,y,z)) / R(x,y,z)) \ (x \ e)
by Theorem 835. We have (((R(x,y,z) * (x \ e)) / R(x,y,z)) \ (x \ e)) \ (x \ e) = (R(x,y,z) * (x \ e)) / R(x,y,z) by Theorem 946. Then (((x \ R(x,y,z)) / R(x,y,z)) \ (x \ e)) \ (x \ e) = (R(x,y,z) * (x \ e)) / R(x,y,z) by Proposition 89. Then (((x \ R(x,y,z)) / R(x,y,z)) \ (x \ e)) \ (x \ e) = (x \ R(x,y,z)) / R(x,y,z) by Proposition 89. Then
(L14) K(x \ e,R(x,y,z)) \ (x \ e) = (x \ R(x,y,z)) / R(x,y,z)
by (L10). We have (x \ e) \ (K(x \ e,R(x,y,z)) \ (x \ e)) = K(x \ e,R(x,y,z)) \ e by Theorem 957. Then
(L16) (x \ e) \ ((x \ R(x,y,z)) / R(x,y,z)) = K(x \ e,R(x,y,z)) \ e
by (L14). We have (x \ e) \ ((x \ R(x,y,z)) / R(x,y,z)) = x * ((x \ R(x,y,z)) / R(x,y,z)) by Theorem 709. Then K(x \ e,R(x,y,z)) \ e = x * ((x \ R(x,y,z)) / R(x,y,z)) by (L16). Then
(L19) K(R(x,y,z),x \ e) = K(x \ e,R(x,y,z)) \ e
by Theorem 781. We have (K(x \ e,R(x,y,z)) \ e) \ e = K(x \ e,R(x,y,z)) by Theorem 900. Then K(R(x,y,z),x \ e) \ e = K(x \ e,R(x,y,z)) by (L19). Then
(L22) a(R(x,y,z),e / x,x) \ e = K(x \ e,R(x,y,z))
by Theorem 782. We have (R(x,y,z) \ T(R(x,y,z),x \ e)) \ e = T(R(x,y,z),x \ e) \ R(x,y,z) by Theorem 749. Then (R(x,y,z) \ R(T(x,x \ e),y,z)) \ e = T(R(x,y,z),x \ e) \ R(x,y,z) by Axiom 9. Then (R(x,y,z) \ R(T(x,x \ e),y,z)) \ e = R(T(x,x \ e),y,z) \ R(x,y,z) by Axiom 9. Then a(R(x,y,z),e / x,x) \ e = R(T(x,x \ e),y,z) \ R(x,y,z) by Theorem 545. Hence we are done by (L22).
Theorem 961 (prov9_e4dadbf002): K(x \ e,R(x,y,z)) = x * T(x \ e,R(x,y,z))
Proof. We have R(R(x,T(e / x,R(x,y,z)),x),y,z) = R(R(x,y,z),T(e / x,R(x,y,z)),x) by Axiom 12. Then
(L2) R(T(x,T(e / x,R(x,y,z))),y,z) = R(R(x,y,z),T(e / x,R(x,y,z)),x)
by Theorem 865. We have R(R(x,y,z),R(x,y,z) \ (R(x,y,z) / x),x) \ R(x,y,z) = (R(x,y,z) \ (R(x,y,z) / x)) * x by Theorem 808. Then R(R(x,y,z),T(e / x,R(x,y,z)),x) \ R(x,y,z) = (R(x,y,z) \ (R(x,y,z) / x)) * x by Theorem 839. Then R(T(x,T(e / x,R(x,y,z))),y,z) \ R(x,y,z) = (R(x,y,z) \ (R(x,y,z) / x)) * x by (L2). Then R(T(x,x \ e),y,z) \ R(x,y,z) = (R(x,y,z) \ (R(x,y,z) / x)) * x by Theorem 886. Then
(L7) R(T(x,x \ e),y,z) \ R(x,y,z) = T(e / x,R(x,y,z)) * x
by Theorem 839. We have T(e / x,R(x,y,z)) * x = x * T(x \ e,R(x,y,z)) by Proposition 61. Then R(T(x,x \ e),y,z) \ R(x,y,z) = x * T(x \ e,R(x,y,z)) by (L7). Hence we are done by Theorem 960.
Theorem 962 (prov9_7847ec3d14): K(R(x,y,z) \ e,R(x,y,z)) * (y * z) = (K(x \ e,R(x,y,z)) * y) * z
Proof. We have R(x * T(x \ e,R(x,y,z)),y,z) * (y * z) = ((x * T(x \ e,R(x,y,z))) * y) * z by Proposition 54. Then K(R(x,y,z) \ e,R(x,y,z)) * (y * z) = ((x * T(x \ e,R(x,y,z))) * y) * z by Theorem 637. Hence we are done by Theorem 961.
Theorem 963 (prov9_580000506a): R(e / x,y,z) \ e = e / R(x \ e,y,z)
Proof. We have R(e / (e / (x \ e)),y,z) \ e = e / R(x \ e,y,z) by Theorem 783. Hence we are done by Proposition 24.
Theorem 964 (prov9_1ea379c9f8): (R(x,y,z) \ e) \ e = R((x \ e) \ e,y,z)
Proof. We have e / R((x \ e) \ e,y,z) = R(e / (x \ e),y,z) \ e by Theorem 963. Then
(L2) e / R((x \ e) \ e,y,z) = R(x,y,z) \ e
by Proposition 24. We have (e / R((x \ e) \ e,y,z)) \ e = R((x \ e) \ e,y,z) by Proposition 25. Hence we are done by (L2).
Theorem 965 (prov9_16f0b963f8): (x * z) * ((((x * z) \ (y * z)) \ e) \ e) = (x * (((x \ y) \ e) \ e)) * z
Proof. We have ((((y * z) / (x * z)) \ e) \ e) * (x * z) = (x * z) * ((((x * z) \ (y * z)) \ e) \ e) by Theorem 921. Then ((R(y / x,x,z) \ e) \ e) * (x * z) = (x * z) * ((((x * z) \ (y * z)) \ e) \ e) by Proposition 55. Then
(L3) R(((y / x) \ e) \ e,x,z) * (x * z) = (x * z) * ((((x * z) \ (y * z)) \ e) \ e)
by Theorem 964. We have R(((y / x) \ e) \ e,x,z) * (x * z) = ((((y / x) \ e) \ e) * x) * z by Proposition 54. Then (x * z) * ((((x * z) \ (y * z)) \ e) \ e) = ((((y / x) \ e) \ e) * x) * z by (L3). Hence we are done by Theorem 921.
Theorem 966 (prov9_e34dffcd6b): K(z \ e,R(z,x,y)) = K(z \ e,z)
Proof. We have (((((z * x) * y) / (x * y)) \ e) \ e) \ ((z * x) * y) = ((z * x) * y) * (((((z * x) * y) \ (x * y)) \ e) \ e) by Theorem 922. Then ((R(z,x,y) \ e) \ e) \ ((z * x) * y) = ((z * x) * y) * (((((z * x) * y) \ (x * y)) \ e) \ e) by Definition 5. Then ((z * x) * ((((z * x) \ x) \ e) \ e)) * y = ((R(z,x,y) \ e) \ e) \ ((z * x) * y) by Theorem 965. Then
(L4) ((R(z,x,y) \ e) \ e) \ ((z * x) * y) = (K(z \ e,z) * x) * y
by Theorem 947. We have K(R(z,x,y) \ e,R(z,x,y)) * (R(z,x,y) \ ((z * x) * y)) = T(R(z,x,y),R(z,x,y) \ e) \ ((z * x) * y) by Theorem 915. Then K(R(z,x,y) \ e,R(z,x,y)) * (x * y) = T(R(z,x,y),R(z,x,y) \ e) \ ((z * x) * y) by Theorem 6. Then T(R(z,x,y),R(z,x,y) \ e) \ ((z * x) * y) = (K(z \ e,R(z,x,y)) * x) * y by Theorem 962. Then ((R(z,x,y) \ e) \ e) \ ((z * x) * y) = (K(z \ e,R(z,x,y)) * x) * y by Proposition 49. Then K(z \ e,z) * x = K(z \ e,R(z,x,y)) * x by (L4) and Proposition 8. Hence we are done by Proposition 10.
Theorem 967 (prov9_4d95a7d5a2): L(z,x,K(y \ e,y)) = z
Proof. We have (R(y,x,z) * (y * T(y \ e,R(y,x,z)))) / R(T(y,y * T(y \ e,R(y,x,z))),x,z) = y * T(y \ e,R(y,x,z)) by Theorem 456. Then (R(y,x,z) * (y * T(y \ e,R(y,x,z)))) / R(y,x,z) = y * T(y \ e,R(y,x,z)) by Theorem 203. Then (y * (R(y,x,z) * T(y \ e,R(y,x,z)))) / R(y,x,z) = y * T(y \ e,R(y,x,z)) by Theorem 227. Then (y * ((y \ e) * R(y,x,z))) / R(y,x,z) = y * T(y \ e,R(y,x,z)) by Proposition 46. Then L(R(y,x,z),y \ e,y) / R(y,x,z) = y * T(y \ e,R(y,x,z)) by Proposition 56. Then
(L9) R(e / (e / y),x,z) / R(y,x,z) = y * T(y \ e,R(y,x,z))
by Theorem 851. We have (((e / y) * R(y,x,z)) / (e / y)) / R(y,x,z) = R(y,x,z) \ (((e / y) * R(y,x,z)) / (e / y)) by Theorem 940. Then R(e / (e / y),x,z) / R(y,x,z) = R(y,x,z) \ (((e / y) * R(y,x,z)) / (e / y)) by Theorem 813. Then R(y,x,z) \ R(e / (e / y),x,z) = R(e / (e / y),x,z) / R(y,x,z) by Theorem 813. Then R(y,x,z) \ R(e / (e / y),x,z) = y * T(y \ e,R(y,x,z)) by (L9). Then
(L11) R(y,x,z) \ R(e / (e / y),x,z) = K(y \ e,R(y,x,z))
by Theorem 961. We have ((R(y,x,z) \ e) / (R(y,x,z) \ L(R(y,x,z),y \ e,y))) \ (R(y,x,z) \ e) = R(y,x,z) \ L(R(y,x,z),y \ e,y) by Proposition 25. Then (L(R(y,x,z),y \ e,y) \ e) \ (R(y,x,z) \ e) = R(y,x,z) \ L(R(y,x,z),y \ e,y) by Theorem 906. Then (R(e / (e / y),x,z) \ e) \ (R(y,x,z) \ e) = R(y,x,z) \ L(R(y,x,z),y \ e,y) by Theorem 851. Then R(y,x,z) \ R(e / (e / y),x,z) = (R(e / (e / y),x,z) \ e) \ (R(y,x,z) \ e) by Theorem 851. Then (R(e / (e / y),x,z) \ e) \ (R(y,x,z) \ e) = K(y \ e,R(y,x,z)) by (L11). Then
(L17) (e / R(y,x,z)) \ (R(y,x,z) \ e) = K(y \ e,R(y,x,z))
by Theorem 783. We have (e / R(y,x,z)) \ (R(y,x,z) \ e) = K(R(y,x,z) \ e,R(y,x,z)) by Proposition 90. Then
(L19) K(y \ e,R(y,x,z)) = K(R(y,x,z) \ e,R(y,x,z))
by (L17). We have K(R(y,x,z) \ e,R(y,x,z)) * (x * z) = (K(y \ e,R(y,x,z)) * x) * z by Theorem 962. Then K(y \ e,R(y,x,z)) * (x * z) = (K(y \ e,R(y,x,z)) * x) * z by (L19). Then L(z,x,K(y \ e,R(y,x,z))) = z by Theorem 54. Hence we are done by Theorem 966.
Theorem 968 (prov9_0a15f82dcb): (K(x \ e,x) * y) * z = K(x \ e,x) * (y * z)
Proof. We have (K(x \ e,x) * y) * L(z,y,K(x \ e,x)) = K(x \ e,x) * (y * z) by Proposition 52. Hence we are done by Theorem 967.
Theorem 969 (prov9_ff0b447b85): (K(x \ e,x) * y) / z = K(x \ e,x) * (y / z)
Proof. We have (K(x \ e,x) * y) / L(z,y / z,K(x \ e,x)) = K(x \ e,x) * (y / z) by Theorem 472. Hence we are done by Theorem 967.
Theorem 970 (prov9_a2c9d723c0): R(K(x \ e,x) * y,z,w) = K(x \ e,x) * R(y,z,w)
Proof. We have ((K(x \ e,x) * (y * z)) * w) / (L(z,y,K(x \ e,x)) * w) = R(K(x \ e,x) * y,L(z,y,K(x \ e,x)),w) by Theorem 797. Then ((K(x \ e,x) * (y * z)) * w) / (z * w) = R(K(x \ e,x) * y,L(z,y,K(x \ e,x)),w) by Theorem 967. Then
(L3) R(K(x \ e,x) * y,L(z,y,K(x \ e,x)),w) = (K(x \ e,x) * ((y * z) * w)) / (z * w)
by Theorem 968. We have (K(x \ e,x) * ((y * z) * w)) / (z * w) = K(x \ e,x) * (((y * z) * w) / (z * w)) by Theorem 969. Then R(K(x \ e,x) * y,L(z,y,K(x \ e,x)),w) = K(x \ e,x) * (((y * z) * w) / (z * w)) by (L3). Then R(K(x \ e,x) * y,L(z,y,K(x \ e,x)),w) = K(x \ e,x) * R(y,z,w) by Definition 5. Hence we are done by Theorem 967.
Theorem 971 (prov9_a8c3c9917a): R(x * w,y,z) = (w \ R(w * x,y,z)) * w
Proof. We have (x * T(w,w \ e)) * K(w \ e,w) = x * w by Theorem 909. Then
(L2) K(w \ e,w) * (x * T(w,w \ e)) = x * w
by Theorem 923. We have K(w \ e,w) * R(x * T(w,w \ e),y,z) = (w \ R(w * x,y,z)) * w by Theorem 927. Then R(K(w \ e,w) * (x * T(w,w \ e)),y,z) = (w \ R(w * x,y,z)) * w by Theorem 970. Hence we are done by (L2).
Theorem 972 (prov9_9fce12ad3d): R(y * x,z,w) / x = x \ R(x * y,z,w)
Proof. We have (x \ R(x * y,z,w)) * x = R(y * x,z,w) by Theorem 971. Hence we are done by Proposition 1.
Theorem 973 (prov9_30a2345e7b): (y \ R(x,z,w)) * y = R((y \ x) * y,z,w)
Proof. We have (y \ R(y * (y \ x),z,w)) * y = R((y \ x) * y,z,w) by Theorem 971. Hence we are done by Axiom 4.
Theorem 974 (prov9_dfbce703b5): R((z \ (x / (e / y))) * z,y \ e,y) = (z \ (x * y)) * z
Proof. We have R((z \ (x / (e / y))) * z,y \ e,y) = (z \ R(x / (e / y),y \ e,y)) * z by Theorem 973. Hence we are done by Theorem 912.
Theorem 975 (prov9_812bd6e16c): y * (R(x,z,w) / y) = R(y * (x / y),z,w)
Proof. We have R((x / y) * y,z,w) / y = y \ R(y * (x / y),z,w) by Theorem 972. Then
(L2) R(x,z,w) / y = y \ R(y * (x / y),z,w)
by Axiom 6. We have y * (y \ R(y * (x / y),z,w)) = R(y * (x / y),z,w) by Axiom 4. Hence we are done by (L2).
Theorem 976 (prov9_bd6c35f159): y * x = (x * (y \ T(y,x))) * y
Proof. We have T((e / (e / (x \ (x / (y \ e))))) * x,K((x \ (x / (y \ e))) \ e,x \ (x / (y \ e)))) = ((x \ (x / (y \ e))) * x) * K((x \ (x / (y \ e))) \ e,x \ (x / (y \ e))) by Theorem 916. Then T((e / (e / (x \ (x / (y \ e))))) * x,K((x \ (x / (y \ e))) \ e,x \ (x / (y \ e)))) = ((x \ (x / (y \ e))) * x) * K((x / (y \ e)) \ x,x \ (x / (y \ e))) by Theorem 666. Then (e / (e / (x \ (x / (y \ e))))) * x = ((x \ (x / (y \ e))) * x) * K((x / (y \ e)) \ x,x \ (x / (y \ e))) by Theorem 717. Then ((x \ (x / (y \ e))) * x) * K(y \ e,x \ (x / (y \ e))) = (e / (e / (x \ (x / (y \ e))))) * x by Proposition 25. Then ((x \ (x / (y \ e))) * x) * K(y \ e,(y \ e) \ e) = (e / (e / (x \ (x / (y \ e))))) * x by Theorem 895. Then ((x \ (x / (y \ e))) * x) * K(y \ e,y) = (e / (e / (x \ (x / (y \ e))))) * x by Theorem 779. Then ((x \ (x / (y \ e))) * x) * K(y \ e,y) = (e / ((x \ (x / (e / y))) \ e)) * x by Theorem 419. Then
(L8) ((x \ (x / (y \ e))) * x) * K(y \ e,y) = (x \ (x / (e / y))) * x
by Proposition 24. We have ((((x \ (x / (y \ e))) * x) * K(y \ e,y)) * (e / y)) * y = R(((x \ (x / (y \ e))) * x) * K(y \ e,y),y \ e,y) by Theorem 911. Then (((x \ (x / (y \ e))) * x) * (y \ e)) * y = R(((x \ (x / (y \ e))) * x) * K(y \ e,y),y \ e,y) by Theorem 411. Then R(((x \ (x / (y \ e))) * x) * K(y \ e,y),y \ e,y) = (x * ((y \ e) * T((y \ e) \ e,x))) * y by Theorem 840. Then R((x \ (x / (e / y))) * x,y \ e,y) = (x * ((y \ e) * T((y \ e) \ e,x))) * y by (L8). Then R((x \ (x / (e / y))) * x,y \ e,y) = (x * (y \ T(y,x))) * y by Theorem 145. Then (x \ (x * y)) * x = (x * (y \ T(y,x))) * y by Theorem 974. Hence we are done by Axiom 3.
Theorem 977 (prov9_040f0caf2a): (x * y) / x = y * (x \ T(x,y))
Proof. We have (y * (x \ T(x,y))) * x = x * y by Theorem 976. Hence we are done by Proposition 1.
Theorem 978 (prov9_5a690dbadd): T(y,x) * (x \ T(x,T(y,x))) = y
Proof. We have
(L1) (T(y,x) * (x \ T(x,T(y,x)))) * x = x * T(y,x)
by Theorem 976. We have y * x = x * T(y,x) by Proposition 46. Hence we are done by (L1) and Proposition 8.
Theorem 979 (prov9_2977ac2465): y \ ((x * y) / x) = x \ T(x,y)
Proof. We have y * (x \ T(x,y)) = (x * y) / x by Theorem 977. Hence we are done by Proposition 2.
Theorem 980 (prov9_4483f3770e): y \ T(y,x \ e) = x * ((y * (x \ e)) / y)
Proof. We have (x \ e) * (((y * (x \ e)) / y) / (x \ e)) = (y * (x \ e)) / y by Theorem 942. Then
(L3) (x \ e) * (x * ((y * (x \ e)) / y)) = (y * (x \ e)) / y
by Theorem 859. We have (x \ e) * (y \ T(y,x \ e)) = (y * (x \ e)) / y by Theorem 977. Hence we are done by (L3) and Proposition 7.
Theorem 981 (prov9_4d4fb7f2ee): x \ T(x,y) = ((x * y) / x) / y
Proof. We have
(L1) y * (x \ T(x,y)) = (x * y) / x
by Theorem 977. We have y * (((x * y) / x) / y) = (x * y) / x by Theorem 942. Hence we are done by (L1) and Proposition 7.
Theorem 982 (prov9_6da06e3ce9): x * (y * T(y \ e,x)) = ((y \ e) * x) / (y \ e)
Proof. We have x * ((y \ e) \ T(y \ e,x)) = ((y \ e) * x) / (y \ e) by Theorem 977. Hence we are done by Theorem 729.
Theorem 983 (prov9_f4ff2a3879): ((x * y) / x) * (y \ e) = x \ T(x,y)
Proof. We have ((L(x,e / x,y) * y) / L(x,e / x,y)) * (y \ e) = y \ ((L(x,e / x,y) * y) / L(x,e / x,y)) by Theorem 195. Then (y \ ((L(x,e / x,y) * y) / L(x,e / x,y))) / (y \ e) = (L(x,e / x,y) * y) / L(x,e / x,y) by Proposition 1. Then (y \ ((L(x,e / x,y) * y) / L(x,e / x,y))) / (y \ e) = y * (T(x,y) / x) by Theorem 883. Then (y \ (y * (T(x,y) / x))) / (y \ e) = y * (T(x,y) / x) by Theorem 883. Then (T(x,y) / x) / (y \ e) = y * (T(x,y) / x) by Axiom 3. Then (x \ T(x,y)) / (y \ e) = y * (T(x,y) / x) by Theorem 758. Then
(L7) (x \ T(x,y)) / (y \ e) = y * (x \ T(x,y))
by Theorem 758. We have ((x \ T(x,y)) / (y \ e)) * (y \ e) = x \ T(x,y) by Axiom 6. Then (y * (x \ T(x,y))) * (y \ e) = x \ T(x,y) by (L7). Hence we are done by Theorem 977.
Theorem 984 (prov9_366a6031e4): ((y * x) / y) \ x = T(y,x) \ y
Proof. We have (x * (y \ T(y,x))) \ x = T(y,x) \ y by Theorem 952. Hence we are done by Theorem 977.
Theorem 985 (prov9_49882f8ca9): T(x,y) \ x = y \ T(y,T(x,y))
Proof. We have T(x,y) * (y \ T(y,T(x,y))) = x by Theorem 978. Hence we are done by Proposition 2.
Theorem 986 (prov9_78f9e18149): x \ T(x,y) = T(y,T(x,y)) \ y
Proof. We have (T(x,y) * (y \ T(y,T(x,y)))) \ T(x,y) = T(y,T(x,y)) \ y by Theorem 952. Hence we are done by Theorem 978.
Theorem 987 (prov9_7bd4cca40a): y \ (x / (x / y)) = (x / y) \ T(x / y,y)
Proof. We have y \ (((x / y) * y) / (x / y)) = (x / y) \ T(x / y,y) by Theorem 979. Hence we are done by Axiom 6.
Theorem 988 (prov9_9e6aa9834e): L(y \ T(y,x),x,y) = K(y,(y * x) / y)
Proof. We have L(x \ ((y * x) / y),x,y) = K(y,(y * x) / y) by Theorem 799. Hence we are done by Theorem 979.
Theorem 989 (prov9_4c7a95813d): (y \ T(y,x)) * y = T(y,R(x,x \ e,y))
Proof. We have (x \ ((y * x) / y)) * y = T(y,R(x,x \ e,y)) by Theorem 589. Hence we are done by Theorem 979.
Theorem 990 (prov9_5eb90bac09): (x / y) \ (y \ x) = T(y,y \ x) \ y
Proof. We have ((y * (y \ x)) / y) \ (y \ x) = T(y,y \ x) \ y by Theorem 984. Hence we are done by Axiom 4.
Theorem 991 (prov9_43d8ad9c8b): y * (T(x,y) \ x) = T(y,T(x,y))
Proof. We have y * (y \ T(y,T(x,y))) = T(y,T(x,y)) by Axiom 4. Hence we are done by Theorem 985.
Theorem 992 (prov9_55a23dd4c9): T(y,T(x,y)) * (x \ T(x,y)) = y
Proof. We have T(y,T(x,y)) * (T(y,T(x,y)) \ y) = y by Axiom 4. Hence we are done by Theorem 986.
Theorem 993 (prov9_f15ff2a237): (x \ T(x,y)) \ y = T(y,T(x,y))
Proof. We have (T(y,T(x,y)) \ y) \ y = T(y,T(x,y)) by Theorem 930. Hence we are done by Theorem 986.
Theorem 994 (prov9_2c206ad8f1): T(y,T(L(x \ e,x,y) \ e,y)) = K(x,y) \ y
Proof. We have y * (T(L(x \ e,x,y) \ e,y) \ (L(x \ e,x,y) \ e)) = K(x,y) \ y by Theorem 959. Hence we are done by Theorem 991.
Theorem 995 (prov9_d683b10914): T(x,y) \ ((x * y) / x) = x \ y
Proof. We have (x \ T(x,y)) * (x * (T(x,y) \ y)) = L(y,x \ T(x,y),x) by Theorem 955. Then
(L2) (x \ T(x,y)) * T(y,T(x,y)) = L(y,x \ T(x,y),x)
by Theorem 177. We have (x \ T(x,y)) * T(T(y,x \ T(x,y)),T(x,y)) = T(y,T(x,y)) * (x \ T(x,y)) by Proposition 50. Then (x \ T(x,y)) * T(y,T(x,y)) = T(y,T(x,y)) * (x \ T(x,y)) by Theorem 936. Then
(L5) T(y,T(x,y)) * (x \ T(x,y)) = L(y,x \ T(x,y),x)
by (L2). We have
(L7) x * (T(x,y) \ ((x \ T(x,y)) * y)) = L(y,x \ T(x,y),x)
by Theorem 832.
x * (T(x,y) \ (y * (x \ T(x,y))))
=L(y,x \ T(x,y),x) by (L7), Theorem 937
=y by (L5), Theorem 992

Then x * (T(x,y) \ (y * (x \ T(x,y)))) = y. Then x \ y = T(x,y) \ (y * (x \ T(x,y))) by Proposition 2. Hence we are done by Theorem 977.
Theorem 996 (prov9_971e821309): T(y,x) * (y \ x) = (y * x) / y
Proof. We have T(y,x) * (T(y,x) \ ((y * x) / y)) = (y * x) / y by Axiom 4. Hence we are done by Theorem 995.
Theorem 997 (prov9_1ba1589be7): T(x,y) = T(x,x / (x / y))
Proof. We have
(L1) T(x,y) * (x \ y) = (x * y) / x
by Theorem 996. We have T(x,x / (x / y)) * (x \ y) = (x * y) / x by Theorem 861. Hence we are done by (L1) and Proposition 8.
Theorem 998 (prov9_99bb1c31ad): T(x,T(y,x)) * (y * x) = x * (x * y)
Proof. We have T(x,x / (x / T(y,x))) * (y * x) = x * (x * y) by Theorem 848. Hence we are done by Theorem 997.
Theorem 999 (prov9_1c5c696be5): T(x,R(y,y \ e,x)) = T(x,y)
Proof. We have (x \ T(x,y)) * x = T(x,y) by Theorem 759. Hence we are done by Theorem 989.
Theorem 1000 (prov9_1cb64dde5d): T(x,R(y \ e,y,x)) = T(x,y \ e)
Proof. We have T(x,R(y \ e,(y \ e) \ e,x)) = T(x,y \ e) by Theorem 999. Hence we are done by Theorem 704.
Theorem 1001 (prov9_157da4e91a): T(y,(x \ y) \ e) = y * K(y,x / y)
Proof. We have T(y,R((x \ y) \ e,((x \ y) \ e) \ e,y)) = y * K(y,x / y) by Theorem 954. Hence we are done by Theorem 999.
Theorem 1002 (prov9_61d4f2a226): T(y,x \ e) = T(y,y / (x * y))
Proof. We have T(y,((y / (x * y)) \ e) \ e) = T(y,y / (x * y)) by Theorem 721. Then T(y,R(x \ e,x,y)) = T(y,y / (x * y)) by Theorem 670. Hence we are done by Theorem 1000.
Theorem 1003 (prov9_9ef8baf693): T(x,y \ e) = T(x,x / (x * y))
Proof. We have T(x,R(e / y,y,x)) = T(x,R(y \ e,y,x)) by Theorem 725. Then T(x,x / (x * y)) = T(x,R(y \ e,y,x)) by Theorem 944. Hence we are done by Theorem 1000.
Theorem 1004 (prov9_1515408e68): y \ T(y,x \ e) = K(y,(y / x) / y)
Proof. We have T(y,R(e / x,x,y)) = T(y,y / (y * x)) by Theorem 944. Then
(L2) T(y,y / (x * y)) = T(y,y / (y * x))
by Proposition 79. We have y \ T(y,y / (y * x)) = x * ((y * (x \ e)) / y) by Theorem 846. Then y \ T(y,y / (y * x)) = K(y,(y / x) / y) by Theorem 333. Then y \ T(y,y / (x * y)) = K(y,(y / x) / y) by (L2). Hence we are done by Theorem 1002.
Theorem 1005 (prov9_cde98ad688): (x * ((x / y) / x)) * y = T(x,y \ e)
Proof. We have (x * ((x / y) / x)) * y = T(x,x / (x * y)) by Theorem 881. Hence we are done by Theorem 1003.
Theorem 1006 (prov9_8eb1da3fa8): L(x \ T(x,y),y,x) = K(T(x,y),y)
Proof. We have L((x \ (T(x,y) * y)) / y,y,x) = (y * ((x * y) \ (T(x,y) * y))) / y by Proposition 75. Then L((x \ (T(x,y) * y)) / y,y,x) = (y * K(T(x,y),y)) / y by Theorem 12. Then L((T(x,y) * (x \ y)) / y,y,x) = (y * K(T(x,y),y)) / y by Theorem 173. Then L((T(x,y) * (x \ y)) / y,y,x) = K(T(x,y),y) by Theorem 934. Then L(((x * y) / x) / y,y,x) = K(T(x,y),y) by Theorem 996. Hence we are done by Theorem 981.
Theorem 1007 (prov9_d9f1d43310): L(T(x \ T(x,y),z),y,x) = T(K(T(x,y),y),z)
Proof. We have L(T(x \ T(x,y),z),y,x) = T(L(x \ T(x,y),y,x),z) by Axiom 8. Hence we are done by Theorem 1006.
Theorem 1008 (prov9_f72c3322fe): K(T(y,x),x) = K(y,(y * x) / y)
Proof. We have L(y \ T(y,x),x,y) = K(y,(y * x) / y) by Theorem 988. Hence we are done by Theorem 1006.
Theorem 1009 (prov9_e5502c247c): (x / y) \ (y \ x) = y \ (x / (x / y))
Proof. We have (x / y) \ T(x / y,y) = y \ (x / (x / y)) by Theorem 987. Hence we are done by Proposition 47.
Theorem 1010 (prov9_bbdf4d3abe): T(y,y \ x) \ y = y \ (x / (x / y))
Proof. We have T(y,y \ x) \ y = (x / y) \ (y \ x) by Theorem 990. Hence we are done by Theorem 1009.
Theorem 1011 (prov9_959da2b0c1): (y * x) / ((y * x) / y) = y * (T(y,x) \ y)
Proof. We have y * (y \ ((y * x) / ((y * x) / y))) = (y * x) / ((y * x) / y) by Axiom 4. Then y * (((y * x) / y) \ (y \ (y * x))) = (y * x) / ((y * x) / y) by Theorem 1009. Then y * (T(y,y \ (y * x)) \ y) = (y * x) / ((y * x) / y) by Theorem 990. Hence we are done by Axiom 3.
Theorem 1012 (prov9_7b2f6e8a5e): T(y,((y * x) \ y) \ K(x,y)) = K(x,y) \ y
Proof. We have T(y,T(L(x \ e,x,y) \ e,y)) = K(x,y) \ y by Theorem 994. Hence we are done by Theorem 540.
Theorem 1013 (prov9_709ad2d871): ((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e) = y * K(x,y)
Proof. We have ((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e) = y * (L(x \ e,x,y) * T(L(x \ e,x,y) \ e,y)) by Theorem 982. Hence we are done by Theorem 542.
Theorem 1014 (prov9_bed53ff55d): T(x \ T(x,y \ e),z) = x \ T(x,T(y,z) \ e)
Proof. We have T(y * ((x * (y \ e)) / x),z) = K(x,(x / T(y,z)) / x) by Theorem 344. Then T(x \ T(x,x / (x * y)),z) = K(x,(x / T(y,z)) / x) by Theorem 846. Then
(L3) T(x \ T(x,y \ e),z) = K(x,(x / T(y,z)) / x)
by Theorem 1003. We have K(x,(x / T(y,z)) / x) = x \ T(x,T(y,z) \ e) by Theorem 1004. Hence we are done by (L3).
Theorem 1015 (prov9_f57976c3e7): T(y,x) * T(z,y) = (T(y,x) * (z \ T(z,y))) * z
Proof. We have R(y,z \ T(z,y),z) = ((y * (z \ T(z,y))) * z) / ((z \ T(z,y)) * z) by Definition 5. Then R(y,z \ T(z,y),z) = (z * y) / ((z \ T(z,y)) * z) by Theorem 976. Then
(L3) (z * y) / T(z,y) = R(y,z \ T(z,y),z)
by Theorem 759. We have (z * y) / T(z,y) = y by Theorem 5. Then
(L5) R(y,z \ T(z,y),z) = y
by (L3). We have R(T(y,x),z \ T(z,y),z) = T(R(y,z \ T(z,y),z),x) by Axiom 9. Then
(L7) R(T(y,x),z \ T(z,y),z) = T(y,x)
by (L5). We have R(T(y,x),z \ T(z,y),z) * ((z \ T(z,y)) * z) = (T(y,x) * (z \ T(z,y))) * z by Proposition 54. Then R(T(y,x),z \ T(z,y),z) * T(z,y) = (T(y,x) * (z \ T(z,y))) * z by Theorem 759. Hence we are done by (L7).
Theorem 1016 (prov9_8dbd2843b1): K(T(y,x),x) * z = K(y,(y * x) / y) * z
Proof. We have (z * L(T(y \ T(y,x),z),x,y)) * K(L(T(y \ T(y,x),z),x,y),z) = L(T(y \ T(y,x),z),x,y) * z by Proposition 82. Then (L(y \ T(y,x),x,y) * z) * K(L(T(y \ T(y,x),z),x,y),z) = L(T(y \ T(y,x),z),x,y) * z by Proposition 58. Then (K(y,(y * x) / y) * z) * K(L(T(y \ T(y,x),z),x,y),z) = L(T(y \ T(y,x),z),x,y) * z by Theorem 988. Then
(L4) (K(y,(y * x) / y) * z) * K(T(K(T(y,x),x),z),z) = L(T(y \ T(y,x),z),x,y) * z
by Theorem 1007. We have
(L6) (z * T(K(T(y,x),x),z)) * K(T(K(T(y,x),x),z),z) = T(K(T(y,x),x),z) * z
by Proposition 82.
(K(T(y,x),x) * z) * K(T(K(T(y,x),x),z),z)
=T(K(T(y,x),x),z) * z by (L6), Proposition 46
=(K(y,(y * x) / y) * z) * K(T(K(T(y,x),x),z),z) by (L4), Theorem 1007

Then (K(T(y,x),x) * z) * K(T(K(T(y,x),x),z),z) = (K(y,(y * x) / y) * z) * K(T(K(T(y,x),x),z),z). Hence we are done by Proposition 10.
Theorem 1017 (prov9_6080c3af78): T(x,y) * z = x * ((x \ T(x,y)) * z)
Proof. We have
(L1) x * (((y \ e) * (x \ (x * z))) * ((x * (((y \ e) * (x \ (x * z))) \ (x \ (x * z)))) / x)) = (x * (((x * z) / ((y \ e) * (x \ (x * z)))) / x)) * ((y \ e) * (x \ (x * z)))
by Theorem 882. Then x * (((y \ e) * (x \ (x * z))) * ((x * (((y \ e) * (x \ (x * z))) \ (x \ (x * z)))) / x)) = (x * (R(x / (y \ e),y \ e,x \ (x * z)) / x)) * ((y \ e) * (x \ (x * z))) by Theorem 794. Then (x * (R(x / (y \ e),y \ e,x \ (x * z)) / x)) * ((y \ e) * (x \ (x * z))) = (x * (((x * z) / ((y \ e) * (x \ (x * z)))) / x)) * ((y \ e) * (x \ (x * z))) by (L1). Then
(L4) R(x * ((x / (y \ e)) / x),y \ e,x \ (x * z)) * ((y \ e) * (x \ (x * z))) = (x * (((x * z) / ((y \ e) * (x \ (x * z)))) / x)) * ((y \ e) * (x \ (x * z)))
by Theorem 975. We have R(x * ((x / (y \ e)) / x),y \ e,x \ (x * z)) * ((y \ e) * (x \ (x * z))) = ((x * ((x / (y \ e)) / x)) * (y \ e)) * (x \ (x * z)) by Proposition 54. Then
(L6) (x * (((x * z) / ((y \ e) * (x \ (x * z)))) / x)) * ((y \ e) * (x \ (x * z))) = ((x * ((x / (y \ e)) / x)) * (y \ e)) * (x \ (x * z))
by (L4). We have x * (((y \ e) * ((x * ((y \ e) \ e)) / x)) * (x \ (x * z))) = (x * (((x * z) / ((y \ e) * (x \ (x * z)))) / x)) * ((y \ e) * (x \ (x * z))) by (L1) and Theorem 343. Then (x * (((x * z) / ((y \ e) * (x \ (x * z)))) / x)) * ((y \ e) * (x \ (x * z))) = x * ((x \ T(x,(y \ e) \ e)) * (x \ (x * z))) by Theorem 980. Then ((x * ((x / (y \ e)) / x)) * (y \ e)) * (x \ (x * z)) = x * ((x \ T(x,(y \ e) \ e)) * (x \ (x * z))) by (L6). Then T(x,(y \ e) \ e) * (x \ (x * z)) = x * ((x \ T(x,(y \ e) \ e)) * (x \ (x * z))) by Theorem 1005. Then x * ((x \ T(x,y)) * (x \ (x * z))) = T(x,(y \ e) \ e) * (x \ (x * z)) by Theorem 721. Then T(x,y) * (x \ (x * z)) = x * ((x \ T(x,y)) * (x \ (x * z))) by Theorem 721. Then x * ((x \ T(x,y)) * z) = T(x,y) * (x \ (x * z)) by Axiom 3. Hence we are done by Axiom 3.
Theorem 1018 (prov9_c0a3d47005): (x \ T(x,y)) * (x * z) = T(x,y) * z
Proof. We have
(L1) x * ((x \ T(x,y)) * (x * z)) = T(x,y) * (x * z)
by Theorem 1017. We have x * (T(x,y) * z) = T(x,y) * (x * z) by Theorem 172. Hence we are done by (L1) and Proposition 7.
Theorem 1019 (prov9_2d7b704219): R(z,K(x,z) \ e,y) = z
Proof. We have z * ((z \ T(z,T(L(x \ e,x,z) \ e,z))) * y) = T(z,T(L(x \ e,x,z) \ e,z)) * y by Theorem 1017. Then
(L2) (T(z,T(L(x \ e,x,z) \ e,z)) * y) / ((z \ T(z,T(L(x \ e,x,z) \ e,z))) * y) = z
by Proposition 1. We have R(z,z \ T(z,T(L(x \ e,x,z) \ e,z)),y) = (T(z,T(L(x \ e,x,z) \ e,z)) * y) / ((z \ T(z,T(L(x \ e,x,z) \ e,z))) * y) by Theorem 785. Then R(z,z \ T(z,T(L(x \ e,x,z) \ e,z)),y) = z by (L2). Then R(z,z \ (K(x,z) \ z),y) = z by Theorem 994. Hence we are done by Theorem 958.
Theorem 1020 (prov9_6d5036c7fc): (T(y,z) * x) / (y * x) = y \ T(y,z)
Proof. We have (y \ T(y,z)) * (y * x) = T(y,z) * x by Theorem 1018. Hence we are done by Proposition 1.
Theorem 1021 (prov9_880aeb5582): L(y \ T(y,T(x,y)),T(x,y),y) = R(K(y,x),y,z)
Proof. We have R(y \ T(y,T(x,y)),y,z) = (T(y,T(x,y)) * z) / (y * z) by Theorem 943. Then
(L2) R(T(x,y) \ x,y,z) = (T(y,T(x,y)) * z) / (y * z)
by Theorem 985. We have L(R(T(x,y) \ x,y,z),T(x,y),y) = R((x * y) \ (y * x),y,z) by Theorem 817. Then L((T(y,T(x,y)) * z) / (y * z),T(x,y),y) = R((x * y) \ (y * x),y,z) by (L2). Then L((T(y,T(x,y)) * z) / (y * z),T(x,y),y) = R(K(y,x),y,z) by Definition 2. Hence we are done by Theorem 1020.
Theorem 1022 (prov9_cc440825cf): (T(y,x) \ y) * (y * x) = x * y
Proof. We have x * ((x \ T(x,T(y,x))) * (y * x)) = T(x,T(y,x)) * (y * x) by Theorem 1017. Then
(L2) x * ((T(y,x) \ y) * (y * x)) = T(x,T(y,x)) * (y * x)
by Theorem 985. We have x * (x * y) = T(x,T(y,x)) * (y * x) by Theorem 998. Hence we are done by (L2) and Proposition 7.
Theorem 1023 (prov9_85d94b16d9): (x * y) / (y * x) = T(y,x) \ y
Proof. We have (T(y,x) \ y) * (y * x) = x * y by Theorem 1022. Hence we are done by Proposition 1.
Theorem 1024 (prov9_8e094d89d8): (x * (y / x)) / y = x \ ((x \ y) \ y)
Proof. We have (y / x) \ ((y / x) * x) = x by Axiom 3. Then (y / x) \ (x * (x \ ((y / x) * x))) = x by Axiom 4. Then
(L7) (y / x) \ (x * (x \ y)) = x
by Axiom 6. We have (y / x) * ((y / x) \ (x * (x \ y))) = x * (x \ y) by Axiom 4. Then
(L9) (y / x) * x = x * (x \ y)
by (L7). We have ((x \ y) * (x \ T(x,x \ y))) * x = x * (x \ y) by Theorem 976. Then (x \ y) * (x \ T(x,x \ y)) = y / x by (L9) and Proposition 8. Then (x \ y) * (x \ ((x \ y) \ y)) = y / x by Proposition 49. Then
(L12) (x \ y) \ (y / x) = x \ ((x \ y) \ y)
by Proposition 2. We have (x * (y / x)) / ((y / x) * x) = T(y / x,x) \ (y / x) by Theorem 1023. Then (x * (y / x)) / y = T(y / x,x) \ (y / x) by Axiom 6. Then (x \ y) \ (y / x) = (x * (y / x)) / y by Proposition 47. Hence we are done by (L12).
Theorem 1025 (prov9_731655c8f1): K(y,x) = R(K(y,x),y,z)
Proof. We have L(y \ T(y,T(x,y)),T(x,y),y) = K(y,(y * T(x,y)) / y) by Theorem 988. Then L(y \ T(y,T(x,y)),T(x,y),y) = K(y,x) by Proposition 48. Hence we are done by Theorem 1021.
Theorem 1026 (prov9_f44dc67883): ((y \ x) / x) \ T((y \ x) / x,z) = y * T(y \ e,z)
Proof. We have R(T((y \ x) / x,z) / ((y \ x) / x),(y \ x) / x,x) = (T((y \ x) / x,z) * x) / (y \ x) by Theorem 67. Then R(((y \ x) / x) \ T((y \ x) / x,z),(y \ x) / x,x) = (T((y \ x) / x,z) * x) / (y \ x) by Theorem 758. Then
(L5) R(((y \ x) / x) \ T((y \ x) / x,z),(y \ x) / x,x) = (x * T(x \ (y \ x),z)) / (y \ x)
by Proposition 61. We have R(((y \ x) / x) \ T((y \ x) / x,z),(y \ x) / x,x) = (T((y \ x) / x,z) * x) / (((y \ x) / x) * x) by Theorem 943. Then R(((y \ x) / x) \ T((y \ x) / x,z),(y \ x) / x,x) = ((y \ x) / x) \ T((y \ x) / x,z) by Theorem 1020. Then
(L6) ((y \ x) / x) \ T((y \ x) / x,z) = (x * T(x \ (y \ x),z)) / (y \ x)
by (L5). We have (x * T(x \ (y \ x),z)) / (y \ x) = y * T(y \ e,z) by Theorem 843. Hence we are done by (L6).
Theorem 1027 (prov9_c9448e05e4): T(x,x \ (y \ x)) = T(x,y \ e)
Proof. We have (x * R(e / y,y,y \ x)) / (y \ x) = ((x * (e / y)) / x) * y by Theorem 828. Then
(L2) (x * ((y \ x) / x)) / (y \ x) = ((x * (e / y)) / x) * y
by Theorem 37. We have ((x * (e / y)) / x) * y = y * ((x * (y \ e)) / x) by Theorem 125. Then (x * ((y \ x) / x)) / (y \ x) = y * ((x * (y \ e)) / x) by (L2). Then (x * ((y \ x) / x)) / (y \ x) = x \ T(x,y \ e) by Theorem 980. Then
(L6) x \ ((x \ (y \ x)) \ (y \ x)) = x \ T(x,y \ e)
by Theorem 1024. We have (x \ T(x,x \ (y \ x))) * x = T(x,x \ (y \ x)) by Theorem 759. Then (x \ ((x \ (y \ x)) \ (y \ x))) * x = T(x,x \ (y \ x)) by Proposition 49. Then
(L9) (x \ T(x,y \ e)) * x = T(x,x \ (y \ x))
by (L6). We have (x \ T(x,y \ e)) * x = T(x,y \ e) by Theorem 759. Hence we are done by (L9).
Theorem 1028 (prov9_d96bcc0089): x * T(x \ e,y) = T(y,x \ e) \ y
Proof. We have
(L1) ((x \ y) / y) \ T((x \ y) / y,y) = y \ ((x \ y) / ((x \ y) / y))
by Theorem 987. We have
(L3) T(y,y \ (x \ y)) \ y = y \ ((x \ y) / ((x \ y) / y))
by Theorem 1010.
x * T(x \ e,y)
=y \ ((x \ y) / ((x \ y) / y)) by (L1), Theorem 1026
=T(y,x \ e) \ y by (L3), Theorem 1027

Hence we are done.
Theorem 1029 (prov9_67e705de1a): K(y \ (y / x),y) = T(y,x \ e) \ y
Proof. We have K(y \ (y / x),y) = x * T(x \ e,y) by Theorem 196. Hence we are done by Theorem 1028.
Theorem 1030 (prov9_f66610a187): T(y,x) \ y = x \ T(x,y)
Proof. We have (x \ e) * T((x \ e) \ e,y) = x \ T(x,y) by Theorem 145. Then T(y,(x \ e) \ e) \ y = x \ T(x,y) by Theorem 1028. Hence we are done by Theorem 721.
Theorem 1031 (prov9_0e7529f187): T(y,x) * (x \ T(x,y)) = y
Proof. We have T(y,x) * (T(y,x) \ y) = y by Axiom 4. Hence we are done by Theorem 1030.
Theorem 1032 (prov9_bb51e72df9): T(x,T(y,x)) = T(x,y)
Proof.
T(x,T(y,x)) * (y \ T(y,x))
=x by Theorem 992
=T(x,y) * (y \ T(y,x)) by Theorem 1031

Then T(x,T(y,x)) * (y \ T(y,x)) = T(x,y) * (y \ T(y,x)). Hence we are done by Proposition 10.
Theorem 1033 (prov9_0f530ca330): y * x = T(y,x) * T(x,y)
Proof. We have (T(y,x) * (x \ T(x,y))) * x = T(y,x) * T(x,y) by Theorem 1015. Hence we are done by Theorem 1031.
Theorem 1034 (prov9_29114f4573): T(x,z / y) = T(x,(y / z) \ e)
Proof. We have ((((y / z) \ y) / y) \ T(((y / z) \ y) / y,x)) \ x = T(x,T(((y / z) \ y) / y,x)) by Theorem 993. Then
(L2) ((y / z) * T((y / z) \ e,x)) \ x = T(x,T(((y / z) \ y) / y,x))
by Theorem 1026. We have (((y / z) \ e) \ T((y / z) \ e,x)) \ x = T(x,T((y / z) \ e,x)) by Theorem 993. Then ((y / z) * T((y / z) \ e,x)) \ x = T(x,T((y / z) \ e,x)) by Theorem 729. Then T(x,T(((y / z) \ y) / y,x)) = T(x,T((y / z) \ e,x)) by (L2). Then
(L6) T(x,((y / z) \ y) / y) = T(x,T((y / z) \ e,x))
by Theorem 1032. We have T(x,T((y / z) \ e,x)) = T(x,(y / z) \ e) by Theorem 1032. Then T(x,((y / z) \ y) / y) = T(x,(y / z) \ e) by (L6). Hence we are done by Proposition 25.
Theorem 1035 (prov9_78d1b8f3a1): K(x,y) \ e = K(y,(y * x) / y)
Proof. We have (T(y,((y * x) \ y) \ e) \ y) \ y = T(y,((y * x) \ y) \ e) by Theorem 930. Then K(y \ (y / ((y * x) \ y)),y) \ y = T(y,((y * x) \ y) \ e) by Theorem 1029. Then
(L3) K(y \ (y * x),y) \ y = T(y,((y * x) \ y) \ e)
by Proposition 24. We have T(y,((y * x) \ y) \ e) = y * K(y,(y * x) / y) by Theorem 1001. Then
(L5) K(y \ (y * x),y) \ y = y * K(y,(y * x) / y)
by (L3). We have y * (K(y \ (y * x),y) \ e) = K(y \ (y * x),y) \ y by Theorem 777. Then y * (K(y \ (y * x),y) \ e) = y * K(y,(y * x) / y) by (L5). Then K(y \ (y * x),y) \ e = K(y,(y * x) / y) by Proposition 9. Hence we are done by Axiom 3.
Theorem 1036 (prov9_ca237a530d): K(y,x) = K(T(x,y),y) \ e
Proof. We have K(y,(y * T(x,y)) / y) = K(T(x,y),y) \ e by Theorem 1035. Hence we are done by Proposition 48.
Theorem 1037 (prov9_d747e6b287): K(T(y,x),x) = K(x,y) \ e
Proof. We have K(T(y,x),x) = K(y,(y * x) / y) by Theorem 1008. Hence we are done by Theorem 1035.
Theorem 1038 (prov9_a4fc24f1b4): y * K(y,x) = K(T(x,y),y) \ y
Proof. We have y * (K(T(x,y),y) \ e) = K(T(x,y),y) \ y by Theorem 777. Hence we are done by Theorem 1036.
Theorem 1039 (prov9_d343b3fe62): R(z,K(z,x),y) = z
Proof. We have R(z,K(T(x,z),z) \ e,y) = z by Theorem 1019. Hence we are done by Theorem 1036.
Theorem 1040 (prov9_46c0c06681): x * (y * K(x,y)) = (x * K(x,y)) * y
Proof. We have R(x,K(x,y),y) * (y * K(x,y)) = (x * K(x,y)) * y by Theorem 938. Hence we are done by Theorem 1039.
Theorem 1041 (prov9_062ec6ce87): R(y \ T(y,z),z,x) = y \ T(y,z)
Proof. We have K(y,(y / R(z \ e,z,x)) / y) = R(z \ e,z,x) * ((y * (R(z \ e,z,x) \ e)) / y) by Theorem 333. Then K(y,(y / R(z \ e,z,x)) / y) = R((z \ e) * ((y * ((z \ e) \ e)) / y),z,x) by Theorem 642. Then
(L5) R(y \ T(y,(z \ e) \ e),z,x) = K(y,(y / R(z \ e,z,x)) / y)
by Theorem 980. We have K(y,(y / R(z \ e,z,x)) / y) = y \ T(y,R(z \ e,z,x) \ e) by Theorem 1004. Then
(L7) R(y \ T(y,(z \ e) \ e),z,x) = y \ T(y,R(z \ e,z,x) \ e)
by (L5). We have y \ T(y,T(x / (z * x),T(z,e)) \ e) = T(y \ T(y,(x / (z * x)) \ e),T(z,e)) by Theorem 1014. Then y \ T(y,R(z \ e,z,x) \ e) = T(y \ T(y,(x / (z * x)) \ e),T(z,e)) by Theorem 945. Then R(y \ T(y,(z \ e) \ e),z,x) = T(y \ T(y,(x / (z * x)) \ e),T(z,e)) by (L7). Then T(y \ T(y,(x / (z * x)) \ e),T(z,e)) = R(y \ T(y,z),z,x) by Theorem 721. Then T(y \ T(y,(z * x) / x),T(z,e)) = R(y \ T(y,z),z,x) by Theorem 1034. Then
(L11) T(y \ T(y,z),T(z,e)) = R(y \ T(y,z),z,x)
by Axiom 5. We have T(K(z \ (z / (y \ e)),z),T(z,e)) = K(z \ (z / (y \ e)),z) by Theorem 939. Then K(K(z \ (z / (y \ e)),z),T(z,e)) = e by Proposition 22. Then K(y \ T(y,z),T(z,e)) = e by Theorem 198. Then T(y \ T(y,z),T(z,e)) = y \ T(y,z) by Proposition 23. Hence we are done by (L11).
Theorem 1042 (prov9_1951f57305): K(x,y) * (T(y,z) * w) = (K(x,y) * T(y,z)) * w
Proof. We have R((((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) / y,y,(y \ e) * ((y \ e) \ w)) \ ((((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) * ((y \ e) * ((y \ e) \ w))) = y * ((y \ e) * ((y \ e) \ w)) by Theorem 800. Then R((L(x \ e,x,y) \ e) \ T(L(x \ e,x,y) \ e,y),y,(y \ e) * ((y \ e) \ w)) \ ((((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) * ((y \ e) * ((y \ e) \ w))) = y * ((y \ e) * ((y \ e) \ w)) by Theorem 981. Then
(L3) ((L(x \ e,x,y) \ e) \ T(L(x \ e,x,y) \ e,y)) \ ((((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) * ((y \ e) * ((y \ e) \ w))) = y * ((y \ e) * ((y \ e) \ w))
by Theorem 1041. We have L((y \ e) \ w,y \ e,((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) = ((((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) * (y \ e)) \ ((((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) * ((y \ e) * ((y \ e) \ w))) by Definition 4. Then L((y \ e) \ w,y \ e,((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) = ((L(x \ e,x,y) \ e) \ T(L(x \ e,x,y) \ e,y)) \ ((((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) * ((y \ e) * ((y \ e) \ w))) by Theorem 983. Then
(L6) y * ((y \ e) * ((y \ e) \ w)) = L((y \ e) \ w,y \ e,((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e))
by (L3). We have y * ((y \ e) * ((y \ e) \ w)) = L((y \ e) \ w,y \ e,y) by Proposition 56. Then L((y \ e) \ w,y \ e,((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) = L((y \ e) \ w,y \ e,y) by (L6). Then
(L9) L((y \ e) \ w,y \ e,y * K(x,y)) = L((y \ e) \ w,y \ e,y)
by Theorem 1013. We have L((y \ e) \ w,y \ e,y) = y * w by Theorem 62. Then
(L11) L((y \ e) \ w,y \ e,y * K(x,y)) = y * w
by (L9). We have (K(x,y) / (y \ e)) * ((y \ e) * T((y \ e) \ e,z)) = K(x,y) * T(K(x,y) \ (K(x,y) / (y \ e)),z) by Theorem 139. Then (y * K(x,y)) * ((y \ e) * T((y \ e) \ e,z)) = K(x,y) * T(K(x,y) \ (K(x,y) / (y \ e)),z) by Theorem 898. Then K(x,y) * T(K(x,y) \ (K(x,y) / (y \ e)),z) = (y * K(x,y)) * (y \ T(y,z)) by Theorem 145. Then K(x,y) * T(K(x,y) \ (y * K(x,y)),z) = (y * K(x,y)) * (y \ T(y,z)) by Theorem 898. Then K(x,y) * T(T(y,K(x,y)),z) = (y * K(x,y)) * (y \ T(y,z)) by Definition 3. Then
(L17) K(x,y) * T(y,z) = (y * K(x,y)) * (y \ T(y,z))
by Theorem 755. We have ((y * K(x,y)) * (y \ e)) * (w * T(w \ L((y \ e) \ w,y \ e,y * K(x,y)),z)) = ((y * K(x,y)) * ((y \ e) * T((y \ e) \ e,z))) * w by Theorem 884. Then K(x,y) * (w * T(w \ L((y \ e) \ w,y \ e,y * K(x,y)),z)) = ((y * K(x,y)) * ((y \ e) * T((y \ e) \ e,z))) * w by Theorem 897. Then K(x,y) * (w * T(w \ L((y \ e) \ w,y \ e,y * K(x,y)),z)) = ((y * K(x,y)) * (y \ T(y,z))) * w by Theorem 145. Then K(x,y) * (w * T(w \ L((y \ e) \ w,y \ e,y * K(x,y)),z)) = (K(x,y) * T(y,z)) * w by (L17). Then K(x,y) * (w * T(w \ (y * w),z)) = (K(x,y) * T(y,z)) * w by (L11). Hence we are done by Theorem 466.
Theorem 1043 (prov9_44f3f26910): L(w,T(z,x),K(y,z)) = w
Proof. We have (K(y,z) * T(z,x)) * w = K(y,z) * (T(z,x) * w) by Theorem 1042. Hence we are done by Theorem 54.
Theorem 1044 (prov9_024ebe7e7c): L(z,y,K(x,y)) = z
Proof. We have L(z,T(y,y),K(x,y)) = z by Theorem 1043. Hence we are done by Theorem 450.
Theorem 1045 (prov9_c33c9320ac): L(z,K(x,y),y) = z
Proof. We have ((y * (((y * x) \ y) \ e)) / ((y * (((y * x) \ y) \ e)) / y)) \ (y * ((T(y,((y * x) \ y) \ e) \ y) * z)) = y * (((y * (((y * x) \ y) \ e)) / ((y * (((y * x) \ y) \ e)) / y)) \ ((T(y,((y * x) \ y) \ e) \ y) * z)) by Theorem 857. Then
(L8) (y * (T(y,((y * x) \ y) \ e) \ y)) \ (y * ((T(y,((y * x) \ y) \ e) \ y) * z)) = y * (((y * (((y * x) \ y) \ e)) / ((y * (((y * x) \ y) \ e)) / y)) \ ((T(y,((y * x) \ y) \ e) \ y) * z))
by Theorem 1011.
L(z,T(y,((y * x) \ y) \ e) \ y,y)
=(y * (T(y,((y * x) \ y) \ e) \ y)) \ (y * ((T(y,((y * x) \ y) \ e) \ y) * z)) by Definition 4
=y * ((y * (T(y,((y * x) \ y) \ e) \ y)) \ ((T(y,((y * x) \ y) \ e) \ y) * z)) by (L8), Theorem 1011

Then
(L11) L(z,T(y,((y * x) \ y) \ e) \ y,y) = y * ((y * (T(y,((y * x) \ y) \ e) \ y)) \ ((T(y,((y * x) \ y) \ e) \ y) * z))
. We have L(y \ z,y,T(y,((y * x) \ y) \ e) \ y) = ((T(y,((y * x) \ y) \ e) \ y) * y) \ ((T(y,((y * x) \ y) \ e) \ y) * z) by Proposition 53. Then L(y \ z,y,T(y,((y * x) \ y) \ e) \ y) = (y * (T(y,((y * x) \ y) \ e) \ y)) \ ((T(y,((y * x) \ y) \ e) \ y) * z) by Theorem 931. Then
(L12) y * L(y \ z,y,T(y,((y * x) \ y) \ e) \ y) = L(z,T(y,((y * x) \ y) \ e) \ y,y)
by (L11). We have L(y \ z,y,K(y \ (y / (((((y * x) \ y) \ e) \ y) / T(y,((y * x) \ y) \ e))),y)) = y \ z by Theorem 1044. Then L(y \ z,y,T(y,((y * x) \ y) \ e) \ y) = y \ z by Theorem 932. Then
(L13) y * (y \ z) = L(z,T(y,((y * x) \ y) \ e) \ y,y)
by (L12). We have y * (y \ z) = z by Axiom 4. Then
(L15) L(z,T(y,((y * x) \ y) \ e) \ y,y) = z
by (L13). We have K(x,y) = ((y * x) \ y) * T(((y * x) \ y) \ e,y) by Theorem 539. Then K(x,y) = T(y,((y * x) \ y) \ e) \ y by Theorem 1028. Hence we are done by (L15).
Theorem 1046 (prov9_5b2a2deaed): (y * K(x,y)) * z = y * (K(x,y) * z)
Proof. We have (y * K(x,y)) * L(z,K(x,y),y) = y * (K(x,y) * z) by Proposition 52. Hence we are done by Theorem 1045.
Theorem 1047 (prov9_62c64cc36e): R(z,K(x,z),y) = z
Proof. We have (z * K(x,z)) * y = z * (K(x,z) * y) by Theorem 1046. Hence we are done by Theorem 64.
Theorem 1048 (prov9_faf1648adc): T(x,y) = T(x,y * K(x,y))
Proof. We have (K(x,y) * T(y,y)) * T(x,y * K(x,y)) = K(x,y) * (T(y,y) * T(x,y * K(x,y))) by Theorem 1042. Then R(K(x,y),T(y,y),T(x,y * K(x,y))) = K(x,y) by Theorem 64. Then
(L3) R(K(x,y),y,T(x,y * K(x,y))) = K(x,y)
by Theorem 450. We have R((y * K(x,y)) / y,y,T(x,y * K(x,y))) \ (x * (y * K(x,y))) = y * T(x,y * K(x,y)) by Theorem 801. Then R(K(x,y),y,T(x,y * K(x,y))) \ (x * (y * K(x,y))) = y * T(x,y * K(x,y)) by Theorem 934. Then
(L6) K(x,y) \ (x * (y * K(x,y))) = y * T(x,y * K(x,y))
by (L3). We have R(K(x,y),x,y) * (x * y) = (K(x,y) * x) * y by Proposition 54. Then K(x,y) * (x * y) = (K(x,y) * x) * y by Theorem 1025. Then (x * K(x,y)) * y = K(x,y) * (x * y) by Theorem 928. Then x * (y * K(x,y)) = K(x,y) * (x * y) by Theorem 1040. Then K(x,y) \ (x * (y * K(x,y))) = x * y by Proposition 2. Then y * T(x,y * K(x,y)) = x * y by (L6). Hence we are done by Theorem 11.
Theorem 1049 (prov9_eb0dabdc53): K(x,y) * T(x,y) = T(x * K(x,y),y)
Proof. We have y \ (L(x,K(x,y),y) * (y * K(x,y))) = K(x,y) * T(x,y * K(x,y)) by Theorem 805. Then
(L2) y \ (x * (y * K(x,y))) = K(x,y) * T(x,y * K(x,y))
by Theorem 1045. We have
(L4) T(x * K(x,y),y) = y \ ((x * K(x,y)) * y)
by Definition 3.
K(x,y) * T(x,y)
=y \ (x * (y * K(x,y))) by (L2), Theorem 1048
=T(x * K(x,y),y) by (L4), Theorem 1040

Hence we are done.
Theorem 1050 (prov9_a5e1a07ed6): K(x,y) \ y = (y * z) / (K(x,y) * z)
Proof. We have R(T(y,((y * x) \ y) \ K(x,y)),K(x,y),z) = T(R(y,K(x,y),z),((y * x) \ y) \ K(x,y)) by Axiom 9. Then
(L2) R(T(y,((y * x) \ y) \ K(x,y)),K(x,y),z) = T(y,((y * x) \ y) \ K(x,y))
by Theorem 1047. We have T(y,((y * x) \ y) \ K(x,y)) = K(x,y) \ y by Theorem 1012. Then R(T(y,((y * x) \ y) \ K(x,y)),K(x,y),z) = K(x,y) \ y by (L2). Then
(L5) R(K(x,y) \ y,K(x,y),z) = K(x,y) \ y
by Theorem 1012. We have R(K(x,y) \ y,K(x,y),z) = (((K(x,y) \ y) * K(x,y)) * z) / (K(x,y) * z) by Definition 5. Then R(K(x,y) \ y,K(x,y),z) = (y * z) / (K(x,y) * z) by Theorem 765. Hence we are done by (L5).
Theorem 1051 (prov9_0aec48ea67): (K(x,y) \ e) * (K(x,y) * z) = z
Proof. We have R(y,K(x,y) \ e,K(x,y) * z) \ ((y * (K(x,y) \ e)) * (K(x,y) * z)) = (K(x,y) \ e) * (K(x,y) * z) by Theorem 6. Then R(y,K(x,y) \ e,K(x,y) * z) \ ((K(x,y) \ y) * (K(x,y) * z)) = (K(x,y) \ e) * (K(x,y) * z) by Theorem 777. Then
(L5) y \ ((K(x,y) \ y) * (K(x,y) * z)) = (K(x,y) \ e) * (K(x,y) * z)
by Theorem 1019. We have y * (y \ ((K(x,y) \ y) * (K(x,y) * z))) = (K(x,y) \ y) * (K(x,y) * z) by Axiom 4. Then
(L7) y * ((K(x,y) \ e) * (K(x,y) * z)) = (K(x,y) \ y) * (K(x,y) * z)
by (L5). We have ((y * z) / (K(x,y) * z)) * (K(x,y) * z) = y * z by Axiom 6. Then (K(x,y) \ y) * (K(x,y) * z) = y * z by Theorem 1050. Then y * ((K(x,y) \ e) * (K(x,y) * z)) = y * z by (L7). Hence we are done by Proposition 9.
Theorem 1052 (prov9_1847dc2c44): K(y,x) \ z = K(T(x,y),y) * z
Proof. We have L(((K(y,x) \ z) * (e / K(T(x,y),y))) * K(T(x,y),y),K(T(x,y),y) \ e,K(T(x,y),y)) / K(T(x,y),y) = (K(T(x,y),y) \ e) * (K(T(x,y),y) * ((K(y,x) \ z) * (e / K(T(x,y),y)))) by Theorem 894. Then
(L3) L(R(K(y,x) \ z,e / K(T(x,y),y),K(T(x,y),y)),K(T(x,y),y) \ e,K(T(x,y),y)) / K(T(x,y),y) = (K(T(x,y),y) \ e) * (K(T(x,y),y) * ((K(y,x) \ z) * (e / K(T(x,y),y))))
by Proposition 69. We have L(R(K(y,x) \ z,e / K(T(x,y),y),K(T(x,y),y)),K(T(x,y),y) \ e,K(T(x,y),y)) / K(T(x,y),y) = L(K(y,x) \ z,K(T(x,y),y) \ e,K(T(x,y),y)) * (e / K(T(x,y),y)) by Theorem 84. Then
(L5) (K(T(x,y),y) \ e) * (K(T(x,y),y) * ((K(y,x) \ z) * (e / K(T(x,y),y)))) = L(K(y,x) \ z,K(T(x,y),y) \ e,K(T(x,y),y)) * (e / K(T(x,y),y))
by (L3). We have (K(T(x,y),y) \ e) * (K(T(x,y),y) * ((K(y,x) \ z) * (e / K(T(x,y),y)))) = (K(y,x) \ z) * (e / K(T(x,y),y)) by Theorem 1051. Then L(K(y,x) \ z,K(T(x,y),y) \ e,K(T(x,y),y)) * (e / K(T(x,y),y)) = (K(y,x) \ z) * (e / K(T(x,y),y)) by (L5). Then L(K(y,x) \ z,K(T(x,y),y) \ e,K(T(x,y),y)) = K(y,x) \ z by Proposition 10. Then
(L8) L(K(y,x) \ z,K(y,x),K(T(x,y),y)) = K(y,x) \ z
by Theorem 1036. We have (e / K(y,x)) \ L(K(y,x) \ z,K(y,x),K(y,x) \ e) = K(y,x) * (K(y,x) \ z) by Theorem 913. Then (K(y,x) \ e) \ L(K(y,x) \ z,K(y,x),K(y,x) \ e) = K(y,x) * (K(y,x) \ z) by Theorem 901. Then
(L11) (K(y,x) \ e) \ L(K(y,x) \ z,K(y,x),K(T(x,y),y)) = K(y,x) * (K(y,x) \ z)
by Theorem 1037. We have K(x,(x * y) / x) * (K(x,(x * y) / x) \ L(K(y,x) \ z,K(y,x),K(T(x,y),y))) = L(K(y,x) \ z,K(y,x),K(T(x,y),y)) by Axiom 4. Then K(T(x,y),y) * (K(x,(x * y) / x) \ L(K(y,x) \ z,K(y,x),K(T(x,y),y))) = L(K(y,x) \ z,K(y,x),K(T(x,y),y)) by Theorem 1016. Then K(T(x,y),y) * ((K(y,x) \ e) \ L(K(y,x) \ z,K(y,x),K(T(x,y),y))) = L(K(y,x) \ z,K(y,x),K(T(x,y),y)) by Theorem 1035. Then K(T(x,y),y) * (K(y,x) * (K(y,x) \ z)) = L(K(y,x) \ z,K(y,x),K(T(x,y),y)) by (L11). Then K(T(x,y),y) * z = L(K(y,x) \ z,K(y,x),K(T(x,y),y)) by Axiom 4. Hence we are done by (L8).
Theorem 1053 (prov9_3146745cfe): K(x,y) = K(T(x,y),y)
Proof. We have T(x,y) * (K(T(x,y),y) * T(K(T(x,y),y) \ y,x)) = (T(y,x) * T(x,y)) * K(T(x,y),y) by Theorem 634. Then T(x,y) * (K(T(x,y),y) * T(K(T(x,y),y) \ y,x)) = (y * x) * K(T(x,y),y) by Theorem 1033. Then T(x,y) * (K(T(x,y),y) * T(y * K(y,x),x)) = (y * x) * K(T(x,y),y) by Theorem 1038. Then
(L6) T(x,y) * (K(y,x) \ T(y * K(y,x),x)) = (y * x) * K(T(x,y),y)
by Theorem 1052. We have K(y,x) * T(y,x) = T(y * K(y,x),x) by Theorem 1049. Then K(y,x) \ T(y * K(y,x),x) = T(y,x) by Proposition 2. Then
(L7) T(x,y) * T(y,x) = (y * x) * K(T(x,y),y)
by (L6). We have T(x,y) * T(y,x) = x * y by Theorem 1033. Then (y * x) * K(T(x,y),y) = x * y by (L7). Hence we are done by Theorem 460.
Theorem 1054 (KxyKyx): K(y,x) * K(x,y) = e
Proof. We have K(T(y,x),x) * (K(T(y,x),x) \ e) = e by Axiom 4. Then K(T(y,x),x) * K(x,y) = e by Theorem 1036. Hence we are done by Theorem 1053.
Theorem 1055 (prov9_035aecae6f): K(x,x \ y) = ((x \ y) * x) \ y
Proof. We have K(x,x \ y) = ((x \ y) * x) \ (x * (x \ y)) by Definition 2. Hence we are done by Axiom 4.
Theorem 1056 (prov9_b807fdcb92): ((x * y) * z) / a(x,y,z) = x * (y * z)
Proof. We have ((x * y) * z) / ((x * (y * z)) \ ((x * y) * z)) = x * (y * z) by Proposition 24. Hence we are done by Definition 1.
Theorem 1057 (prov9_f838cee35a): T(R(x,y,z),y * z) = (y * z) \ ((x * y) * z)
Proof. We have T(((x * y) * z) / (y * z),y * z) = (y * z) \ ((x * y) * z) by Proposition 47. Hence we are done by Definition 5.
Theorem 1058 (prov9_9744abd3aa): L(K(y,z),z * y,x) = (x * (z * y)) \ (x * (y * z))
Proof. We have L(K(y,z),z * y,x) = (x * (z * y)) \ (x * ((z * y) * K(y,z))) by Definition 4. Hence we are done by Proposition 82.
Theorem 1059 (prov9_75268842cb): T(y * x,K(x,y)) = K(x,y) \ (x * y)
Proof. We have T(y * x,K(x,y)) = K(x,y) \ ((y * x) * K(x,y)) by Definition 3. Hence we are done by Proposition 82.
Theorem 1060 (prov9_52d0413c02): L(K(z \ y,z),y,x) = (x * y) \ (x * ((z \ y) * z))
Proof. We have L(K(z \ y,z),y,x) = (x * y) \ (x * (y * K(z \ y,z))) by Definition 4. Hence we are done by Theorem 17.
Theorem 1061 (prov9_29b4497041): L(L(w,z,y),y * z,x) = (x * (y * z)) \ (x * (y * (z * w)))
Proof. We have L(L(w,z,y),y * z,x) = (x * (y * z)) \ (x * ((y * z) * L(w,z,y))) by Definition 4. Hence we are done by Proposition 52.
Theorem 1062 (prov9_da5eb7ad50): y \ e = (x * y) \ R(x,y,y \ e)
Proof. We have R(x,y,y \ e) \ (R(x,y,y \ e) * e) = e by Axiom 3. Then R(x,y,y \ e) \ ((x * y) * ((x * y) \ (R(x,y,y \ e) * e))) = e by Axiom 4. Then
(L4) R(x,y,y \ e) \ ((x * y) * ((x * y) \ R(x,y,y \ e))) = e
by Axiom 2. We have R(x,y,y \ e) * (R(x,y,y \ e) \ ((x * y) * ((x * y) \ R(x,y,y \ e)))) = (x * y) * ((x * y) \ R(x,y,y \ e)) by Axiom 4. Then R(x,y,y \ e) * e = (x * y) * ((x * y) \ R(x,y,y \ e)) by (L4). Then
(L7) (x * y) * ((x * y) \ R(x,y,y \ e)) = R(x,y,y \ e) * (y * (y \ e))
by Axiom 4. We have (x * y) * (y \ e) = R(x,y,y \ e) * (y * (y \ e)) by Proposition 54. Hence we are done by (L7) and Proposition 7.
Theorem 1063 (prov9_ff95dc5b6e): R(x,z,T(y,z)) * (y * z) = (x * z) * T(y,z)
Proof. We have R(x,z,T(y,z)) * (z * T(y,z)) = (x * z) * T(y,z) by Proposition 54. Hence we are done by Proposition 46.
Theorem 1064 (prov9_74d263068f): ((x * y) * z) * L(w,y * z,R(x,y,z)) = R(x,y,z) * ((y * z) * w)
Proof. We have (R(x,y,z) * (y * z)) * L(w,y * z,R(x,y,z)) = R(x,y,z) * ((y * z) * w) by Proposition 52. Hence we are done by Proposition 54.
Theorem 1065 (prov9_9ea13440b0): R(y / x,x,y \ z) \ z = x * (y \ z)
Proof. We have y * (y \ z) = z by Axiom 4. Then ((y * (y \ z)) / (x * (y \ z))) * (x * (y \ z)) = z by Axiom 6. Then ((y * (y \ z)) / (x * (y \ z))) \ z = x * (y \ z) by Proposition 2. Hence we are done by Proposition 55.
Theorem 1066 (prov9_e989e7152a): (y * (x * z)) / L(T(x,z),z,y) = y * z
Proof. We have (y * (z * T(x,z))) / L(T(x,z),z,y) = y * z by Theorem 453. Hence we are done by Proposition 46.
Theorem 1067 (prov9_f496c1db78): R(x,x \ y,z) \ (y * z) = (x \ y) * z
Proof. We have R(x,x \ y,z) \ ((x * (x \ y)) * z) = (x \ y) * z by Theorem 6. Hence we are done by Axiom 4.
Theorem 1068 (prov9_e2b7485f97): R(y * x,K(x,y),z) \ ((x * y) * z) = K(x,y) * z
Proof. We have R(y * x,K(x,y),z) \ (((y * x) * K(x,y)) * z) = K(x,y) * z by Theorem 6. Hence we are done by Proposition 82.
Theorem 1069 (prov9_495d46b148): R(x,x \ e,x) = x / K(x \ e,x)
Proof. We have R(x,x \ e,x) = x / ((x \ e) * x) by Proposition 66. Hence we are done by Proposition 76.
Theorem 1070 (prov9_18cd75688a): L((u * w) \ x,y,z) = L(L(w \ (u \ x),y,z),w,u)
Proof. We have L(L(w \ (u \ x),w,u),y,z) = L(L(w \ (u \ x),y,z),w,u) by Axiom 11. Hence we are done by Proposition 70.
Theorem 1071 (prov9_d9c4a91972): R(T(w,w * z),x,y) = L(R(T(w,z),x,y),z,w)
Proof. We have R(L(T(w,z),z,w),x,y) = L(R(T(w,z),x,y),z,w) by Axiom 10. Hence we are done by Proposition 63.
Theorem 1072 (prov9_07212a2af2): L(T(x \ y,z),x,e / x) = T((e / x) * y,z)
Proof. We have L(T(x \ y,z),x,e / x) = T(L(x \ y,x,e / x),z) by Axiom 8. Hence we are done by Theorem 63.
Theorem 1073 (prov9_1d8b73dc9a): y * ((y \ z) * T(x,z)) = L(x,y \ z,y) * z
Proof. We have z * L(T(x,z),y \ z,y) = L(x,y \ z,y) * z by Proposition 58. Hence we are done by Theorem 52.
Theorem 1074 (prov9_7c8af251e2): y * ((e / y) * (x * y)) = L(x,y,e / y) * y
Proof. We have y * L(T(x,y),y,e / y) = L(x,y,e / y) * y by Proposition 58. Hence we are done by Theorem 72.
Theorem 1075 (prov9_7592f11855): (x \ e) * L(y,K(x \ e,x),e / x) = (e / x) * (K(x \ e,x) * y)
Proof. We have ((e / x) * K(x \ e,x)) * L(y,K(x \ e,x),e / x) = (e / x) * (K(x \ e,x) * y) by Proposition 52. Hence we are done by Theorem 109.
Theorem 1076 (prov9_5e16f340ea): ((z * y) * R(x,z,y)) / y = (((z * y) * x) / (z * y)) * z
Proof. We have (((((z * y) * x) / (z * y)) * z) * y) / y = (((z * y) * x) / (z * y)) * z by Axiom 5. Then (((((((z * y) * x) / (z * y)) * z) * y) / (z * y)) * (z * y)) / y = (((z * y) * x) / (z * y)) * z by Axiom 6. Then (R(((z * y) * x) / (z * y),z,y) * (z * y)) / y = (((z * y) * x) / (z * y)) * z by Definition 5. Hence we are done by Theorem 87.
Theorem 1077 (prov9_cdf310a89f): T(R(x,y,T(x,y)),x * y) = T(x,y)
Proof. We have T(((x * y) * T(x,y)) / (x * y),x * y) = T(x,y) by Theorem 7. Hence we are done by Theorem 787.
Theorem 1078 (prov9_8652cdd8d9): R(T(x,x * y),y,T(x,y)) = T(x,y)
Proof. We have R(T(x,x * y),y,T(x,y)) = T(R(x,y,T(x,y)),x * y) by Axiom 9. Hence we are done by Theorem 1077.
Theorem 1079 (prov9_6ea2224951): T(x,y) * (x * y) = (T(x,x * y) * y) * T(x,y)
Proof. We have R(T(x,x * y),y,T(x,y)) * (x * y) = (T(x,x * y) * y) * T(x,y) by Theorem 1063. Hence we are done by Theorem 1078.
Theorem 1080 (prov9_b129633854): (e / x) \ T((e / x) * y,z) = x * T(x \ y,z)
Proof. We have (e / x) \ L(T(x \ y,z),x,e / x) = x * T(x \ y,z) by Theorem 71. Hence we are done by Theorem 1072.
Theorem 1081 (prov9_80144387c5): x \ (L(y,x \ z,x) * z) = (x \ z) * T(y,z)
Proof. We have x * ((x \ z) * T(y,z)) = L(y,x \ z,x) * z by Theorem 1073. Hence we are done by Proposition 2.
Theorem 1082 (prov9_158a5a8fd1): x \ ((y * (x * y)) / y) = T(y,x * y)
Proof. We have ((x * T(y,x * y)) * y) / y = x * T(y,x * y) by Axiom 5. Then ((((x * T(y,x * y)) * y) / (x * y)) * (x * y)) / y = x * T(y,x * y) by Axiom 6. Then (y * (x * y)) / y = x * T(y,x * y) by Theorem 492. Hence we are done by Proposition 2.
Theorem 1083 (prov9_d0c7417702): T(x / y,T(y,x)) = T(y,x) \ ((y * x) / y)
Proof. We have T(x / y,T(y,x)) = T(y,x) \ ((x / y) * T(y,x)) by Definition 3. Hence we are done by Theorem 818.
Theorem 1084 (prov9_9c5d2c7e23): (T(x,y) * z) \ ((x * y) * z) = R(T(y,T(x,y) * z),T(x,y),z)
Proof. We have (T(x,y) * z) \ ((y * T(x,y)) * z) = R(T(y,T(x,y) * z),T(x,y),z) by Proposition 81. Hence we are done by Proposition 46.
Theorem 1085 (prov9_2f08434849): x \ ((z \ (x * y)) * z) = (x \ z) * T((x \ z) \ y,z)
Proof. We have x \ (L((x \ z) \ y,x \ z,x) * z) = (x \ z) * T((x \ z) \ y,z) by Theorem 1081. Hence we are done by Theorem 60.
Theorem 1086 (prov9_e3a7705f6a): (x * (y \ x)) / x = y \ T(x,x / y)
Proof. We have (x * T(y \ T(x,x / y),x)) / x = y \ T(x,x / y) by Proposition 48. Hence we are done by Theorem 513.
Theorem 1087 (prov9_c4c4ae9f9c): (y / x) \ T(y,y / (y / x)) = (y * x) / y
Proof. We have (y / x) \ T(y,y / (y / x)) = (y * ((y / x) \ y)) / y by Theorem 1086. Hence we are done by Proposition 25.
Theorem 1088 (prov9_1471236799): (y * (T(y,z) * x)) / (y * x) = T(y,z)
Proof. We have T(y,z) * (y * x) = y * (T(y,z) * x) by Theorem 172. Hence we are done by Proposition 1.
Theorem 1089 (prov9_080a8fa319): (y * x) / (y * (T(y,z) \ x)) = T(y,z)
Proof. We have (y * (T(y,z) * (T(y,z) \ x))) / (y * (T(y,z) \ x)) = T(y,z) by Theorem 1088. Hence we are done by Axiom 4.
Theorem 1090 (prov9_d7b5a725d1): T(x,x \ y) \ (x * z) = x * (((x \ y) \ y) \ z)
Proof. We have T(x,x \ y) \ (x * z) = x * (T(x,x \ y) \ z) by Theorem 174. Hence we are done by Proposition 49.
Theorem 1091 (prov9_ec8a5ec723): T(y,x) / (x \ y) = (y * x) / y
Proof. We have ((y * x) / y) * (x \ y) = T(y,x) by Theorem 179. Hence we are done by Proposition 1.
Theorem 1092 (prov9_45fa76b7ec): T(y,x) * K(x \ y,(y * x) / y) = (x \ y) * ((y * x) / y)
Proof. We have (((y * x) / y) * (x \ y)) * K(x \ y,(y * x) / y) = (x \ y) * ((y * x) / y) by Proposition 82. Hence we are done by Theorem 179.
Theorem 1093 (prov9_430ffd450f): (y / (y / x)) * (x \ (y / x)) = x \ y
Proof. We have T(y / x,x) = (((y / x) * x) / (y / x)) * (x \ (y / x)) by Theorem 179. Then
(L2) T(y / x,x) = (y / (y / x)) * (x \ (y / x))
by Axiom 6. We have T(y / x,x) = x \ y by Proposition 47. Hence we are done by (L2).
Theorem 1094 (prov9_fcbe49b83a): (z / x) * (x * T(x \ z,y)) = T(z / x,y) * z
Proof. We have T(z / x,y) * ((z / x) * x) = (z / x) * (T(z / x,y) * x) by Theorem 172. Then T(z / x,y) * z = (z / x) * (T(z / x,y) * x) by Axiom 6. Hence we are done by Proposition 61.
Theorem 1095 (prov9_5a20b5b6ab): ((x \ y) \ y) \ (x * z) = x * (((x \ y) \ y) \ z)
Proof. We have T(x,x \ y) \ (x * z) = x * (((x \ y) \ y) \ z) by Theorem 1090. Hence we are done by Proposition 49.
Theorem 1096 (prov9_1476924788): (z \ (x * (y \ z))) * z = (y * T(y \ x,z)) * (y \ z)
Proof. We have R(T(x / y,z),y,y \ z) * z = (T(x / y,z) * y) * (y \ z) by Theorem 25. Then (z \ (x * (y \ z))) * z = (T(x / y,z) * y) * (y \ z) by Theorem 142. Hence we are done by Proposition 61.
Theorem 1097 (prov9_b1f9a2404e): x \ L(x,x \ e,R(x,y,z)) = (x \ e) * ((x \ R(x,y,z)) \ R(x,y,z))
Proof. We have (x \ e) * L((x \ e) \ e,x \ e,R(x,y,z)) = x \ L(x,x \ e,R(x,y,z)) by Theorem 836. Then (x \ e) * ((R(x,y,z) * (x \ e)) \ R(x,y,z)) = x \ L(x,x \ e,R(x,y,z)) by Proposition 78. Hence we are done by Proposition 89.
Theorem 1098 (prov9_0a46952b79): (y * (y * K(x,y))) / y = y * K(x,y)
Proof. We have (y * (y * K(x,y))) / T(y,y * K(x,y)) = y * K(x,y) by Theorem 5. Hence we are done by Theorem 541.
Theorem 1099 (prov9_3325121f5d): (y \ (x \ y)) * y = (x * T(x \ e,y)) * (x \ y)
Proof. We have (y \ (e * (x \ y))) * y = (x * T(x \ e,y)) * (x \ y) by Theorem 1096. Hence we are done by Axiom 1.
Theorem 1100 (prov9_dc88275694): R(T(x,y),x \ e,x) = T(T(x,x \ e),y)
Proof. We have R(T(T(x,(x \ e) * x),y),x \ e,x) = T(R(T(x,(x \ e) * x),x \ e,x),y) by Axiom 9. Then R(T(T(x,(x \ e) * x),y),x \ e,x) = T(T(x,x \ e),y) by Theorem 113. Hence we are done by Theorem 204.
Theorem 1101 (prov9_7444758e7c): L(T(z,z \ e),x,y) / z = L(z,x,y) * (e / z)
Proof. We have L(T(z,e / z),x,y) / z = L(z,x,y) * (e / z) by Theorem 164. Hence we are done by Theorem 211.
Theorem 1102 (prov9_1b53e66c3d): (x * y) * ((w * ((x * y) \ (x * z))) / w) = x * (y * ((w * (y \ z)) / w))
Proof. We have (x * y) * ((w * L(y \ z,y,x)) / w) = x * (y * ((w * (y \ z)) / w)) by Theorem 815. Hence we are done by Proposition 53.
Theorem 1103 (prov9_2c4336d238): x \ ((z * (x * y)) / z) = (x \ e) * ((z * ((x \ e) \ y)) / z)
Proof. We have L((z * ((x \ e) \ y)) / z,x \ e,x) = (z * L((x \ e) \ y,x \ e,x)) / z by Theorem 122. Then
(L2) L((z * ((x \ e) \ y)) / z,x \ e,x) = (z * (x * y)) / z
by Theorem 62. We have x \ L((z * ((x \ e) \ y)) / z,x \ e,x) = (x \ e) * ((z * ((x \ e) \ y)) / z) by Proposition 57. Hence we are done by (L2).
Theorem 1104 (prov9_49d57790c7): (e / x) \ ((z * ((e / x) * y)) / z) = x * ((z * (x \ y)) / z)
Proof. We have L((z * (x \ y)) / z,x,e / x) = (z * L(x \ y,x,e / x)) / z by Theorem 122. Then
(L2) L((z * (x \ y)) / z,x,e / x) = (z * ((e / x) * y)) / z
by Theorem 63. We have (e / x) \ L((z * (x \ y)) / z,x,e / x) = x * ((z * (x \ y)) / z) by Theorem 71. Hence we are done by (L2).
Theorem 1105 (prov9_5d7763782b): (x * T(z,y)) \ ((x * z) * T(z,y)) = z
Proof. We have (x * T(z,y)) * z = (x * z) * T(z,y) by Theorem 238. Hence we are done by Proposition 2.
Theorem 1106 (prov9_b0261ed9d4): (x * (y \ z)) / (z / y) = (x / (z / y)) * (y \ z)
Proof. We have (x * (y \ z)) / (y \ z) = x by Axiom 5. Then
(L3) (((x * (y \ z)) / (z / y)) * (z / y)) / (y \ z) = x
by Axiom 6. We have ((((x * (y \ z)) / (z / y)) * (z / y)) / (y \ z)) * (y \ z) = ((x * (y \ z)) / (z / y)) * (z / y) by Axiom 6. Then ((((((x * (y \ z)) / (z / y)) * (z / y)) / (y \ z)) / (z / y)) * (z / y)) * (y \ z) = ((x * (y \ z)) / (z / y)) * (z / y) by Axiom 6. Then
(L6) ((x / (z / y)) * (z / y)) * (y \ z) = ((x * (y \ z)) / (z / y)) * (z / y)
by (L3). We have ((x / (z / y)) * (z / y)) * (y \ z) = ((x / (z / y)) * (y \ z)) * (z / y) by Theorem 241. Then ((x * (y \ z)) / (z / y)) * (z / y) = ((x / (z / y)) * (y \ z)) * (z / y) by (L6). Hence we are done by Proposition 10.
Theorem 1107 (prov9_647287c32a): x \ T(e / (e / x),y) = (x \ e) * T(x,y)
Proof. We have x \ L(T(x,y),x \ e,x) = (x \ e) * T(x,y) by Proposition 57. Then x \ T(L(x,x \ e,x),y) = (x \ e) * T(x,y) by Axiom 8. Then x \ T(x * K(x \ e,x),y) = (x \ e) * T(x,y) by Theorem 70. Hence we are done by Theorem 262.
Theorem 1108 (prov9_989acd928a): K(x \ e,x) * x = e / (e / x)
Proof. We have K(x \ e,x) * x = x * K(x \ e,x) by Theorem 933. Hence we are done by Theorem 262.
Theorem 1109 (prov9_46ac0726b7): (e / (e / x)) / x = K(x \ e,x)
Proof. We have K(x \ e,x) * x = e / (e / x) by Theorem 1108. Hence we are done by Proposition 1.
Theorem 1110 (prov9_7dca9947f5): (x \ e) \ T(e / x,y) = ((x \ e) \ e) * T(x \ e,y)
Proof. We have (x \ e) \ T(e / (e / (x \ e)),y) = ((x \ e) \ e) * T(x \ e,y) by Theorem 1107. Hence we are done by Proposition 24.
Theorem 1111 (prov9_c8bde11eae): (T(y / (x / z),w) * (x / z)) * z = x * T(x \ (y * z),w)
Proof. We have T((y * z) / x,w) * x = x * T(x \ (y * z),w) by Proposition 61. Hence we are done by Theorem 169.
Theorem 1112 (prov9_a7a01c3199): ((x * (x * y)) / x) / y = T(x,x / (e / y))
Proof. We have ((x * (x / (e / y))) / x) * (e / y) = T(x,x / (e / y)) by Theorem 512. Hence we are done by Theorem 232.
Theorem 1113 (prov9_8ff8a6851f): y * (x * T(x \ T(x,y),z)) = (y * x) * T(K(x,y),z)
Proof. We have (y * x) * T((y * x) \ (x * y),z) = y * (x * T(x \ (y \ (x * y)),z)) by Theorem 136. Then (y * x) * T(K(x,y),z) = y * (x * T(x \ (y \ (x * y)),z)) by Definition 2. Hence we are done by Definition 3.
Theorem 1114 (prov9_09c9f16a39): ((x * y) / x) / (x \ y) = T(x,x / (x / y))
Proof. We have T(x,x / (x / y)) * (x \ y) = (x * y) / x by Theorem 861. Hence we are done by Proposition 1.
Theorem 1115 (prov9_039e2d4d58): z * K(y \ (y / T(x,z)),y) = (x * T(x \ e,y)) * z
Proof. We have z * (z \ ((x * T(x \ e,y)) * z)) = (x * T(x \ e,y)) * z by Axiom 4. Hence we are done by Theorem 295.
Theorem 1116 (prov9_b1e98a0c79): (x * T(x \ (z * y),w)) / y = (x / y) * T((x / y) \ z,w)
Proof. We have (T(z / (x / y),w) * (x / y)) * y = x * T(x \ (z * y),w) by Theorem 1111. Then
(L2) (x * T(x \ (z * y),w)) / y = T(z / (x / y),w) * (x / y)
by Proposition 1. We have T(z / (x / y),w) * (x / y) = (x / y) * T((x / y) \ z,w) by Proposition 61. Hence we are done by (L2).
Theorem 1117 (prov9_5da5947276): (z * (x * T(x \ y,w))) / y = ((z * x) / y) * T(((z * x) / y) \ z,w)
Proof. We have ((z * x) * T((z * x) \ (z * y),w)) / y = ((z * x) / y) * T(((z * x) / y) \ z,w) by Theorem 1116. Hence we are done by Theorem 486.
Theorem 1118 (prov9_364e048997): L(K(x,x \ e) \ y,K(x,x \ e),x) = x * (((x \ e) \ e) \ y)
Proof. We have L(K(x,x \ e) \ y,K(x,x \ e),x) = (x * K(x,x \ e)) \ (x * y) by Proposition 53. Then
(L2) L(K(x,x \ e) \ y,K(x,x \ e),x) = T(x,x \ e) \ (x * y)
by Theorem 562. We have T(x,x \ e) \ (x * y) = x * (((x \ e) \ e) \ y) by Theorem 1090. Hence we are done by (L2).
Theorem 1119 (prov9_e01cd88e04): T(y,R(y,z,w) * x) = L(T(y,x),x,R(y,z,w))
Proof. We have (R(y,z,w) * x) \ (R(y,z,w) * (y * x)) = L(T(y,x),x,R(y,z,w)) by Proposition 62. Then
(L2) (R(y,z,w) * x) \ (y * (R(y,z,w) * x)) = L(T(y,x),x,R(y,z,w))
by Theorem 227. We have T(y,R(y,z,w) * x) = (R(y,z,w) * x) \ (y * (R(y,z,w) * x)) by Definition 3. Hence we are done by (L2).
Theorem 1120 (prov9_a9748f4f19): (x * L(x \ y,w,u)) * z = x * (z * L(z \ (x \ (y * z)),w,u))
Proof. We have (x * z) * L((x * z) \ (y * z),w,u) = x * (z * L(z \ (x \ (y * z)),w,u)) by Theorem 155. Hence we are done by Theorem 332.
Theorem 1121 (prov9_d4177e635b): L(x,x \ e,R(x,y,z)) = T(e / (e / x),x \ R(x,y,z))
Proof. We have L(T(x,x \ e),x \ e,R(x,y,z)) / x = L(x,x \ e,R(x,y,z)) * (e / x) by Theorem 1101. Then T(x,R(x,y,z) * (x \ e)) / x = L(x,x \ e,R(x,y,z)) * (e / x) by Theorem 1119. Then
(L3) L(x,x \ e,R(x,y,z)) * (e / x) = T(x,x \ R(x,y,z)) / x
by Proposition 89. We have T(e / (e / x),x \ R(x,y,z)) * (e / x) = T(x,x \ R(x,y,z)) / x by Theorem 147. Hence we are done by (L3) and Proposition 8.
Theorem 1122 (prov9_7186be15d4): x \ L(x,x \ e,R(x,y,z)) = (x \ e) * T(x,x \ R(x,y,z))
Proof. We have x \ T(e / (e / x),x \ R(x,y,z)) = (x \ e) * T(x,x \ R(x,y,z)) by Theorem 1107. Hence we are done by Theorem 1121.
Theorem 1123 (prov9_6f29d2c27b): T(x,y \ x) = T(x,x * (y \ e))
Proof. We have T(T(((x / y) * y) * (y \ e),(x / y) * y) / (y \ e),((x / y) * y) * (y \ e)) = (x / y) * y by Theorem 509. Then T(T(R(x / y,y,y \ e),(x / y) * y) / (y \ e),((x / y) * y) * (y \ e)) = (x / y) * y by Proposition 68. Then T(R(T(x / y,(x / y) * y),y,y \ e) / (y \ e),((x / y) * y) * (y \ e)) = (x / y) * y by Axiom 9. Then T(T(x / y,(x / y) * y) * y,((x / y) * y) * (y \ e)) = (x / y) * y by Theorem 73. Then
(L7) T(T(x / y,(x / y) * y) * y,R(x / y,y,y \ e)) = (x / y) * y
by Proposition 68. We have T(T(T(x / y,(x / y) * y) * y,R(x / y,y,y \ e)),T(x / y,y)) = T(T(T(x / y,(x / y) * y) * y,T(x / y,y)),R(x / y,y,y \ e)) by Axiom 7. Then
(L9) T((x / y) * y,T(x / y,y)) = T(T(T(x / y,(x / y) * y) * y,T(x / y,y)),R(x / y,y,y \ e))
by (L7). We have T(x / y,y) * ((x / y) * y) = (T(x / y,(x / y) * y) * y) * T(x / y,y) by Theorem 1079. Then T(T(x / y,(x / y) * y) * y,T(x / y,y)) = (x / y) * y by Theorem 11. Then T((x / y) * y,R(x / y,y,y \ e)) = T((x / y) * y,T(x / y,y)) by (L9). Then T(x,R(x / y,y,y \ e)) = T((x / y) * y,T(x / y,y)) by Axiom 6. Then T((x / y) * y,T(x / y,y)) = T(x,x * (y \ e)) by Proposition 71. Then T(x,T(x / y,y)) = T(x,x * (y \ e)) by Axiom 6. Hence we are done by Proposition 47.
Theorem 1124 (prov9_cfb233f3b2): T(y * T(x,x \ e),z) / ((x \ e) \ e) = x \ T(x * y,z)
Proof. We have T(y * ((x \ e) \ e),z) / ((x \ e) \ e) = (x \ e) * T((x \ e) \ y,z) by Theorem 497. Then
(L2) T(y * T(x,x \ e),z) / ((x \ e) \ e) = (x \ e) * T((x \ e) \ y,z)
by Proposition 49. We have (x \ e) * T((x \ e) \ y,z) = x \ T(x * y,z) by Theorem 140. Hence we are done by (L2).
Theorem 1125 (prov9_062f48aede): T(y * T(x,x \ e),z) / T(x,x \ e) = x \ T(x * y,z)
Proof. We have T(y * T(x,x \ e),z) / ((x \ e) \ e) = x \ T(x * y,z) by Theorem 1124. Hence we are done by Proposition 49.
Theorem 1126 (prov9_56b897a521): T(y * ((x \ e) \ e),z) / T(x,x \ e) = x \ T(x * y,z)
Proof. We have T(y * T(x,x \ e),z) / T(x,x \ e) = x \ T(x * y,z) by Theorem 1125. Hence we are done by Proposition 49.
Theorem 1127 (prov9_cd18a97994): (y * x) \ ((x * y) * T(K(y,x),z)) = K(x,y) * T(K(x,y) \ e,z)
Proof. We have (x * y) * T((x * y) \ (y * x),z) = (y * x) * K(z \ (z / ((y * x) \ (x * y))),z) by Theorem 284. Then (x * y) * T((x * y) \ (y * x),z) = (y * x) * K(z \ (z / K(x,y)),z) by Definition 2. Then (x * y) * T(K(y,x),z) = (y * x) * K(z \ (z / K(x,y)),z) by Definition 2. Then (y * x) * (K(x,y) * T(K(x,y) \ e,z)) = (x * y) * T(K(y,x),z) by Theorem 196. Hence we are done by Proposition 2.
Theorem 1128 (prov9_d79913fd10): y * ((y * ((x * (y \ e)) / x)) * z) = (y * ((x * (y \ e)) / x)) * (y * z)
Proof. We have (y * (y * z)) * ((x * ((y * (y * z)) \ (y * z))) / x) = y * ((y * z) * ((x * ((y * z) \ z)) / x)) by Theorem 1102. Then (y * ((x * (y \ e)) / x)) * (y * z) = y * ((y * z) * ((x * ((y * z) \ z)) / x)) by Theorem 343. Hence we are done by Theorem 343.
Theorem 1129 (prov9_5f2a2806f9): (y \ e) * T(y,x) = T((y \ e) * T(y,x),y \ e)
Proof. We have T((y \ e) * T((y \ e) \ K(y \ e,y),x),y \ e) = (y \ e) * T((y \ e) \ T(K(y \ e,y),y \ e),x) by Theorem 309. Then T((y \ e) * T(y,x),y \ e) = (y \ e) * T((y \ e) \ T(K(y \ e,y),y \ e),x) by Theorem 21. Then (y \ e) * T((y \ e) \ K(y \ e,y),x) = T((y \ e) * T(y,x),y \ e) by Theorem 734. Hence we are done by Theorem 21.
Theorem 1130 (prov9_b44bd934d5): T(y \ e,(y \ e) * T(y,x)) = y \ e
Proof. We have T((y \ e) * T(y,x),y \ e) = (y \ e) * T(y,x) by Theorem 1129. Hence we are done by Proposition 21.
Theorem 1131 (prov9_7d8c114b93): T(y,y * T(e / y,x)) = y
Proof. We have (e / y) \ e = y by Proposition 25. Then T((e / y) \ e,((e / y) \ e) * T(e / y,x)) = y by Theorem 1130. Then T(y,((e / y) \ e) * T(e / y,x)) = y by Proposition 25. Hence we are done by Proposition 25.
Theorem 1132 (prov9_f6b7b6b7e4): T(x * T(e / x,y),x) = x * T(e / x,y)
Proof. We have T(x,x * T(e / x,y)) = x by Theorem 1131. Hence we are done by Proposition 21.
Theorem 1133 (prov9_7b32c94d31): R(e / y,y,(x * y) \ x) = y \ e
Proof. We have (y \ e) * (y * ((x * y) \ x)) = (x * y) \ x by Theorem 379. Then
(L2) ((x * y) \ x) / (y * ((x * y) \ x)) = y \ e
by Proposition 1. We have R(e / y,y,(x * y) \ x) = ((x * y) \ x) / (y * ((x * y) \ x)) by Proposition 79. Hence we are done by (L2).
Theorem 1134 (prov9_d84ca93df4): R(e / y,y,x \ (x / y)) = y \ e
Proof. We have R(e / y,y,((x / y) * y) \ (x / y)) = y \ e by Theorem 1133. Hence we are done by Axiom 6.
Theorem 1135 (prov9_2de431642a): K(y,y \ e) * T(x,y) = T(K(y,y \ e) * x,y)
Proof. We have K(y,y \ e) * T(K(y,y \ e) \ (K(y,y \ e) * x),y) = T(K(y,y \ e) * x,y) by Theorem 620. Hence we are done by Axiom 3.
Theorem 1136 (prov9_459a215163): K(y,y \ e) * (x * y) = (K(y,y \ e) * x) * y
Proof. We have y * (K(y,y \ e) * (y \ (x * y))) = K(y,y \ e) * (x * y) by Theorem 327. Then y * (K(y,y \ e) * T(x,y)) = K(y,y \ e) * (x * y) by Definition 3. Then
(L3) y * T(K(y,y \ e) * x,y) = K(y,y \ e) * (x * y)
by Theorem 1135. We have y * T(K(y,y \ e) * x,y) = (K(y,y \ e) * x) * y by Proposition 46. Hence we are done by (L3).
Theorem 1137 (prov9_be6fb1b620): L(y,x,K(y,y \ e)) = y
Proof. We have (K(y,y \ e) * x) * y = K(y,y \ e) * (x * y) by Theorem 1136. Hence we are done by Theorem 54.
Theorem 1138 (prov9_28752303bf): (e / y) * (x * K(y \ e,y)) = ((e / y) * x) * K(y \ e,y)
Proof. We have R(e / y,x,K(e / y,(e / y) \ e)) = e / y by Theorem 624. Then
(L2) R(e / y,x,K(y \ e,y)) = e / y
by Theorem 272. We have R(e / y,x,K(y \ e,y)) * (x * K(y \ e,y)) = ((e / y) * x) * K(y \ e,y) by Proposition 54. Hence we are done by (L2).
Theorem 1139 (prov9_3c9714161c): R(x * T(x \ e,w),y,z) = R(x,y,z) * T(R(x,y,z) \ e,w)
Proof. We have K(w \ (w / R(x,y,z)),w) = R(x,y,z) * T(R(x,y,z) \ e,w) by Theorem 196. Hence we are done by Theorem 636.
Theorem 1140 (prov9_bfec1d03ca): T(T(x,y),z) * (x * w) = x * (T(T(x,y),z) * w)
Proof. We have ((x * w) / (x * (T(x,y) \ w))) * ((x * (T(x,y) \ w)) * T((x * (T(x,y) \ w)) \ (x * w),z)) = T((x * w) / (x * (T(x,y) \ w)),z) * (x * w) by Theorem 1094. Then T(x,y) * ((x * (T(x,y) \ w)) * T((x * (T(x,y) \ w)) \ (x * w),z)) = T((x * w) / (x * (T(x,y) \ w)),z) * (x * w) by Theorem 1089. Then T(x,y) * (x * ((T(x,y) \ w) * T((T(x,y) \ w) \ w,z))) = T((x * w) / (x * (T(x,y) \ w)),z) * (x * w) by Theorem 486. Then
(L4) T(x,y) * (x * (T(T(x,y),z) * (T(x,y) \ w))) = T((x * w) / (x * (T(x,y) \ w)),z) * (x * w)
by Theorem 48. We have T(x,y) * (x * (T(T(x,y),z) * (T(x,y) \ w))) = x * (T(x,y) * (T(T(x,y),z) * (T(x,y) \ w))) by Theorem 172. Then T((x * w) / (x * (T(x,y) \ w)),z) * (x * w) = x * (T(x,y) * (T(T(x,y),z) * (T(x,y) \ w))) by (L4). Then T((x * w) / (x * (T(x,y) \ w)),z) * (x * w) = x * (T(T(x,y),z) * w) by Theorem 830. Hence we are done by Theorem 1089.
Theorem 1141 (prov9_ca21842b8c): T(T(z,x),y) * w = z * (T(T(z,x),y) * (z \ w))
Proof. We have T(T(z,x),y) * (z * (z \ w)) = z * (T(T(z,x),y) * (z \ w)) by Theorem 1140. Hence we are done by Axiom 4.
Theorem 1142 (prov9_31580da1a3): x * K(y,y \ e) = (x / y) * T(y,y \ e)
Proof. We have ((x / y) * y) * K(y,y \ e) = (x / y) * T(y,y \ e) by Theorem 406. Hence we are done by Axiom 6.
Theorem 1143 (prov9_b285117f18): x * K(y,y \ e) = (x / y) * ((y \ e) \ e)
Proof. We have ((x / y) * y) * K(y,y \ e) = (x / y) * ((y \ e) \ e) by Theorem 407. Hence we are done by Axiom 6.
Theorem 1144 (prov9_65aa58aba2): (x * (y \ e)) / (e / y) = x * K(y \ e,y)
Proof. We have (x * (K(y \ e,y) * (e / y))) / L(e / y,K(y \ e,y),x) = x * K(y \ e,y) by Theorem 453. Then (x * (y \ e)) / L(e / y,K(y \ e,y),x) = x * K(y \ e,y) by Theorem 255. Hence we are done by Theorem 644.
Theorem 1145 (prov9_3cf7cfc8a9): (x * K(y \ e,y)) / (y \ e) = x / (e / y)
Proof. We have (x / (e / y)) * (y \ e) = (x * (y \ e)) / (e / y) by Theorem 1106. Then ((x * (y \ e)) / (e / y)) / (y \ e) = x / (e / y) by Proposition 1. Hence we are done by Theorem 1144.
Theorem 1146 (prov9_e40320f74f): x * T(x \ (x / y),y) = x / T(y,y \ e)
Proof. We have x * T(x \ (x / y),y) = x / ((y \ e) \ e) by Theorem 422. Hence we are done by Proposition 49.
Theorem 1147 (prov9_02a42aeca5): (x / y) * K(y \ e,y) = x / T(y,y \ e)
Proof. We have (x / y) * K(y \ e,y) = x * T(x \ (x / y),y) by Theorem 154. Hence we are done by Theorem 1146.
Theorem 1148 (prov9_55851b2bf4): (x / T(y,y \ e)) * y = x * K(y \ e,y)
Proof. We have ((x / y) * K(y \ e,y)) * y = x * K(y \ e,y) by Theorem 315. Hence we are done by Theorem 1147.
Theorem 1149 (prov9_868b90d608): (z \ T(z * x,y)) * z = T(x * ((z \ e) \ e),y) * K(z \ e,z)
Proof. We have (T(x * ((z \ e) \ e),y) / T(z,z \ e)) * z = T(x * ((z \ e) \ e),y) * K(z \ e,z) by Theorem 1148. Hence we are done by Theorem 1126.
Theorem 1150 (prov9_92c074f51c): ((x \ y) \ e) \ K(y \ x,x \ y) = x \ y
Proof. We have ((x \ y) \ e) \ K((x \ y) \ e,x \ y) = x \ y by Theorem 21. Hence we are done by Theorem 666.
Theorem 1151 (prov9_93c53dab30): R(x,y \ e,y) = R(x,e / y,y)
Proof. We have (x * (e / y)) * y = R(x,e / y,y) by Proposition 69. Hence we are done by Theorem 911.
Theorem 1152 (prov9_20ad2cfde2): ((x \ e) \ e) \ (K(x,x \ e) * y) = x \ y
Proof. We have
(L1) T(x,x \ e) * (x \ y) = K(x,x \ e) * y
by Theorem 431. We have T(x,x \ e) * (x \ y) = ((x \ e) \ e) * (x \ y) by Theorem 464. Then ((x \ e) \ e) * (x \ y) = K(x,x \ e) * y by (L1). Hence we are done by Proposition 2.
Theorem 1153 (prov9_a6b181bb05): x * (K(x,x \ e) * y) = T(x,x \ e) * y
Proof. We have x * (T(x,x \ e) * (x \ y)) = T(x,x \ e) * y by Theorem 830. Hence we are done by Theorem 431.
Theorem 1154 (prov9_eeb314bed4): L(y,K(x,x \ e),x) = y
Proof. We have L(y,K(x,x \ e),x) = L(y,x,K(x,x \ e)) by Theorem 587. Hence we are done by Theorem 432.
Theorem 1155 (prov9_914af4ecfc): x = L(x,K(y \ e,y),e / y)
Proof. We have L(x,e / y,K(e / y,(e / y) \ e)) = L(x,K(y \ e,y),e / y) by Theorem 588. Hence we are done by Theorem 432.
Theorem 1156 (prov9_ff2896cfc0): K(x,x \ e) \ y = x * (((x \ e) \ e) \ y)
Proof. We have L(K(x,x \ e) \ y,K(x,x \ e),x) = x * (((x \ e) \ e) \ y) by Theorem 1118. Hence we are done by Theorem 1154.
Theorem 1157 (prov9_03fd06bd49): (x \ e) * y = (e / x) * (K(x \ e,x) * y)
Proof. We have (x \ e) * L(y,K(x \ e,x),e / x) = (e / x) * (K(x \ e,x) * y) by Theorem 1075. Hence we are done by Theorem 1155.
Theorem 1158 (prov9_156579241d): L(y,x,x \ e) = (e / x) * (x * y)
Proof. We have L(L(y,x,x \ e),K(x \ e,x),e / x) = (e / x) * (x * y) by Theorem 618. Hence we are done by Theorem 1155.
Theorem 1159 (prov9_28e1bc5716): T(x,(e / x) * y) = T(x,(x \ e) * y)
Proof. We have T(x,K(x \ e,x) * L(x \ y,x,x \ e)) = T(x,L(x \ y,x,x \ e)) by Theorem 630. Then T(x,(x \ e) * y) = T(x,L(x \ y,x,x \ e)) by Theorem 156. Hence we are done by Theorem 434.
Theorem 1160 (prov9_e840a45f6d): R(e / y,K(y \ e,y),x) = e / y
Proof. We have R(e / y,K(e / y,(e / y) \ e),x) = e / y by Theorem 1039. Hence we are done by Theorem 272.
Theorem 1161 (prov9_b4c2a223d5): K(x \ e,x) * y = (x \ e) * ((e / x) \ y)
Proof. We have K(x \ e,x) * ((e / x) * ((e / x) \ y)) = (x \ e) * ((e / x) \ y) by Theorem 682. Hence we are done by Axiom 4.
Theorem 1162 (prov9_179f7ceb13): (e / y) * (x * K(y \ e,y)) = (y \ e) * T(x,K(y \ e,y))
Proof. We have (e / y) * (K(y \ e,y) * T(x,K(y \ e,y))) = (y \ e) * T(x,K(y \ e,y)) by Theorem 1157. Hence we are done by Proposition 46.
Theorem 1163 (prov9_a9817820cb): (e / (e / x)) \ (x * y) = K(x,x \ e) * y
Proof. We have (e / (e / x)) * (K(x,x \ e) * y) = x * y by Theorem 684. Hence we are done by Proposition 2.
Theorem 1164 (prov9_118b0b05ad): K(x,x \ e) * ((y \ (e / (e / x))) * y) = x * K(y \ (e / (e / x)),y)
Proof. We have K(x,x \ e) * ((e / (e / x)) * K(y \ (e / (e / x)),y)) = x * K(y \ (e / (e / x)),y) by Theorem 685. Hence we are done by Theorem 17.
Theorem 1165 (prov9_2f9e41d687): (x * (y \ (x \ y))) / x = y \ ((e / (e / x)) \ y)
Proof. We have T(y \ (K(x,x \ e) * (x \ y)),x) = y \ (x \ y) by Theorem 324. Then
(L2) T(y \ ((e / (e / x)) \ y),x) = y \ (x \ y)
by Theorem 690. We have (x * T(y \ ((e / (e / x)) \ y),x)) / x = y \ ((e / (e / x)) \ y) by Proposition 48. Hence we are done by (L2).
Theorem 1166 (prov9_a29eae70e2): K(x,x \ e) * y = (x \ e) \ ((e / x) * y)
Proof. We have K(x,x \ e) * ((e / x) \ ((e / x) * y)) = (x \ e) \ ((e / x) * y) by Theorem 440. Hence we are done by Axiom 3.
Theorem 1167 (prov9_804a5c87b0): (e / x) * y = (x \ e) * (K(x,x \ e) * y)
Proof. We have (e / x) * ((e / x) \ ((x \ e) * (K(x,x \ e) * y))) = (x \ e) * (K(x,x \ e) * y) by Axiom 4. Hence we are done by Theorem 441.
Theorem 1168 (prov9_bb66bdc24d): L(y,K(x \ e,x),(x \ e) \ e) = y
Proof. We have L(y,K(x \ e,(x \ e) \ e),(x \ e) \ e) = y by Theorem 1045. Hence we are done by Theorem 779.
Theorem 1169 (prov9_34175b7970): x * y = ((x \ e) \ e) * (K(x \ e,x) * y)
Proof. We have T(x,x \ e) = (x \ e) \ e by Proposition 49. Then
(L2) R(x,x \ e,x) = (x \ e) \ e
by Theorem 207. We have x / K(x \ e,x) = R(x,x \ e,x) by Theorem 1069. Then
(L4) x / K(x \ e,x) = (x \ e) \ e
by (L2). We have (x / K(x \ e,x)) * K(x \ e,x) = x by Axiom 6. Then
(L6) ((x \ e) \ e) * K(x \ e,x) = x
by (L4). We have (((x \ e) \ e) * K(x \ e,x)) * L(y,K(x \ e,x),(x \ e) \ e) = ((x \ e) \ e) * (K(x \ e,x) * y) by Proposition 52. Then x * L(y,K(x \ e,x),(x \ e) \ e) = ((x \ e) \ e) * (K(x \ e,x) * y) by (L6). Hence we are done by Theorem 1168.
Theorem 1170 (prov9_71871e3c34): x * y = T(x,x \ e) * (K(x \ e,x) * y)
Proof. We have x / K(x \ e,x) = R(x,x \ e,x) by Theorem 1069. Then
(L4) x / K(x \ e,x) = T(x,x \ e)
by Theorem 207. We have (x / K(x \ e,x)) * K(x \ e,x) = x by Axiom 6. Then
(L6) T(x,x \ e) * K(x \ e,x) = x
by (L4). We have (T(x,x \ e) * K(x \ e,x)) * L(y,K(x \ e,x),T(x,x \ e)) = T(x,x \ e) * (K(x \ e,x) * y) by Proposition 52. Then
(L8) x * L(y,K(x \ e,x),T(x,x \ e)) = T(x,x \ e) * (K(x \ e,x) * y)
by (L6). We have L(y,K(x \ e,x),(x \ e) \ e) = y by Theorem 1168. Then L(y,K(x \ e,x),T(x,x \ e)) = y by Proposition 49. Hence we are done by (L8).
Theorem 1171 (prov9_f21825dcf5): K(x \ e,x) * (T(x,x \ e) * y) = x * y
Proof. We have
(L1) K(x,x \ e) * (K(x \ e,x) * (T(x,x \ e) * y)) = T(x,x \ e) * y
by Theorem 442. We have K(x,x \ e) * (x * y) = T(x,x \ e) * y by Theorem 676. Hence we are done by (L1) and Proposition 7.
Theorem 1172 (prov9_acc408bf47): T(x,x \ e) \ y = x \ (K(x \ e,x) * y)
Proof. We have T(x,x \ e) * (x \ (K(x \ e,x) * y)) = K(x,x \ e) * (K(x \ e,x) * y) by Theorem 431. Then T(x,x \ e) \ (K(x,x \ e) * (K(x \ e,x) * y)) = x \ (K(x \ e,x) * y) by Proposition 2. Hence we are done by Theorem 442.
Theorem 1173 (prov9_f73f3e51e9): (e / (e / x)) * y = x * (K(x \ e,x) * y)
Proof. We have (e / (e / x)) * (K(x,x \ e) * (K(x \ e,x) * y)) = x * (K(x \ e,x) * y) by Theorem 684. Hence we are done by Theorem 442.
Theorem 1174 (prov9_6efad61ef6): (x \ e) * (x * y) = (e / x) * ((e / (e / x)) * y)
Proof. We have (x \ e) * (K(x,x \ e) * ((e / (e / x)) * y)) = (e / x) * ((e / (e / x)) * y) by Theorem 1167. Hence we are done by Theorem 685.
Theorem 1175 (prov9_b866a328c5): x \ L(y,e / x,x) = (x \ e) * y
Proof. We have
(L2) x * (K(x,x \ e) * (x \ L(y,e / x,x))) = (x * (e / x)) * L(y,e / x,x)
by Theorem 326. We have x * ((e / x) * y) = (x * (e / x)) * L(y,e / x,x) by Proposition 52. Then
(L4) K(x,x \ e) * (x \ L(y,e / x,x)) = (e / x) * y
by (L2) and Proposition 7. We have K(x,x \ e) * ((x \ e) * y) = (e / x) * y by Theorem 697. Then K(x,x \ e) * (x \ L(y,e / x,x)) = K(x,x \ e) * ((x \ e) * y) by (L4). Hence we are done by Proposition 9.
Theorem 1176 (prov9_66eb7bf087): L(x,e / y,y) = L(x,y \ e,y)
Proof. We have
(L1) y \ L(L(x,y \ e,y),e / y,y) = (y \ e) * L(x,y \ e,y)
by Theorem 1175. We have y \ L(L(x,y \ e,y),e / y,y) = (y \ e) * L(x,e / y,y) by Theorem 803. Then (y \ e) * L(x,e / y,y) = (y \ e) * L(x,y \ e,y) by (L1). Hence we are done by Proposition 9.
Theorem 1177 (prov9_6ce83ad2e4): L(x,y,y \ e) = L(x,(y \ e) \ e,y \ e)
Proof. We have L(x,e / (y \ e),y \ e) = L(x,(y \ e) \ e,y \ e) by Theorem 1176. Hence we are done by Proposition 24.
Theorem 1178 (prov9_871c919872): K(y,y \ e) * x = T(x,K(y \ e,y)) * K(y,y \ e)
Proof. We have
(L1) T(T(x,K(y \ e,y)),K(y,y \ e)) * K(y \ e,y) = K(y \ e,y) * T(x,K(y \ e,y))
by Theorem 699. We have x * K(y \ e,y) = K(y \ e,y) * T(x,K(y \ e,y)) by Proposition 46. Then
(L3) T(T(x,K(y \ e,y)),K(y,y \ e)) = x
by (L1) and Proposition 8. We have K(y,y \ e) * T(T(x,K(y \ e,y)),K(y,y \ e)) = T(x,K(y \ e,y)) * K(y,y \ e) by Proposition 46. Hence we are done by (L3).
Theorem 1179 (prov9_5d6c478806): L(z,K(x \ e,x),T(x,y)) = z
Proof. We have T(R(x,K(x \ e,x),z),y) * (K(x \ e,x) * z) = (T(x,y) * K(x \ e,x)) * z by Proposition 65. Then T(x,y) * (K(x \ e,x) * z) = (T(x,y) * K(x \ e,x)) * z by Theorem 1047. Hence we are done by Theorem 54.
Theorem 1180 (prov9_629e73d66a): (((y \ e) \ e) * x) \ x = T((y * x) \ x,y)
Proof. We have T(x / (((y \ e) \ e) * x),((y \ e) \ e) * x) = (((y \ e) \ e) * x) \ x by Proposition 47. Then
(L4) T(R(y \ e,y,x),((y \ e) \ e) * x) = (((y \ e) \ e) * x) \ x
by Theorem 703. We have R(T(y \ e,((y \ e) \ e) * x),y,x) = T(R(y \ e,y,x),((y \ e) \ e) * x) by Axiom 9. Then
(L6) R(T(y \ e,((y \ e) \ e) * x),y,x) = (((y \ e) \ e) * x) \ x
by (L4). We have T(y \ e,(e / (y \ e)) * x) = T(y \ e,((y \ e) \ e) * x) by Theorem 1159. Then T(y \ e,y * x) = T(y \ e,((y \ e) \ e) * x) by Proposition 24. Then
(L7) R(T(y \ e,y * x),y,x) = (((y \ e) \ e) * x) \ x
by (L6). We have T(T(x / (y * x),y),y * x) = T((y * x) \ x,y) by Proposition 51. Then
(L9) T(R(y \ e,y,x),y * x) = T((y * x) \ x,y)
by Theorem 89. We have R(T(y \ e,y * x),y,x) = T(R(y \ e,y,x),y * x) by Axiom 9. Then R(T(y \ e,y * x),y,x) = T((y * x) \ x,y) by (L9). Hence we are done by (L7).
Theorem 1181 (prov9_20979601b5): e / (e / ((y * x) \ x)) = (y * ((y * x) \ x)) / y
Proof. We have (y * x) \ ((e / (e / y)) \ (y * x)) = (y * ((y * x) \ (y \ (y * x)))) / y by Theorem 1165. Then (y * x) \ ((e / (e / y)) \ (y * x)) = (y * ((y * x) \ x)) / y by Axiom 3. Then (y * x) \ (K(y,y \ e) * x) = (y * ((y * x) \ x)) / y by Theorem 1163. Then T(e / (e / (x / (y * x))),y * x) = (y * ((y * x) \ x)) / y by Theorem 710. Hence we are done by Theorem 713.
Theorem 1182 (prov9_37d1453950): K(x \ e,x) * (y * K(x,x \ e)) = y
Proof. We have
(L1) T((K(x \ e,x) * (y * K(x,x \ e))) * K(x,x \ e),K(x \ e,x)) = (K(x \ e,x) * (y * K(x,x \ e))) * K(x,x \ e)
by Theorem 717. We have T((K(x \ e,x) * (y * K(x,x \ e))) * K(x,x \ e),K(x \ e,x)) = y * K(x,x \ e) by Theorem 658. Then (K(x \ e,x) * (y * K(x,x \ e))) * K(x,x \ e) = y * K(x,x \ e) by (L1). Hence we are done by Proposition 10.
Theorem 1183 (prov9_5bc599a814): y * K(x,x \ e) = K(x,x \ e) * y
Proof. We have T(y,K(x \ e,x)) * K(x,x \ e) = K(x,x \ e) * y by Theorem 1178. Hence we are done by Theorem 717.
Theorem 1184 (prov9_50e57350b1): ((y \ e) * x) / (y \ e) = ((e / y) * x) / (e / y)
Proof. We have ((e / y) * x) * K(y \ e,y) = (e / y) * (x * K(y \ e,y)) by Theorem 1138. Then
(L2) ((e / y) * x) * K(y \ e,y) = (y \ e) * T(x,K(y \ e,y))
by Theorem 1162. We have (((e / y) * x) * K(y \ e,y)) / (y \ e) = ((e / y) * x) / (e / y) by Theorem 1145. Then ((y \ e) * T(x,K(y \ e,y))) / (y \ e) = ((e / y) * x) / (e / y) by (L2). Hence we are done by Theorem 717.
Theorem 1185 (prov9_a205f5761b): (e / y) * x = (y \ e) * (x * K(y,y \ e))
Proof. We have (y \ e) * (K(y,y \ e) * T(x,K(y,y \ e))) = (e / y) * T(x,K(y,y \ e)) by Theorem 1167. Then (y \ e) * (x * K(y,y \ e)) = (e / y) * T(x,K(y,y \ e)) by Proposition 46. Hence we are done by Theorem 718.
Theorem 1186 (prov9_8dd2b0e262): (e / y) * x = ((y \ e) * x) * K(y,y \ e)
Proof. We have T((e / y) * x,K(y,y \ e)) = ((y \ e) * x) * K(y,y \ e) by Theorem 700. Hence we are done by Theorem 718.
Theorem 1187 (prov9_3e97369866): (y \ x) * y = (((y \ e) \ e) \ x) * ((y \ e) \ e)
Proof. We have
(L1) ((((y \ e) \ e) \ x) * ((y \ e) \ e)) * (y \ e) = T(x * (y \ e),(y \ e) \ e)
by Theorem 887. We have (y * T(y \ x,(y \ e) \ e)) * (y \ e) = T(x * (y \ e),(y \ e) \ e) by Theorem 506. Then (((y \ e) \ e) \ x) * ((y \ e) \ e) = y * T(y \ x,(y \ e) \ e) by (L1) and Proposition 8. Then (((y \ e) \ e) \ x) * ((y \ e) \ e) = T((y \ x) * y,K(y,y \ e)) by Theorem 705. Hence we are done by Theorem 718.
Theorem 1188 (prov9_cf7bfa459e): T(x,T((y \ e) \ e,z)) = T(x,T(y,z))
Proof. We have T(x,(T(y,z) \ e) \ e) = T(x,T(y,z)) by Theorem 721. Hence we are done by Theorem 919.
Theorem 1189 (prov9_4e76f528f0): R(x,x \ y,K(z \ e,z)) \ (K(z \ e,z) * y) = (x \ y) * K(z \ e,z)
Proof. We have R(x,x \ y,K(z \ e,z)) \ (y * K(z \ e,z)) = (x \ y) * K(z \ e,z) by Theorem 1067. Hence we are done by Theorem 923.
Theorem 1190 (prov9_99d8470148): (y \ e) * ((e / y) \ x) = x * K(y \ e,y)
Proof. We have (y \ e) * ((e / y) \ x) = K(y \ e,y) * x by Theorem 1161. Hence we are done by Theorem 923.
Theorem 1191 (prov9_6163afa805): x * (e / y) = (K(y,y \ e) * x) * (y \ e)
Proof. We have ((K(y,y \ e) * x) * K(y \ e,y)) * (e / y) = (K(y,y \ e) * x) * (y \ e) by Theorem 411. Hence we are done by Theorem 924.
Theorem 1192 (prov9_22710280c0): L((x * K(y,y \ e)) \ z,x * K(y,y \ e),K(y \ e,y)) = x \ (K(y \ e,y) * z)
Proof. We have L((x * K(y,y \ e)) \ z,x * K(y,y \ e),K(y \ e,y)) = (K(y \ e,y) * (x * K(y,y \ e))) \ (K(y \ e,y) * z) by Proposition 53. Hence we are done by Theorem 1182.
Theorem 1193 (prov9_c984d84d04): ((y \ e) \ e) * x = y * (x * K(y,y \ e))
Proof. We have ((y \ e) \ e) * (K(y \ e,y) * (x * K(y,y \ e))) = y * (x * K(y,y \ e)) by Theorem 1169. Hence we are done by Theorem 1182.
Theorem 1194 (prov9_da7730a919): T(x,T(y \ e,z)) = T(x,T(e / y,z))
Proof. We have T(x,T(((e / y) \ e) \ e,z)) = T(x,T(e / y,z)) by Theorem 1188. Hence we are done by Proposition 25.
Theorem 1195 (prov9_151986bf55): L(y * K(x \ e,x),x,x \ e) = (x \ e) * (x * y)
Proof. We have
(L3) ((x \ e) \ e) \ (x * y) = x * (((x \ e) \ e) \ y)
by Theorem 1095. We have
(L1) T(y,K(x,x \ e)) * K(x \ e,x) = K(x,x \ e) \ y
by Theorem 659.
((x \ e) \ e) \ (x * y)
=K(x,x \ e) \ y by (L3), Theorem 1156
=y * K(x \ e,x) by (L1), Theorem 718

Then
(L5) ((x \ e) \ e) \ (x * y) = y * K(x \ e,x)
. We have L(((x \ e) \ e) \ (x * y),(x \ e) \ e,x \ e) = (x \ e) * (x * y) by Theorem 62. Then L(((x \ e) \ e) \ (x * y),x,x \ e) = (x \ e) * (x * y) by Theorem 1177. Hence we are done by (L5).
Theorem 1196 (prov9_f00c834340): (x * y) / x = T((x * y) / x,K(y,z))
Proof. We have (x * T(y,K(y,z))) / x = T((x * y) / x,K(y,z)) by Theorem 50. Hence we are done by Theorem 736.
Theorem 1197 (prov9_f5a4d0671d): (T(x,y) \ x) \ e = x \ T(x,y)
Proof. We have T(y \ T(y,R(T(x,y),T(x,y) \ e,y)),K(y \ (y / (x \ e)),y)) = y \ T(y,R(T(x,y),T(x,y) \ e,y)) by Theorem 619. Then K(y \ T(y,R(T(x,y),T(x,y) \ e,y)),K(y \ (y / (x \ e)),y)) = e by Proposition 22. Then K(y \ ((T(x,y) \ x) * y),K(y \ (y / (x \ e)),y)) = e by Theorem 591. Then K(T(T(x,y) \ x,y),K(y \ (y / (x \ e)),y)) = e by Definition 3. Then K(T(x,y) \ x,K(y \ (y / (x \ e)),y)) = e by Theorem 738. Then
(L6) K(T(x,y) \ x,x \ T(x,y)) = e
by Theorem 198. We have ((x \ T(x,y)) \ e) \ K(T(x,y) \ x,x \ T(x,y)) = x \ T(x,y) by Theorem 1150. Then ((x \ T(x,y)) \ e) \ e = x \ T(x,y) by (L6). Hence we are done by Theorem 749.
Theorem 1198 (prov9_5b98a42731): x * ((T(x,y) \ x) \ e) = T(x,y)
Proof. We have x * (x \ T(x,y)) = T(x,y) by Axiom 4. Hence we are done by Theorem 1197.
Theorem 1199 (prov9_ccbd95461c): (e / x) * T((e / x) \ e,y) = x \ T(x,y)
Proof. We have (e / x) * T((e / x) \ e,y) = T(x,y) / x by Theorem 149. Hence we are done by Theorem 758.
Theorem 1200 (prov9_fd7b96c581): T(y,T(y \ e,x)) = (y \ e) \ e
Proof. We have T(e / (y \ e),T(y \ e,x)) = (y \ e) \ e by Theorem 762. Hence we are done by Proposition 24.
Theorem 1201 (prov9_3e3e5e5b06): T(x \ e,x) = T(x \ e,T(x,y))
Proof. We have T(T(e / x,T(x,y)),x) = T(x \ e,T(x,y)) by Proposition 51. Hence we are done by Theorem 762.
Theorem 1202 (prov9_262d9aef69): K(x \ e,T(x,y)) = x * T(x \ e,T(x,y))
Proof. We have K(T(x,y) \ (T(x,y) / x),T(x,y)) = x * T(x \ e,T(x,y)) by Theorem 196. Then K(T(e / x,T(x,y)),T(x,y)) = x * T(x \ e,T(x,y)) by Theorem 501. Hence we are done by Theorem 762.
Theorem 1203 (prov9_fdbf63b5e8): T(x,x \ e) = T(x,T(x \ e,y))
Proof. We have T(x,x \ e) = (x \ e) \ e by Proposition 49. Hence we are done by Theorem 1200.
Theorem 1204 (prov9_704620eed9): (x \ e) \ T(e / x,y) = x * T(e / x,y)
Proof. We have (e / x) * ((e / (e / x)) * T(e / x,y)) = (x \ e) * (x * T(e / x,y)) by Theorem 1174. Then (e / x) * (T(e / x,y) / (e / x)) = (x \ e) * (x * T(e / x,y)) by Theorem 150. Then T(e / x,y) = (x \ e) * (x * T(e / x,y)) by Theorem 757. Hence we are done by Proposition 2.
Theorem 1205 (prov9_343d31d9a0): K(x \ e,x) = K(x \ e,T(x,y))
Proof. We have x * T(x \ e,T(x,y)) = K(x \ e,T(x,y)) by Theorem 1202. Then
(L2) x * T(x \ e,x) = K(x \ e,T(x,y))
by Theorem 1201. We have x * T(x \ e,x) = (x \ e) * x by Proposition 46. Then
(L4) K(x \ e,T(x,y)) = (x \ e) * x
by (L2). We have K(x \ e,x) = (x \ e) * x by Proposition 76. Hence we are done by (L4).
Theorem 1206 (prov9_d8d2bc18b1): ((x \ e) \ e) * T(x \ e,y) = x * T(e / x,y)
Proof. We have ((x \ e) \ e) * T(x \ e,y) = (x \ e) \ T(e / x,y) by Theorem 1110. Hence we are done by Theorem 1204.
Theorem 1207 (prov9_f15dd106da): K(x,x \ e) * T(x \ e,y) = T(e / x,y)
Proof. We have R(x,T(e / x,y),x) * T(T(e / x,y),x) = T(x * T(e / x,y),x) by Theorem 230. Then T(x,T(e / x,y)) * T(T(e / x,y),x) = T(x * T(e / x,y),x) by Theorem 865. Then T(x,T(e / x,y)) * T(x \ e,y) = T(x * T(e / x,y),x) by Proposition 51. Then
(L6) T(x,T(e / x,y)) * T(x \ e,y) = x * T(e / x,y)
by Theorem 1132. We have T(x,T(e / x,y)) = T(x,T(x \ e,y)) by Theorem 1194. Then T(x,T(e / x,y)) = T(x,x \ e) by Theorem 1203. Then
(L7) T(x,x \ e) * T(x \ e,y) = x * T(e / x,y)
by (L6). We have x * (K(x,x \ e) * T(x \ e,y)) = T(x,x \ e) * T(x \ e,y) by Theorem 1153. Then x * (K(x,x \ e) * T(x \ e,y)) = x * T(e / x,y) by (L7). Hence we are done by Proposition 9.
Theorem 1208 (prov9_e74374252e): T(e / y,x) = T(y \ e,x) * K(y,y \ e)
Proof. We have K(y,y \ e) * T(y \ e,x) = T(y \ e,x) * K(y,y \ e) by Theorem 1183. Hence we are done by Theorem 1207.
Theorem 1209 (prov9_6b32b514e3): T(x \ e,y) * (K(x,x \ e) * z) = T(e / x,y) * z
Proof. We have R(T(x \ e,y),K(x,x \ e),z) = T(R(x \ e,K(x,x \ e),z),y) by Axiom 9. Then
(L2) R(T(x \ e,y),K(x,x \ e),z) = T(x \ e,y)
by Theorem 1047. We have R(T(x \ e,y),K(x,x \ e),z) \ ((T(x \ e,y) * K(x,x \ e)) * z) = K(x,x \ e) * z by Theorem 6. Then T(x \ e,y) \ ((T(x \ e,y) * K(x,x \ e)) * z) = K(x,x \ e) * z by (L2). Then
(L5) T(x \ e,y) \ (T(e / x,y) * z) = K(x,x \ e) * z
by Theorem 1208. We have T(x \ e,y) * (T(x \ e,y) \ (T(e / x,y) * z)) = T(e / x,y) * z by Axiom 4. Hence we are done by (L5).
Theorem 1210 (prov9_9611015fc9): T(y \ e,x) * ((y \ e) \ z) = T(e / y,x) * ((e / y) \ z)
Proof. We have T(y \ e,x) * (K(y,y \ e) * ((e / y) \ z)) = T(e / y,x) * ((e / y) \ z) by Theorem 1209. Hence we are done by Theorem 440.
Theorem 1211 (prov9_9b9e967150): (y * (x \ T(x,y))) * (T(x,y) \ x) = y
Proof. We have (y * (x \ T(x,y))) * ((y * (x \ T(x,y))) \ y) = y by Axiom 4. Hence we are done by Theorem 952.
Theorem 1212 (prov9_a58e537efb): ((((e / x) * y) \ y) \ e) \ e = ((x \ e) * y) \ y
Proof. We have (e / (e / (((e / x) * y) \ y))) \ e = e / (((e / x) * y) \ y) by Proposition 25. Then
(L2) (((e / x) * (((e / x) * y) \ y)) / (e / x)) \ e = e / (((e / x) * y) \ y)
by Theorem 1181. We have (((((e / x) * (((e / x) * y) \ y)) / (e / x)) \ e) \ e) * (e / x) = (e / x) * ((((e / x) \ ((e / x) * (((e / x) * y) \ y))) \ e) \ e) by Theorem 921. Then ((e / (((e / x) * y) \ y)) \ e) * (e / x) = (e / x) * ((((e / x) \ ((e / x) * (((e / x) * y) \ y))) \ e) \ e) by (L2). Then (e / x) * ((((e / x) \ ((e / x) * (((e / x) * y) \ y))) \ e) \ e) = (((e / x) * y) \ y) * (e / x) by Proposition 25. Then
(L6) (e / x) * (((((e / x) * y) \ y) \ e) \ e) = (((e / x) * y) \ y) * (e / x)
by Axiom 3. We have T(((e / x) * y) \ y,e / x) = ((((e / x) \ e) \ e) * y) \ y by Theorem 1180. Then
(L8) T(((e / x) * y) \ y,e / x) = ((x \ e) * y) \ y
by Proposition 25. We have (e / x) * T(((e / x) * y) \ y,e / x) = (((e / x) * y) \ y) * (e / x) by Proposition 46. Then (e / x) * (((x \ e) * y) \ y) = (((e / x) * y) \ y) * (e / x) by (L8). Hence we are done by (L6) and Proposition 7.
Theorem 1213 (prov9_ce79effa97): K(((x \ e) * y) \ y,z) = K(((e / x) * y) \ y,z)
Proof. We have K(((((e / x) * y) \ y) \ e) \ e,z) = K(((e / x) * y) \ y,z) by Theorem 733. Hence we are done by Theorem 1212.
Theorem 1214 (prov9_3b936cebc2): (e / x) * (T(x \ e,y) * z) = (x \ e) * (T(e / x,y) * z)
Proof. We have ((T(e / x,y) * ((e / x) \ e)) * (e / x)) * L(z,((e / x) \ e) * (e / x),R(T(e / x,y),(e / x) \ e,e / x)) = R(T(e / x,y),(e / x) \ e,e / x) * ((((e / x) \ e) * (e / x)) * z) by Theorem 1064. Then ((T(e / x,y) * ((e / x) \ e)) * (e / x)) * L(z,((e / x) \ e) * (e / x),T(T(e / x,(e / x) \ e),y)) = R(T(e / x,y),(e / x) \ e,e / x) * ((((e / x) \ e) * (e / x)) * z) by Theorem 1100. Then (((e / x) \ T(e / x,y)) * (e / x)) * L(z,((e / x) \ e) * (e / x),T(T(e / x,(e / x) \ e),y)) = R(T(e / x,y),(e / x) \ e,e / x) * ((((e / x) \ e) * (e / x)) * z) by Theorem 146. Then
(L6) (((e / x) \ T(e / x,y)) * (e / x)) * L(z,K((e / x) \ e,e / x),T(T(e / x,(e / x) \ e),y)) = R(T(e / x,y),(e / x) \ e,e / x) * ((((e / x) \ e) * (e / x)) * z)
by Proposition 76. We have L(z,K(T(e / x,(e / x) \ e) \ e,T(e / x,(e / x) \ e)),T(T(e / x,(e / x) \ e),y)) = z by Theorem 1179. Then L(z,K((e / x) \ e,e / x),T(T(e / x,(e / x) \ e),y)) = z by Theorem 929. Then (((e / x) \ T(e / x,y)) * (e / x)) * z = R(T(e / x,y),(e / x) \ e,e / x) * ((((e / x) \ e) * (e / x)) * z) by (L6). Then T(T(e / x,(e / x) \ e),y) * ((((e / x) \ e) * (e / x)) * z) = (((e / x) \ T(e / x,y)) * (e / x)) * z by Theorem 1100. Then
(L9) T(T(e / x,(e / x) \ e),y) * (K((e / x) \ e,e / x) * z) = (((e / x) \ T(e / x,y)) * (e / x)) * z
by Proposition 76. We have K((e / x) \ e,e / x) * (T(e / x,(e / x) \ e) * (T(T(e / x,(e / x) \ e),y) * (T(e / x,(e / x) \ e) \ z))) = (e / x) * (T(T(e / x,(e / x) \ e),y) * (T(e / x,(e / x) \ e) \ z)) by Theorem 1171. Then
(L11) K((e / x) \ e,e / x) * (T(T(e / x,(e / x) \ e),y) * z) = (e / x) * (T(T(e / x,(e / x) \ e),y) * (T(e / x,(e / x) \ e) \ z))
by Theorem 830. We have (e / x) * (T(T(e / x,(e / x) \ e),y) * ((e / x) \ (K((e / x) \ e,e / x) * z))) = T(T(e / x,(e / x) \ e),y) * (K((e / x) \ e,e / x) * z) by Theorem 1141. Then (e / x) * (T(T(e / x,(e / x) \ e),y) * (T(e / x,(e / x) \ e) \ z)) = T(T(e / x,(e / x) \ e),y) * (K((e / x) \ e,e / x) * z) by Theorem 1172. Then K((e / x) \ e,e / x) * (T(T(e / x,(e / x) \ e),y) * z) = T(T(e / x,(e / x) \ e),y) * (K((e / x) \ e,e / x) * z) by (L11). Then
(L15) (((e / x) \ T(e / x,y)) * (e / x)) * z = K((e / x) \ e,e / x) * (T(T(e / x,(e / x) \ e),y) * z)
by (L9). We have T(e / x,(e / x) \ e) * (K((e / x) \ e,e / x) * (T(T(e / x,(e / x) \ e),y) * z)) = (e / x) * (T(T(e / x,(e / x) \ e),y) * z) by Theorem 1170. Then T(e / x,(e / x) \ e) * ((((e / x) \ T(e / x,y)) * (e / x)) * z) = (e / x) * (T(T(e / x,(e / x) \ e),y) * z) by (L15). Then T(e / x,(e / x) \ e) * (T(e / x,y) * z) = (e / x) * (T(T(e / x,(e / x) \ e),y) * z) by Theorem 759. Then
(L19) T(e / x,(e / x) \ e) * (T(e / x,y) * z) = (e / x) * (T(((e / x) \ e) \ e,y) * z)
by Proposition 49. We have T(e / x,(e / x) \ e) * (T(e / x,y) * z) = (((e / x) \ e) \ e) * (T(e / x,y) * z) by Theorem 464. Then T(e / x,(e / x) \ e) * (T(e / x,y) * z) = (x \ e) * (T(e / x,y) * z) by Proposition 25. Then (e / x) * (T(((e / x) \ e) \ e,y) * z) = (x \ e) * (T(e / x,y) * z) by (L19). Hence we are done by Proposition 25.
Theorem 1215 (prov9_8bfceee0fe): (T(x \ e,y) * z) * K(x,x \ e) = T(e / x,y) * z
Proof.
(x \ e) * ((T(x \ e,y) * z) * K(x,x \ e))
=(e / x) * (T(x \ e,y) * z) by Theorem 1185
=(x \ e) * (T(e / x,y) * z) by Theorem 1214

Then (x \ e) * ((T(x \ e,y) * z) * K(x,x \ e)) = (x \ e) * (T(e / x,y) * z). Hence we are done by Proposition 9.
Theorem 1216 (prov9_5dcd115d8f): y * (((y * x) / y) \ x) = (x \ ((y * x) / y)) \ y
Proof. We have (y \ T(y,R(x,x \ e,y))) \ y = y * (T(y,R(x,x \ e,y)) \ y) by Theorem 956. Then (x \ ((y * x) / y)) \ y = y * (T(y,R(x,x \ e,y)) \ y) by Theorem 760. Hence we are done by Theorem 769.
Theorem 1217 (prov9_b11d024f63): L(x,T(e / z,y),z) = L(x,T(z \ e,y),(z \ e) \ e)
Proof. We have z * ((T(z \ e,y) * x) * K(z,z \ e)) = ((z \ e) \ e) * (T(z \ e,y) * x) by Theorem 1193. Then
(L2) z * (T(e / z,y) * x) = ((z \ e) \ e) * (T(z \ e,y) * x)
by Theorem 1215. We have L(x,T(z \ e,y),(z \ e) \ e) = (((z \ e) \ e) * T(z \ e,y)) \ (((z \ e) \ e) * (T(z \ e,y) * x)) by Definition 4. Then L(x,T(z \ e,y),(z \ e) \ e) = (z * T(e / z,y)) \ (((z \ e) \ e) * (T(z \ e,y) * x)) by Theorem 1206. Then
(L5) (z * T(e / z,y)) \ (z * (T(e / z,y) * x)) = L(x,T(z \ e,y),(z \ e) \ e)
by (L2). We have L(x,T(e / z,y),z) = (z * T(e / z,y)) \ (z * (T(e / z,y) * x)) by Definition 4. Hence we are done by (L5).
Theorem 1218 (prov9_c50328b2f2): L(x,y,z \ e) = L(L(x,y,e / z),(e / z) * y,K(z \ e,z))
Proof. We have (K(z \ e,z) * ((e / z) * y)) \ (K(z \ e,z) * ((e / z) * (y * x))) = L(L(x,y,e / z),(e / z) * y,K(z \ e,z)) by Theorem 1061. Then ((z \ e) * y) \ (K(z \ e,z) * ((e / z) * (y * x))) = L(L(x,y,e / z),(e / z) * y,K(z \ e,z)) by Theorem 682. Then
(L3) ((z \ e) * y) \ ((z \ e) * (y * x)) = L(L(x,y,e / z),(e / z) * y,K(z \ e,z))
by Theorem 682. We have L(x,y,z \ e) = ((z \ e) * y) \ ((z \ e) * (y * x)) by Definition 4. Hence we are done by (L3).
Theorem 1219 (prov9_87cab01bf9): K(x,((y \ e) * z) \ z) = K(x,((e / y) * z) \ z)
Proof. We have K(x,((((e / y) * z) \ z) \ e) \ e) = K(x,((e / y) * z) \ z) by Theorem 779. Hence we are done by Theorem 1212.
Theorem 1220 (prov9_7a67dbe7ce): x * K(x \ e,R(x,y,z)) = L(x,x \ e,R(x,y,z))
Proof. We have R(e / (x \ e),x \ e,(x \ R(x,y,z)) \ ((x \ R(x,y,z)) / (x \ e))) = ((x \ R(x,y,z)) \ ((x \ R(x,y,z)) / (x \ e))) / ((x \ e) * ((x \ R(x,y,z)) \ ((x \ R(x,y,z)) / (x \ e)))) by Proposition 79. Then (x \ e) \ e = ((x \ R(x,y,z)) \ ((x \ R(x,y,z)) / (x \ e))) / ((x \ e) * ((x \ R(x,y,z)) \ ((x \ R(x,y,z)) / (x \ e)))) by Theorem 1134. Then ((x \ R(x,y,z)) \ R(x,y,z)) / ((x \ e) * ((x \ R(x,y,z)) \ ((x \ R(x,y,z)) / (x \ e)))) = (x \ e) \ e by Theorem 834. Then T(x,x \ R(x,y,z)) / ((x \ e) * ((x \ R(x,y,z)) \ ((x \ R(x,y,z)) / (x \ e)))) = (x \ e) \ e by Proposition 49. Then T(x,x \ R(x,y,z)) / ((x \ e) * ((x \ R(x,y,z)) \ R(x,y,z))) = (x \ e) \ e by Theorem 834. Then
(L6) T(x,x \ R(x,y,z)) / (x \ L(x,x \ e,R(x,y,z))) = (x \ e) \ e
by Theorem 1097. We have R(R(x,x \ e,T(x,x \ R(x,y,z))),y,z) = R(R(x,y,z),x \ e,T(x,x \ R(x,y,z))) by Axiom 12. Then R(T(x,x \ R(x,y,z)) / ((x \ e) * T(x,x \ R(x,y,z))),y,z) = R(R(x,y,z),x \ e,T(x,x \ R(x,y,z))) by Proposition 66. Then R(R(x,y,z),x \ e,T(x,x \ R(x,y,z))) = R(T(x,x \ R(x,y,z)) / (x \ L(x,x \ e,R(x,y,z))),y,z) by Theorem 1122. Then
(L10) R((x \ e) \ e,y,z) = R(R(x,y,z),x \ e,T(x,x \ R(x,y,z)))
by (L6). We have R((x \ R(x,y,z)) / (x \ e),x \ e,T(x,x \ R(x,y,z))) \ (x * (x \ R(x,y,z))) = (x \ e) * T(x,x \ R(x,y,z)) by Theorem 801. Then R((x \ R(x,y,z)) / (x \ e),x \ e,T(x,x \ R(x,y,z))) \ (x * (x \ R(x,y,z))) = x \ L(x,x \ e,R(x,y,z)) by Theorem 1122. Then R(R(x,y,z),x \ e,T(x,x \ R(x,y,z))) \ (x * (x \ R(x,y,z))) = x \ L(x,x \ e,R(x,y,z)) by Theorem 834. Then R((x \ e) \ e,y,z) \ (x * (x \ R(x,y,z))) = x \ L(x,x \ e,R(x,y,z)) by (L10). Then R((x \ e) \ e,y,z) \ R(x,y,z) = x \ L(x,x \ e,R(x,y,z)) by Axiom 4. Then R(T(x,x \ e),y,z) \ R(x,y,z) = x \ L(x,x \ e,R(x,y,z)) by Proposition 49. Then
(L17) K(x \ e,R(x,y,z)) = x \ L(x,x \ e,R(x,y,z))
by Theorem 960. We have x * (x \ L(x,x \ e,R(x,y,z))) = L(x,x \ e,R(x,y,z)) by Axiom 4. Hence we are done by (L17).
Theorem 1221 (prov9_9d04e8aff8): L(z,x,K(y,y \ e)) = z
Proof. We have L(z,x,K((e / y) \ e,e / y)) = z by Theorem 967. Hence we are done by Theorem 253.
Theorem 1222 (prov9_9b7ef95f90): (y * K(x \ e,x)) * z = K(x \ e,x) * (y * z)
Proof. We have (K(x \ e,x) * y) * L(z,y,K(x \ e,x)) = K(x \ e,x) * (y * z) by Proposition 52. Then (y * K(x \ e,x)) * L(z,y,K(x \ e,x)) = K(x \ e,x) * (y * z) by Theorem 923. Hence we are done by Theorem 967.
Theorem 1223 (prov9_f0e4611299): (x * K(y,y \ e)) \ z = x \ (K(y \ e,y) * z)
Proof. We have L((x * K(y,y \ e)) \ z,x * K(y,y \ e),K(y \ e,y)) = x \ (K(y \ e,y) * z) by Theorem 1192. Hence we are done by Theorem 967.
Theorem 1224 (prov9_799bf182ab): L(x,y,e / z) = L(x,y,z \ e)
Proof. We have L(L(x,y,e / z),(e / z) * y,K(z \ e,z)) = L(x,y,z \ e) by Theorem 1218. Hence we are done by Theorem 967.
Theorem 1225 (prov9_23464de584): (K(x,x \ e) * y) * z = K(x,x \ e) * (y * z)
Proof. We have (K(x,x \ e) * y) * L(z,y,K(x,x \ e)) = K(x,x \ e) * (y * z) by Proposition 52. Hence we are done by Theorem 1221.
Theorem 1226 (prov9_174267dd07): (K(x,x \ e) * y) / z = K(x,x \ e) * (y / z)
Proof. We have (K(x,x \ e) * y) / L(z,y / z,K(x,x \ e)) = K(x,x \ e) * (y / z) by Theorem 472. Hence we are done by Theorem 1221.
Theorem 1227 (prov9_06ae109092): L(x,y,z) = L(x,y,(z \ e) \ e)
Proof. We have L(x,y,e / (z \ e)) = L(x,y,(z \ e) \ e) by Theorem 1224. Hence we are done by Proposition 24.
Theorem 1228 (prov9_863f904565): L(x,T(z \ e,y),z) = L(x,T(e / z,y),z)
Proof. We have L(x,T(z \ e,y),(z \ e) \ e) = L(x,T(e / z,y),z) by Theorem 1217. Hence we are done by Theorem 1227.
Theorem 1229 (prov9_8e0babc9ba): (e / y) / x = ((y \ e) / x) * K(y,y \ e)
Proof. We have (K(y,y \ e) * (y \ e)) / L(x,(y \ e) / x,K(y,y \ e)) = K(y,y \ e) * ((y \ e) / x) by Theorem 472. Then (e / y) / L(x,(y \ e) / x,K(y,y \ e)) = K(y,y \ e) * ((y \ e) / x) by Theorem 849. Then
(L3) (e / y) / x = K(y,y \ e) * ((y \ e) / x)
by Theorem 1221. We have K(y,y \ e) * ((y \ e) / x) = ((y \ e) / x) * K(y,y \ e) by Theorem 1183. Hence we are done by (L3).
Theorem 1230 (prov9_e95d554a07): R(K(z \ e,z),x,y) = K(z \ e,z)
Proof. We have (K(z \ e,z) * x) * y = K(z \ e,z) * (x * y) by Theorem 968. Hence we are done by Theorem 64.
Theorem 1231 (prov9_82b586fafc): (y * K(x,x \ e)) * z = K(x,x \ e) * (y * z)
Proof. We have (K(x,x \ e) * y) * z = K(x,x \ e) * (y * z) by Theorem 1225. Hence we are done by Theorem 1183.
Theorem 1232 (prov9_3183e7210b): a((y * x) \ e,y,x) / x = (e / (y * x)) * y
Proof. We have a((y * x) \ e,y,x) = (((y * x) \ e) * (y * x)) \ ((((y * x) \ e) * y) * x) by Definition 1. Then a((y * x) \ e,y,x) = K((y * x) \ e,y * x) \ ((((y * x) \ e) * y) * x) by Proposition 76. Then
(L5) K(y * x,(y * x) \ e) * ((((y * x) \ e) * y) * x) = a((y * x) \ e,y,x)
by Theorem 696. We have (K(y * x,(y * x) \ e) * (((y * x) \ e) * y)) * x = K(y * x,(y * x) \ e) * ((((y * x) \ e) * y) * x) by Theorem 1225. Then ((e / (y * x)) * y) * x = K(y * x,(y * x) \ e) * ((((y * x) \ e) * y) * x) by Theorem 697. Then ((e / (y * x)) * y) * x = a((y * x) \ e,y,x) by (L5). Hence we are done by Proposition 1.
Theorem 1233 (prov9_c3bcf6c573): T(y * (T(y,T(x,y)) \ y),x) = y
Proof. We have (y * (x \ T(x,y))) * x = x * y by Theorem 976. Then (y * (T(y,T(x,y)) \ y)) * x = x * y by Theorem 986. Hence we are done by Theorem 11.
Theorem 1234 (prov9_c864ba6855): x \ y = T(y / x,T(x,y))
Proof. We have T(x,y) \ ((x * y) / x) = T(y / x,T(x,y)) by Theorem 1083. Hence we are done by Theorem 995.
Theorem 1235 (prov9_1b82868990): T(x \ y,x) = T(x \ y,T(x,y))
Proof. We have T(T(y / x,T(x,y)),x) = T(x \ y,T(x,y)) by Proposition 51. Hence we are done by Theorem 1234.
Theorem 1236 (prov9_8d92bc856b): x * ((x * y) / x) = T(x,y) * y
Proof. We have x * (T(x,y) * (x \ y)) = T(x,y) * y by Theorem 830. Hence we are done by Theorem 996.
Theorem 1237 (prov9_73f4ef16e3): T(y,y / x) = T(y,x \ y)
Proof. We have T(y,y / (y / (x \ y))) = T(y,x \ y) by Theorem 997. Hence we are done by Proposition 24.
Theorem 1238 (prov9_9213193e91): (y / x) \ T(y,x) = (y * x) / y
Proof. We have (y / x) \ T(y,y / (y / x)) = (y * x) / y by Theorem 1087. Hence we are done by Theorem 997.
Theorem 1239 (prov9_9a0da00a65): T(x,y) / ((x * y) / x) = x / y
Proof. We have T(x,x / (x / y)) / ((x / y) \ T(x,x / (x / y))) = x / y by Proposition 24. Then T(x,x / (x / y)) / ((x * y) / x) = x / y by Theorem 1087. Hence we are done by Theorem 997.
Theorem 1240 (prov9_dcf30dce64): T(x,x * y) = T(x,x / (e / y))
Proof. We have ((x * (x * (x \ (x * y)))) / x) / (x \ (x * y)) = T(x,x / (e / (x \ (x * y)))) by Theorem 1112. Then ((x * (x * y)) / x) / (x \ (x * y)) = T(x,x / (e / (x \ (x * y)))) by Axiom 4. Then T(x,x / (e / (x \ (x * y)))) = T(x,x / (x / (x * y))) by Theorem 1114. Then T(x,x / (e / y)) = T(x,x / (x / (x * y))) by Axiom 3. Hence we are done by Theorem 997.
Theorem 1241 (prov9_5c5a805e3d): T(y,x) = T(y,(x \ y) \ y)
Proof. We have T(y,y / (x \ y)) = T(y,(x \ y) \ y) by Theorem 1237. Hence we are done by Proposition 24.
Theorem 1242 (prov9_acef6a61cd): T(y,(x \ e) \ y) = T(y,x * y)
Proof. We have (x \ e) * ((y * ((x \ e) \ y)) / y) = T(y,y / (x \ e)) by Theorem 515. Then
(L2) x \ ((y * (x * y)) / y) = T(y,y / (x \ e))
by Theorem 1103. We have x \ ((y * (x * y)) / y) = T(y,x * y) by Theorem 1082. Then T(y,y / (x \ e)) = T(y,x * y) by (L2). Hence we are done by Theorem 1237.
Theorem 1243 (prov9_05260cd4f9): T(x,T(y,y \ x)) = T(x,y)
Proof. We have T(x,(y \ x) \ x) = T(x,y) by Theorem 1241. Hence we are done by Proposition 49.
Theorem 1244 (prov9_ca5a444a0a): T(y * x,T(y,x)) = T(y * x,y)
Proof. We have T(y * x,T(y,y \ (y * x))) = T(y * x,y) by Theorem 1243. Hence we are done by Axiom 3.
Theorem 1245 (prov9_2ff785cd1f): T(y,y \ x) * (y \ x) = y * (x / y)
Proof. We have T(y,y \ x) * (y \ x) = y * ((y * (y \ x)) / y) by Theorem 1236. Hence we are done by Axiom 4.
Theorem 1246 (prov9_7879ffbaf0): x \ T(x * y,y) = ((x * y) * y) / (x * y)
Proof. We have ((x * y) / y) \ T(x * y,y) = ((x * y) * y) / (x * y) by Theorem 1238. Hence we are done by Axiom 5.
Theorem 1247 (prov9_091801b8c3): T(x * y,x \ e) = x * ((x * y) * (y / (x * y)))
Proof. We have (x * y) * (((x * y) * ((x * y) \ y)) / (x * y)) = (x * (((x * y) * (x \ e)) / (x * y))) * y by Theorem 343. Then
(L2) (x * y) * (y / (x * y)) = (x * (((x * y) * (x \ e)) / (x * y))) * y
by Axiom 4. We have (x * (((x * y) * (x \ e)) / (x * y))) * (x * y) = x * ((x * (((x * y) * (x \ e)) / (x * y))) * y) by Theorem 1128. Then T(x * y,R(x \ e,x,x * y)) = x * ((x * (((x * y) * (x \ e)) / (x * y))) * y) by Theorem 926. Then x * ((x * y) * (y / (x * y))) = T(x * y,R(x \ e,x,x * y)) by (L2). Hence we are done by Theorem 1000.
Theorem 1248 (prov9_7d99e23e2c): x \ T(y,x \ e) = y * ((x \ y) / y)
Proof. We have (x * y) * ((y * ((x * y) \ y)) / y) = T(y,y / (x * y)) by Theorem 515. Then
(L2) (x * y) * L((x \ y) / y,y,x) = T(y,y / (x * y))
by Proposition 75. We have x \ ((x * y) * L((x \ y) / y,y,x)) = y * ((x \ y) / y) by Theorem 790. Then x \ T(y,y / (x * y)) = y * ((x \ y) / y) by (L2). Hence we are done by Theorem 1002.
Theorem 1249 (prov9_7d2580a5cb): y \ (x \ T(y,x \ e)) = (x \ y) / y
Proof. We have y * ((x \ y) / y) = x \ T(y,x \ e) by Theorem 1248. Hence we are done by Proposition 2.
Theorem 1250 (prov9_68808b0ac9): K(T(y,y \ x),y \ x) = K(y,x / y)
Proof. We have K(T(y,y \ x),y \ x) = K(y,(y * (y \ x)) / y) by Theorem 1008. Hence we are done by Axiom 4.
Theorem 1251 (prov9_8636ef63ae): K((y \ x) \ x,y \ x) = K(y,x / y)
Proof. We have K(T(y,y \ x),y \ x) = K(y,x / y) by Theorem 1250. Hence we are done by Proposition 49.
Theorem 1252 (prov9_1f7fd4c3a5): R(T(y,y * x),x,y) = T(y,T(x,y))
Proof. We have T(y,y / (y / T(x,y))) * (x * y) = y * (y * x) by Theorem 848. Then
(L4) (y * (y * x)) / (x * y) = T(y,y / (y / T(x,y)))
by Proposition 1. We have
(L5) T(y,y / (y / T(x,y))) = (y / T(x,y)) * x
by Theorem 825. Then
(L6) (y * (y * x)) / (x * y) = (y / T(x,y)) * x
by (L4). We have R(((y * (y * x)) / y) / x,x,y) = (y * (y * x)) / (x * y) by Proposition 74. Then R(T(y,y / (e / x)),x,y) = (y * (y * x)) / (x * y) by Theorem 1112. Then
(L7) R(T(y,y / (e / x)),x,y) = (y / T(x,y)) * x
by (L6).
R(T(y,y * x),x,y)
=(y / T(x,y)) * x by (L7), Theorem 1240
=T(y,T(x,y)) by (L5), Theorem 997

Hence we are done.
Theorem 1253 (prov9_8ad30157e4): y * T(x,T(y,x)) = L(R(x,y,x),y,x) * y
Proof. We have y * L(T(R(x,y,x),y),y,x) = L(R(x,y,x),y,x) * y by Proposition 58. Then y * L(R(T(x,y),y,x),y,x) = L(R(x,y,x),y,x) * y by Axiom 9. Then y * R(T(x,x * y),y,x) = L(R(x,y,x),y,x) * y by Theorem 1071. Hence we are done by Theorem 1252.
Theorem 1254 (prov9_927d97e7ed): L(R(y,x,y),x,y) = T((x * y) / x,T(x,y))
Proof. We have
(L1) L(R(y,x,y),x,y) * x = x * T(y,T(x,y))
by Theorem 1253. We have T((x * y) / x,T(x,y)) * x = x * T(y,T(x,y)) by Theorem 47. Hence we are done by (L1) and Proposition 8.
Theorem 1255 (prov9_d74ee94cf8): y * (T(y,x) \ y) = x / (y \ ((y * x) / y))
Proof. We have ((y * x) / ((y * x) / y)) * (y \ ((y * x) / y)) = y \ (y * x) by Theorem 1093. Then (y \ (y * x)) / (y \ ((y * x) / y)) = (y * x) / ((y * x) / y) by Proposition 1. Then x / (y \ ((y * x) / y)) = (y * x) / ((y * x) / y) by Axiom 3. Hence we are done by Theorem 1011.
Theorem 1256 (prov9_9a5715169b): K(y \ x,T(y,x)) = K(y \ x,y)
Proof. We have (x \ y) \ K(T(y,x) \ (T(y,x) / (x \ y)),T(y,x)) = T((x \ y) \ e,T(y,x)) by Theorem 197. Then (x \ y) \ K(T(y,x) \ ((y * x) / y),T(y,x)) = T((x \ y) \ e,T(y,x)) by Theorem 1091. Then
(L3) (x \ y) \ K(T(x / y,T(y,x)),T(y,x)) = T((x \ y) \ e,T(y,x))
by Theorem 1083. We have (x \ y) * ((x \ y) \ K(T(x / y,T(y,x)),T(y,x))) = K(T(x / y,T(y,x)),T(y,x)) by Axiom 4. Then
(L5) (x \ y) * T((x \ y) \ e,T(y,x)) = K(T(x / y,T(y,x)),T(y,x))
by (L3). We have (x \ y) * T((x \ y) \ e,T(y,x)) = x \ (y * T(y \ x,T(y,x))) by Theorem 568. Then K(T(x / y,T(y,x)),T(y,x)) = x \ (y * T(y \ x,T(y,x))) by (L5). Then
(L8) K(y \ x,T(y,x)) = x \ (y * T(y \ x,T(y,x)))
by Theorem 1234. We have x \ (y * T(y \ x,y)) = K(y \ x,y) by Theorem 46. Then x \ (y * T(y \ x,T(y,x))) = K(y \ x,y) by Theorem 1235. Hence we are done by (L8).
Theorem 1257 (prov9_bbfe9a0a00): K(y,x) = K(y,T(x,x * y))
Proof. We have K(x \ (x * y),T(x,x * y)) = K(x \ (x * y),x) by Theorem 1256. Then K(y,T(x,x * y)) = K(x \ (x * y),x) by Axiom 3. Hence we are done by Axiom 3.
Theorem 1258 (prov9_efcb3e589e): K(x \ e,T(y,x \ y)) = K(x \ e,y)
Proof. We have K(x \ e,T(y,y * (x \ e))) = K(x \ e,y) by Theorem 1257. Hence we are done by Theorem 1123.
Theorem 1259 (prov9_71f7ec1efe): K(x,T(y,x * y)) = K(x,y)
Proof. We have K((x \ e) \ e,T(y,(x \ e) \ y)) = K((x \ e) \ e,y) by Theorem 1258. Then
(L2) K((x \ e) \ e,T(y,x * y)) = K((x \ e) \ e,y)
by Theorem 1242. We have K((x \ e) \ e,T(y,x * y)) = K(x,T(y,x * y)) by Theorem 733. Then
(L4) K((x \ e) \ e,y) = K(x,T(y,x * y))
by (L2). We have K((x \ e) \ e,y) = K(x,y) by Theorem 733. Hence we are done by (L4).
Theorem 1260 (prov9_3b65accee4): K(x,T(x \ y,y)) = K(x,x \ y)
Proof. We have K(x,T(x \ y,x * (x \ y))) = K(x,x \ y) by Theorem 1259. Hence we are done by Axiom 4.
Theorem 1261 (prov9_990620f35d): T(K(y,z),y * K(x,y)) = K(y,z)
Proof. We have T(((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e),K(y,z)) = ((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e) by Theorem 1196. Then T(K(y,z),((L(x \ e,x,y) \ e) * y) / (L(x \ e,x,y) \ e)) = K(y,z) by Proposition 21. Hence we are done by Theorem 1013.
Theorem 1262 (prov9_b5ae4ca02c): y / (y * K(x,y)) = (y * K(x,y)) \ y
Proof. We have (y * x) \ ((x * y) * T(K(y,x),y * K(x,y))) = K(x,y) * T(K(x,y) \ e,y * K(x,y)) by Theorem 1127. Then (y * x) \ ((x * y) * K(y,x)) = K(x,y) * T(K(x,y) \ e,y * K(x,y)) by Theorem 1261. Then
(L3) (y * x) \ (y * x) = K(x,y) * T(K(x,y) \ e,y * K(x,y))
by Proposition 82. We have (y * x) \ (y * x) = e by Proposition 28. Then
(L5) K(x,y) * T(K(x,y) \ e,y * K(x,y)) = e
by (L3). We have K((y * K(x,y)) \ y,y * K(x,y)) = K(x,y) * T(K(x,y) \ e,y * K(x,y)) by Theorem 184. Then
(L7) K((y * K(x,y)) \ y,y * K(x,y)) = e
by (L5). We have T(y,y * K(x,y)) * K((y * K(x,y)) \ y,(y * (y * K(x,y))) / y) = ((y * K(x,y)) \ y) * ((y * (y * K(x,y))) / y) by Theorem 1092. Then T(y,y * K(x,y)) * K((y * K(x,y)) \ y,y * K(x,y)) = ((y * K(x,y)) \ y) * ((y * (y * K(x,y))) / y) by Theorem 1098. Then y * K((y * K(x,y)) \ y,y * K(x,y)) = ((y * K(x,y)) \ y) * ((y * (y * K(x,y))) / y) by Theorem 541. Then
(L11) ((y * K(x,y)) \ y) * ((y * (y * K(x,y))) / y) = y * e
by (L7). We have (((y * K(x,y)) \ y) * ((y * (y * K(x,y))) / y)) * e = ((y * K(x,y)) \ y) * ((y * (y * K(x,y))) / y) by Axiom 2. Then y * e = (((y * K(x,y)) \ y) * ((y * (y * K(x,y))) / y)) * e by (L11). Then y = ((y * K(x,y)) \ y) * ((y * (y * K(x,y))) / y) by Proposition 10. Then ((y * K(x,y)) \ y) * (y * K(x,y)) = y by Theorem 1098. Hence we are done by Proposition 1.
Theorem 1263 (prov9_085215ab73): (K(x,y) \ e) \ y = y * K(x,y)
Proof. We have (y * K(x,y)) \ R(y,K(x,y),K(x,y) \ e) = K(x,y) \ e by Theorem 1062. Then (y * K(x,y)) \ y = K(x,y) \ e by Theorem 1047. Then
(L3) y / (y * K(x,y)) = K(x,y) \ e
by Theorem 1262. We have (y / (y * K(x,y))) \ y = y * K(x,y) by Proposition 25. Hence we are done by (L3).
Theorem 1264 (prov9_cddc9753bd): (x \ ((y \ e) \ e)) * K(y \ e,y) = ((x \ y) / y) * y
Proof. We have (K(y,y \ e) * y) / L((y / x) \ y,y / x,K(y,y \ e)) = K(y,y \ e) * (y / x) by Theorem 61. Then ((y \ e) \ e) / L((y / x) \ y,y / x,K(y,y \ e)) = K(y,y \ e) * (y / x) by Theorem 552. Then
(L3) ((y \ e) \ e) / ((y / x) \ y) = K(y,y \ e) * (y / x)
by Theorem 1221. We have K(y,y \ e) * (y / x) = (y / x) * K(y,y \ e) by Theorem 1183. Then ((y \ e) \ e) / ((y / x) \ y) = (y / x) * K(y,y \ e) by (L3). Then
(L6) ((y \ e) \ e) / x = (y / x) * K(y,y \ e)
by Proposition 25. We have T(y * ((y / x) / y),x) = x \ ((y * ((y / x) / y)) * x) by Definition 3. Then T(y * ((y / x) / y),x) = x \ T(y,y / (y * x)) by Theorem 881. Then
(L9) x \ T(y,x \ e) = T(y * ((y / x) / y),x)
by Theorem 1003. We have T(((y / x) / y) * ((y \ e) \ e),x) * K(y \ e,y) = (y \ T(y * ((y / x) / y),x)) * y by Theorem 1149. Then T(((y / x) / y) * ((y \ e) \ e),x) * K(y \ e,y) = (y \ (x \ T(y,x \ e))) * y by (L9). Then T((y / x) * K(y,y \ e),x) * K(y \ e,y) = (y \ (x \ T(y,x \ e))) * y by Theorem 1143. Then T(((y \ e) \ e) / x,x) * K(y \ e,y) = (y \ (x \ T(y,x \ e))) * y by (L6). Then (x \ ((y \ e) \ e)) * K(y \ e,y) = (y \ (x \ T(y,x \ e))) * y by Proposition 47. Hence we are done by Theorem 1249.
Theorem 1265 (prov9_8d4c2e8718): (x \ ((y \ e) \ e)) * z = K(y,y \ e) * ((x \ y) * z)
Proof. We have (((x \ y) * K(y,y \ e)) / K(y,y \ e)) * K(y,y \ e) = (x \ y) * K(y,y \ e) by Axiom 6. Then (((((x \ y) * K(y,y \ e)) / K(y,y \ e)) / y) * y) * K(y,y \ e) = (x \ y) * K(y,y \ e) by Axiom 6. Then (((x \ y) / y) * y) * K(y,y \ e) = (x \ y) * K(y,y \ e) by Axiom 5. Then ((x \ y) * K(y,y \ e)) / K(y,y \ e) = ((x \ y) / y) * y by Proposition 1. Then
(L6) ((x \ y) * K(y,y \ e)) * K(y \ e,y) = ((x \ y) / y) * y
by Theorem 425. We have (x \ ((y \ e) \ e)) * K(y \ e,y) = ((x \ y) / y) * y by Theorem 1264. Then
(L7) x \ ((y \ e) \ e) = (x \ y) * K(y,y \ e)
by (L6) and Proposition 8. We have ((x \ y) * K(y,y \ e)) * z = K(y,y \ e) * ((x \ y) * z) by Theorem 1231. Hence we are done by (L7).
Theorem 1266 (prov9_9ec6586e37): (e / (e / y)) / T(x,x \ y) = x \ (e / (e / y))
Proof. We have x * (x \ T(e / (e / y),(e / (e / y)) \ e)) = ((e / (e / y)) \ e) \ e by Theorem 795. Then
(L4) x \ (((e / (e / y)) \ e) \ e) = x \ T(e / (e / y),(e / (e / y)) \ e)
by Proposition 2. We have (((e / (e / y)) \ e) \ e) / T(x,x \ (((e / (e / y)) \ e) \ e)) = x \ (((e / (e / y)) \ e) \ e) by Theorem 15. Then
(L6) (((e / (e / y)) \ e) \ e) / T(x,x \ T(e / (e / y),(e / (e / y)) \ e)) = x \ (((e / (e / y)) \ e) \ e)
by (L4). We have ((((e / (e / y)) \ e) \ e) / T(x,x \ T(e / (e / y),(e / (e / y)) \ e))) * K((e / (e / y)) \ e,((e / (e / y)) \ e) \ e) = (e / ((e / (e / y)) \ e)) / T(x,x \ T(e / (e / y),(e / (e / y)) \ e)) by Theorem 1229. Then (x \ (((e / (e / y)) \ e) \ e)) * K((e / (e / y)) \ e,((e / (e / y)) \ e) \ e) = (e / ((e / (e / y)) \ e)) / T(x,x \ T(e / (e / y),(e / (e / y)) \ e)) by (L6). Then (e / ((e / (e / y)) \ e)) / T(x,x \ T(e / (e / y),(e / (e / y)) \ e)) = (x \ (((e / (e / y)) \ e) \ e)) * K((e / (e / y)) \ e,e / (e / y)) by Theorem 779. Then
(L10) (e / (e / y)) / T(x,x \ T(e / (e / y),(e / (e / y)) \ e)) = (x \ (((e / (e / y)) \ e) \ e)) * K((e / (e / y)) \ e,e / (e / y))
by Proposition 24. We have ((x \ (e / (e / y))) / (e / (e / y))) * (e / (e / y)) = x \ (e / (e / y)) by Axiom 6. Then (x \ (((e / (e / y)) \ e) \ e)) * K((e / (e / y)) \ e,e / (e / y)) = x \ (e / (e / y)) by Theorem 1264. Then (e / (e / y)) / T(x,x \ T(e / (e / y),(e / (e / y)) \ e)) = x \ (e / (e / y)) by (L10). Then (e / (e / y)) / T(x,x \ T(e / (e / y),e / y)) = x \ (e / (e / y)) by Proposition 25. Hence we are done by Theorem 14.
Theorem 1267 (prov9_db2839a4c3): ((y \ (e / x)) * y) * x = x * ((y \ (x \ e)) * y)
Proof. We have ((y \ (y / (x * y))) * y) * (x * y) = y * ((x * y) * T((x * y) \ e,y)) by Theorem 840. Then ((y \ R(e / x,x,y)) * y) * (x * y) = y * ((x * y) * T((x * y) \ e,y)) by Proposition 79. Then
(L8) R((y \ (e / x)) * y,x,y) * (x * y) = y * ((x * y) * T((x * y) \ e,y))
by Theorem 973. We have R((y \ (e / x)) * y,x,y) * (x * y) = (((y \ (e / x)) * y) * x) * y by Proposition 54. Then y * ((x * y) * T((x * y) \ e,y)) = (((y \ (e / x)) * y) * x) * y by (L8). Then
(L11) y * (x * (y * T(y \ (x \ e),y))) = (((y \ (e / x)) * y) * x) * y
by Theorem 136. We have ((e / (y * x)) \ T(e / (y * x),y)) * y = y * ((e / (y * x)) \ T(e / (y * x),y)) by Theorem 937. Then ((e / (y * x)) \ T(e / (y * x),y)) * y = ((e / (y * x)) * y) / (e / (y * x)) by Theorem 977. Then ((e / (y * x)) \ T(e / (y * x),y)) * y = (a((y * x) \ e,y,x) / x) / (e / (y * x)) by Theorem 1232. Then
(L4) (a((y * x) \ e,y,x) / x) / (e / (y * x)) = ((y * x) * T((y * x) \ e,y)) * y
by Theorem 102.
(a((y * x) \ e,y,x) / x) / (e / (y * x))
=(((y \ (e / x)) * y) * x) * y by (L4), Theorem 530
=y * (x * ((y \ (x \ e)) * y)) by (L11), Proposition 46

Then
(L13) (a((y * x) \ e,y,x) / x) / (e / (y * x)) = y * (x * ((y \ (x \ e)) * y))
. We have (((y * x) \ e) * y) / ((y * x) \ e) = y * ((y * x) * T((y * x) \ e,y)) by Theorem 982. Then
(L15) (((y * x) \ e) * y) / ((y * x) \ e) = y * (((y \ (e / x)) * y) * x)
by Theorem 530. We have ((e / (y * x)) * y) / (e / (y * x)) = (((y * x) \ e) * y) / ((y * x) \ e) by Theorem 1184. Then ((e / (y * x)) * y) / (e / (y * x)) = y * (((y \ (e / x)) * y) * x) by (L15). Then (a((y * x) \ e,y,x) / x) / (e / (y * x)) = y * (((y \ (e / x)) * y) * x) by Theorem 1232. Then y * (((y \ (e / x)) * y) * x) = y * (x * ((y \ (x \ e)) * y)) by (L13). Hence we are done by Proposition 9.
Theorem 1268 (prov9_7fc39aa35f): ((y \ x) * y) * (x \ e) = (e / x) * ((y \ x) * y)
Proof. We have ((y \ (e / (x \ e))) * y) * (x \ e) = (x \ e) * ((y \ ((x \ e) \ e)) * y) by Theorem 1267. Then
(L2) ((y \ x) * y) * (x \ e) = (x \ e) * ((y \ ((x \ e) \ e)) * y)
by Proposition 24. We have (x \ e) * (K(x,x \ e) * ((y \ x) * y)) = (e / x) * ((y \ x) * y) by Theorem 1167. Then (x \ e) * ((y \ ((x \ e) \ e)) * y) = (e / x) * ((y \ x) * y) by Theorem 1265. Hence we are done by (L2).
Theorem 1269 (prov9_9bbc44b93c): L(z,y \ T(y,x),y) = z
Proof. We have
(L1) y * ((y \ T(y,x)) * z) = T(y,x) * z
by Theorem 1017. We have T(y,x) * L(z,y \ T(y,x),y) = y * ((y \ T(y,x)) * z) by Theorem 52. Then T(y,x) * L(z,y \ T(y,x),y) = T(y,x) * z by (L1). Hence we are done by Proposition 9.
Theorem 1270 (prov9_5a27ba27a7): (y \ T(y,x)) * z = T(y,x) * (y \ z)
Proof. We have
(L1) y * ((y \ T(y,x)) * z) = T(y,x) * z
by Theorem 1017. We have y * (T(y,x) * (y \ z)) = T(y,x) * z by Theorem 830. Hence we are done by (L1) and Proposition 7.
Theorem 1271 (prov9_b705ce5192): L(z,T(x,T(y,x)) \ x,y) = z
Proof. We have L(z,y \ T(y,x),y) = z by Theorem 1269. Hence we are done by Theorem 986.
Theorem 1272 (prov9_797b673cf9): (x \ T(x,y)) \ z = x * (T(x,y) \ z)
Proof. We have L((x \ T(x,y)) \ z,x \ T(x,y),x) = T(x,y) \ (x * z) by Theorem 60. Then
(L2) L((x \ T(x,y)) \ z,T(y,T(x,y)) \ y,x) = T(x,y) \ (x * z)
by Theorem 986. We have T(x,y) \ (x * z) = x * (T(x,y) \ z) by Theorem 174. Then L((x \ T(x,y)) \ z,T(y,T(x,y)) \ y,x) = x * (T(x,y) \ z) by (L2). Hence we are done by Theorem 1271.
Theorem 1273 (prov9_88cd95e86a): K(y,x) = T(T(x,y) \ x,x * y)
Proof. We have
(L1) T(T(x,y) \ x,x * y) = (x * y) \ ((T(x,y) \ x) * (x * y))
by Definition 3.
K(y,x)
=(x * y) \ (y * x) by Definition 2
=T(T(x,y) \ x,x * y) by (L1), Theorem 1022

Hence we are done.
Theorem 1274 (prov9_9c39e6cf62): T(x,y) * K(y,x) = y \ (K(y,x) * (x * y))
Proof. We have (y \ (x * y)) * T((y \ (x * y)) \ x,x * y) = y \ (((x * y) \ (y * x)) * (x * y)) by Theorem 1085. Then (y \ (x * y)) * T((y \ (x * y)) \ x,x * y) = y \ (K(y,x) * (x * y)) by Definition 2. Then T(x,y) * T((y \ (x * y)) \ x,x * y) = y \ (K(y,x) * (x * y)) by Definition 3. Then T(x,y) * T(T(x,y) \ x,x * y) = y \ (K(y,x) * (x * y)) by Definition 3. Hence we are done by Theorem 1273.
Theorem 1275 (prov9_70bc47641e): (T(x,y) \ x) * z = T(y,x) * (y \ z)
Proof. We have (y \ T(y,x)) * z = T(y,x) * (y \ z) by Theorem 1270. Hence we are done by Theorem 1030.
Theorem 1276 (prov9_7780a9f47f): T(x \ y,x) = T(x,x \ y) \ y
Proof. We have T(x \ y,T(x,x \ y)) = T(x,x \ y) \ y by Theorem 463. Hence we are done by Theorem 1032.
Theorem 1277 (prov9_01b89ea753): y = L(R(y,x,y),x,y)
Proof. We have y * (x \ T(x,y)) = (x * y) / x by Theorem 977. Then
(L4) y * (T(y,T(x,y)) \ y) = (x * y) / x
by Theorem 986. We have T((x * y) / x,T(x,y)) = L(R(y,x,y),x,y) by Theorem 1254. Then
(L6) T(y * (T(y,T(x,y)) \ y),T(x,y)) = L(R(y,x,y),x,y)
by (L4). We have T(y * (T(y,T(T(x,y),y)) \ y),T(x,y)) = y by Theorem 1233. Then T(y * (T(y,T(x,y)) \ y),T(x,y)) = y by Theorem 1032. Hence we are done by (L6).
Theorem 1278 (prov9_e80238cfcd): T(x \ T(x * y,y),T(x * y,y)) = y
Proof. We have T(((x * y) * y) / (x * y),T(x * y,y)) = L(R(y,x * y,y),x * y,y) by Theorem 1254. Then T(x \ T(x * y,y),T(x * y,y)) = L(R(y,x * y,y),x * y,y) by Theorem 1246. Hence we are done by Theorem 1277.
Theorem 1279 (prov9_3ee299d77e): ((x * y) / x) \ x = y \ ((y * x) / y)
Proof. We have y * (x \ T(x,y)) = (x * y) / x by Theorem 977. Then
(L2) y * (T(y,x) \ y) = (x * y) / x
by Theorem 1030. We have (x / (y \ ((y * x) / y))) \ x = y \ ((y * x) / y) by Proposition 25. Then (y * (T(y,x) \ y)) \ x = y \ ((y * x) / y) by Theorem 1255. Hence we are done by (L2).
Theorem 1280 (prov9_6ad3105263): K(x,y) * (y * x) = y * (x * K(x,y))
Proof. We have y * (x * T(x \ T(x,y),y * x)) = (y * x) * T(K(x,y),y * x) by Theorem 1113. Then y * (x * T(T(y,T(x,y)) \ y,y * x)) = (y * x) * T(K(x,y),y * x) by Theorem 986. Then L(T(T(y,T(x,y)) \ y,y * x),x,y) = T(K(x,y),y * x) by Theorem 54. Then L(T(T(y,x) \ y,y * x),x,y) = T(K(x,y),y * x) by Theorem 1032. Then
(L5) L(K(x,y),x,y) = T(K(x,y),y * x)
by Theorem 1273. We have (y * x) * T(K(x,y),y * x) = K(x,y) * (y * x) by Proposition 46. Then
(L7) (y * x) * L(K(x,y),x,y) = K(x,y) * (y * x)
by (L5). We have (y * x) * L(K(x,y),x,y) = y * (x * K(x,y)) by Proposition 52. Hence we are done by (L7).
Theorem 1281 (prov9_f0fc53e6ca): (y \ T(y,x)) \ z = (T(y,x) \ y) * z
Proof. We have ((y * x) / y) * (x * (x \ ((((y * x) / y) \ x) * z))) = x * (((y * x) / y) * (x \ ((((y * x) / y) \ x) * z))) by Theorem 854. Then
(L4) ((y * x) / y) * ((((y * x) / y) \ x) * z) = x * (((y * x) / y) * (x \ ((((y * x) / y) \ x) * z)))
by Axiom 4. We have x * L(z,((y * x) / y) \ x,(y * x) / y) = ((y * x) / y) * ((((y * x) / y) \ x) * z) by Theorem 52. Then ((y * x) / y) * (x \ ((((y * x) / y) \ x) * z)) = L(z,((y * x) / y) \ x,(y * x) / y) by (L4) and Proposition 7. Then ((y * x) / y) * (x \ ((T(y,x) \ y) * z)) = L(z,((y * x) / y) \ x,(y * x) / y) by Theorem 984. Then ((y * x) / y) * (x \ ((T(y,x) \ y) * z)) = L(z,T(y,x) \ y,(y * x) / y) by Theorem 984. Then
(L9) ((y * x) / y) \ L(z,T(y,x) \ y,(y * x) / y) = x \ ((T(y,x) \ y) * z)
by Proposition 2. We have L(z,T(y,T((y * x) / y,y)) \ y,(y * x) / y) = z by Theorem 1271. Then L(z,T(y,x) \ y,(y * x) / y) = z by Theorem 7. Then
(L10) ((y * x) / y) \ z = x \ ((T(y,x) \ y) * z)
by (L9). We have x * (x \ ((T(y,x) \ y) * z)) = (T(y,x) \ y) * z by Axiom 4. Then
(L12) x * (((y * x) / y) \ z) = (T(y,x) \ y) * z
by (L10). We have R(((y * x) / y) / x,x,((y * x) / y) \ z) \ z = x * (((y * x) / y) \ z) by Theorem 1065. Then R(y \ T(y,x),x,((y * x) / y) \ z) \ z = x * (((y * x) / y) \ z) by Theorem 981. Then R(y \ T(y,x),x,((y * x) / y) \ z) \ z = (T(y,x) \ y) * z by (L12). Hence we are done by Theorem 1041.
Theorem 1282 (prov9_f5e09a7cae): (T(y,x) \ y) * (T(y,x) * z) = y * z
Proof. We have (y \ T(y,x)) * (y * z) = T(y,x) * z by Theorem 1018. Then (y \ T(y,x)) \ (T(y,x) * z) = y * z by Proposition 2. Hence we are done by Theorem 1281.
Theorem 1283 (prov9_1e31e144c5): K(y,x) \ x = x * K(x,y)
Proof. We have K(T(y,x),x) \ x = x * K(x,y) by Theorem 1038. Hence we are done by Theorem 1053.
Theorem 1284 (prov9_4a386abea2): T(x * y,K(y,x)) = y * (x * K(x,y))
Proof. We have (T(y,R(x,y,(x * y) \ y)) \ y) * y = y * (T(y,R(x,y,(x * y) \ y)) \ y) by Theorem 931. Then T(y,T(y,R(x,y,(x * y) \ y)) \ y) = y by Theorem 11. Then T(y,K(y,x) \ e) = y by Theorem 949. Then
(L14) T(K(y,x) \ e,y) = K(y,x) \ e
by Proposition 21. We have (y \ (K(y,x) \ y)) * y = (K(y,x) * T(K(y,x) \ e,y)) * (K(y,x) \ y) by Theorem 1099. Then
(L16) (y \ (K(y,x) \ y)) * y = (K(y,x) * (K(y,x) \ e)) * (K(y,x) \ y)
by (L14). We have K(y,x) * (K(y,x) \ e) = e by Axiom 4. Then
(L18) e * (K(y,x) \ y) = (y \ (K(y,x) \ y)) * y
by (L16). We have e * ((y \ (K(y,x) \ y)) * y) = (y \ (K(y,x) \ y)) * y by Axiom 1. Then e * (K(y,x) \ y) = e * ((y \ (K(y,x) \ y)) * y) by (L18). Then
(L21) K(y,x) \ y = (y \ (K(y,x) \ y)) * y
by Proposition 9. We have (((y \ (K(y,x) \ y)) * y) * x) / a(y \ (K(y,x) \ y),y,x) = (y \ (K(y,x) \ y)) * (y * x) by Theorem 1056. Then ((K(y,x) \ y) * x) / a(y \ (K(y,x) \ y),y,x) = (y \ (K(y,x) \ y)) * (y * x) by (L21). Then
(L24) ((K(y,x) \ y) * x) / a(K(y,x) \ e,y,x) = (y \ (K(y,x) \ y)) * (y * x)
by Theorem 957. We have L(x,y,K(T(x,y),y)) = x by Theorem 1044. Then a(K(T(x,y),y),y,x) = e by Proposition 20. Then a(K(y,x) \ e,y,x) = e by Theorem 1037. Then
(L25) ((K(y,x) \ y) * x) / e = (y \ (K(y,x) \ y)) * (y * x)
by (L24). We have ((K(y,x) \ y) * x) / e = (K(y,x) \ y) * x by Proposition 27. Then (y \ (K(y,x) \ y)) * (y * x) = (K(y,x) \ y) * x by (L25). Then
(L28) (K(y,x) \ e) * (y * x) = (K(y,x) \ y) * x
by Theorem 957. We have (K(y,x) \ e) * (K(y,x) * (K(y,x) \ (y * x))) = K(y,x) \ (y * x) by Theorem 1051. Then (K(y,x) \ e) * (y * x) = K(y,x) \ (y * x) by Axiom 4. Then
(L29) K(y,x) \ (y * x) = (K(y,x) \ y) * x
by (L28). We have (K(T(x,y),y) \ e) \ y = y * K(T(x,y),y) by Theorem 1263. Then
(L2) K(y,x) \ y = y * K(T(x,y),y)
by Theorem 1036. We have (y * K(T(x,y),y)) * x = y * (K(T(x,y),y) * x) by Theorem 1046. Then (K(y,x) \ y) * x = y * (K(T(x,y),y) * x) by (L2). Then y * (K(y,x) \ x) = (K(y,x) \ y) * x by Theorem 1052. Then
(L30) K(y,x) \ (y * x) = y * (K(y,x) \ x)
by (L29). We have K(y,x) \ (y * x) = T(x * y,K(y,x)) by Theorem 1059. Then y * (K(y,x) \ x) = T(x * y,K(y,x)) by (L30). Hence we are done by Theorem 1283.
Theorem 1285 (prov9_3e158803ed): T(x * y,x) / R(x,y,x) = y
Proof. We have (x * y) * R(T(x,x * y),y,x) = R(x,y,x) * (x * y) by Proposition 59. Then
(L7) (x * y) * T(x,T(y,x)) = R(x,y,x) * (x * y)
by Theorem 1252. We have R(x,y,x) * (x * y) = x * (R(x,y,x) * y) by Theorem 227. Then
(L9) (x * y) * T(x,T(y,x)) = x * (R(x,y,x) * y)
by (L7). We have T(x * y,T(x,y)) = T(x * y,x) by Theorem 1244. Then
(L11) T(x * y,T(x,T(y,x))) = T(x * y,x)
by Theorem 1032. We have T(x * y,T(x,T(y,x))) / (((x * y) * T(x,T(y,x))) / (x * y)) = (x * y) / T(x,T(y,x)) by Theorem 1239. Then T(x * y,x) / (((x * y) * T(x,T(y,x))) / (x * y)) = (x * y) / T(x,T(y,x)) by (L11). Then
(L14) T(x * y,x) / ((x * (R(x,y,x) * y)) / (x * y)) = (x * y) / T(x,T(y,x))
by (L9). We have R((x * (x * y)) / (x * y),y,x) * (x * y) = (x * y) * R((x * y) \ (x * (x * y)),y,x) by Proposition 83. Then R((x * (x * y)) / (x * y),y,x) * (x * y) = x * (y * R(y \ (x * y),y,x)) by Theorem 516. Then R((x * (x * y)) / (x * y),y,x) * (x * y) = x * (R((x * y) / y,y,x) * y) by Proposition 83. Then R((x * (x * y)) / (x * y),y,x) * (x * y) = x * (R(x,y,x) * y) by Axiom 5. Then (x * (R(x,y,x) * y)) / (x * y) = R((x * (x * y)) / (x * y),y,x) by Proposition 1. Then T(x * y,x) / R((x * (x * y)) / (x * y),y,x) = (x * y) / T(x,T(y,x)) by (L14). Then
(L16) T(x * y,x) / R(x,y,x) = (x * y) / T(x,T(y,x))
by Axiom 5. We have T((T(y,x) * T(x,y)) \ ((T(y,x) * x) * T(x,y)),T(y,x)) = R(T(x,T(y,x) * T(x,y)),T(y,x),T(x,y)) by Theorem 519. Then T(x,T(y,x)) = R(T(x,T(y,x) * T(x,y)),T(y,x),T(x,y)) by Theorem 1105. Then (T(y,x) * T(x,y)) \ ((y * x) * T(x,y)) = T(x,T(y,x)) by Theorem 1084. Then
(L21) (T(y,x) * T(x,y)) \ ((x * y) * x) = T(x,T(y,x))
by Theorem 114. We have ((x * y) * x) / ((T(y,x) * T(x,y)) \ ((x * y) * x)) = T(y,x) * T(x,y) by Proposition 24. Then
(L23) ((x * y) * x) / T(x,T(y,x)) = T(y,x) * T(x,y)
by (L21). We have ((x * y) * x) / T(x,T(y,x)) = ((x * y) / T(x,T(y,x))) * x by Theorem 239. Then
(L25) T(y,x) * T(x,y) = ((x * y) / T(x,T(y,x))) * x
by (L23). We have (T(y,x) * (x \ T(x,y))) * x = T(y,x) * T(x,y) by Theorem 1015. Then T(y,x) * (x \ T(x,y)) = (x * y) / T(x,T(y,x)) by (L25) and Proposition 8. Then
(L27) T(x * y,x) / R(x,y,x) = T(y,x) * (x \ T(x,y))
by (L16). We have T(y,x) * (x \ T(x,y)) = y by Theorem 1031. Hence we are done by (L27).
Theorem 1286 (prov9_b0e2751627): x * R(y,x,y) = T(y * x,y)
Proof. We have (T(y * x,y) / R(y,x,y)) * R(y,x,y) = T(y * x,y) by Axiom 6. Hence we are done by Theorem 1285.
Theorem 1287 (prov9_22ed0c9349): K(y \ T(x,y),y) = K(y \ x,y)
Proof. We have ((((y \ x) * y) * y) / ((y \ x) * y)) * (y \ x) = (y \ x) * R(y,y \ x,y) by Theorem 482. Then
(L4) ((y \ x) \ T((y \ x) * y,y)) * (y \ x) = (y \ x) * R(y,y \ x,y)
by Theorem 1246. We have (((y \ x) \ T((y \ x) * y,y)) * (y \ x)) \ T((y \ x) * y,y) = K(y \ x,(y \ x) \ T((y \ x) * y,y)) by Theorem 1055. Then
(L6) ((y \ x) * R(y,y \ x,y)) \ T((y \ x) * y,y) = K(y \ x,(y \ x) \ T((y \ x) * y,y))
by (L4). We have K(y \ x,T((y \ x) \ T((y \ x) * y,y),T((y \ x) * y,y))) = K(y \ x,(y \ x) \ T((y \ x) * y,y)) by Theorem 1260. Then K(y \ x,y) = K(y \ x,(y \ x) \ T((y \ x) * y,y)) by Theorem 1278. Then ((y \ x) * R(y,y \ x,y)) \ T((y \ x) * y,y) = K(y \ x,y) by (L6). Then
(L8) K(y \ x,y) = T(y * (y \ x),y) \ T((y \ x) * y,y)
by Theorem 1286. We have T(y * (y \ x),y) \ ((y \ T(y * (y \ x),y)) * y) = K(y \ T(y * (y \ x),y),y) by Theorem 2. Then T(y * (y \ x),y) \ T((y \ (y * (y \ x))) * y,y) = K(y \ T(y * (y \ x),y),y) by Theorem 310. Then T(y * (y \ x),y) \ T((y \ x) * y,y) = K(y \ T(y * (y \ x),y),y) by Axiom 3. Then K(y \ x,y) = K(y \ T(y * (y \ x),y),y) by (L8). Hence we are done by Axiom 4.
Theorem 1288 (prov9_dd775886e1): K(((x * y) / x) \ x,y) = K(y \ x,y)
Proof. We have K(y \ T((y * x) / y,y),y) = K(y \ ((y * x) / y),y) by Theorem 1287. Then K(y \ x,y) = K(y \ ((y * x) / y),y) by Theorem 7. Hence we are done by Theorem 1279.
Theorem 1289 (prov9_6180d9a2f5): T(y,(x \ y) * x) = y
Proof. We have (y / ((y * x) / y)) * T((y * x) / y,y) = (((y * x) / y) * y) / ((y * x) / y) by Theorem 818. Then (y / ((y * x) / y)) * T((y * x) / y,y) = (y * x) / ((y * x) / y) by Axiom 6. Then (y / ((y * x) / y)) * x = (y * x) / ((y * x) / y) by Theorem 7. Then
(L4) y * (T(y,x) \ y) = (y / ((y * x) / y)) * x
by Theorem 1011. We have T(y / ((y * x) / y),x) = x \ ((y / ((y * x) / y)) * x) by Definition 3. Then
(L6) T(y / ((y * x) / y),x) = x \ (y * (T(y,x) \ y))
by (L4). We have T(y / ((y * x) / y),T((y * x) / y,y)) = ((y * x) / y) \ y by Theorem 1234. Then T(y / ((y * x) / y),x) = ((y * x) / y) \ y by Theorem 7. Then
(L9) x \ (y * (T(y,x) \ y)) = ((y * x) / y) \ y
by (L6). We have ((y * x) / y) * T(((y * x) / y) \ y,x) = y * K(x \ (x / (y \ ((y * x) / y))),x) by Theorem 284. Then
(L11) ((y * x) / y) * T(((y * x) / y) \ y,x) = y * K(x \ (y * (T(y,x) \ y)),x)
by Theorem 1255. We have ((y * x) / y) * T(((y * x) / y) \ y,x) = (y * (x * T(x \ y,x))) / y by Theorem 1117. Then y * K(x \ (y * (T(y,x) \ y)),x) = (y * (x * T(x \ y,x))) / y by (L11). Then y * K(x \ (y * (T(y,x) \ y)),x) = (y * ((x \ y) * x)) / y by Proposition 46. Then
(L15) y * K(((y * x) / y) \ y,x) = (y * ((x \ y) * x)) / y
by (L9). We have
(L17) T(y,(x \ y) * x) * (y \ ((x \ y) * x)) = (y * ((x \ y) * x)) / y
by Theorem 996.
T(y,(x \ y) * x) * K(x \ y,x)
=(y * ((x \ y) * x)) / y by (L17), Theorem 2
=y * K(x \ y,x) by (L15), Theorem 1288

Then T(y,(x \ y) * x) * K(x \ y,x) = y * K(x \ y,x). Hence we are done by Proposition 10.
Theorem 1290 (prov9_c28a218b94): T((y \ x) * y,x) = (y \ x) * y
Proof. We have T(x,(y \ x) * y) = x by Theorem 1289. Hence we are done by Proposition 21.
Theorem 1291 (prov9_e1471f8f54): T(y * (x / y),x) = y * (x / y)
Proof. We have (x / y) * y = x by Axiom 6. Then
(L2) (y \ (y * (x / y))) * y = x
by Axiom 3. We have T(y * (x / y),(y \ (y * (x / y))) * y) = y * (x / y) by Theorem 1289. Hence we are done by (L2).
Theorem 1292 (prov9_bad944e60c): R(z,K(x \ e,x),y) = z
Proof. We have R(e / (x \ e),K((x \ e) \ e,x \ e),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) * T(R(e / (x \ e),K((x \ e) \ e,x \ e),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) \ e,z) = R((e / (x \ e)) * T((e / (x \ e)) \ e,z),K((x \ e) \ e,x \ e),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 1139. Then (e / (x \ e)) * T(R(e / (x \ e),K((x \ e) \ e,x \ e),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) \ e,z) = R((e / (x \ e)) * T((e / (x \ e)) \ e,z),K((x \ e) \ e,x \ e),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 1160. Then
(L34) R((e / (x \ e)) * T((e / (x \ e)) \ e,z),K((x \ e) \ e,x \ e),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) = (e / (x \ e)) * T((e / (x \ e)) \ e,z)
by Theorem 1160. We have (e / (x \ e)) * T((e / (x \ e)) \ e,z) = (x \ e) \ T(x \ e,z) by Theorem 1199. Then R((e / (x \ e)) * T((e / (x \ e)) \ e,z),K((x \ e) \ e,x \ e),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) = (x \ e) \ T(x \ e,z) by (L34). Then
(L37) R((x \ e) \ T(x \ e,z),K((x \ e) \ e,x \ e),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) = (x \ e) \ T(x \ e,z)
by Theorem 1199. We have R(T(x \ e,z) * ((x \ e) \ e),K((x \ e) \ e,T(x \ e,z)),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) \ ((((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = K((x \ e) \ e,T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 1068. Then R((x \ e) \ T(x \ e,z),K((x \ e) \ e,T(x \ e,z)),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) \ ((((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = K((x \ e) \ e,T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 146. Then R((x \ e) \ T(x \ e,z),K((x \ e) \ e,x \ e),(x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) \ ((((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = K((x \ e) \ e,T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 1205. Then ((x \ e) \ T(x \ e,z)) \ ((((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = K((x \ e) \ e,T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by (L37). Then
(L42) ((x \ e) \ T(x \ e,z)) \ ((((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = K((x \ e) \ e,x \ e) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))
by Theorem 1205. We have ((x \ e) \ T(x \ e,z)) * (((x \ e) \ T(x \ e,z)) \ ((((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) = (((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Axiom 4. Then ((x \ e) \ T(x \ e,z)) * (K((x \ e) \ e,x \ e) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = (((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by (L42). Then ((x \ e) \ T(x \ e,z)) * (((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) * K((x \ e) \ e,x \ e)) = (((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 923. Then
(L46) T(x \ e,z) * ((x \ e) \ (((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) * K((x \ e) \ e,x \ e))) = (((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))
by Theorem 1270. We have (x \ e) \ (K((x \ e) \ e,x \ e) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = T(x \ e,(x \ e) \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 1172. Then (x \ e) \ (((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) * K((x \ e) \ e,x \ e)) = T(x \ e,(x \ e) \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 923. Then
(L47) T(x \ e,z) * (T(x \ e,(x \ e) \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = (((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))
by (L46). We have (((x \ e) \ e) \ e) \ (K(x \ e,(x \ e) \ e) * (K((x \ e) \ e,x \ e) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) = (x \ e) \ (K((x \ e) \ e,x \ e) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) by Theorem 1152. Then (((x \ e) \ e) \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) = (x \ e) \ (K((x \ e) \ e,x \ e) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) by Theorem 442. Then
(L50) (x \ e) \ (K(x,x \ e) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = (((x \ e) \ e) \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))
by Theorem 733. We have T(x \ e,z) * (K(x,x \ e) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) = T(e / x,z) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) by Theorem 1209. Then T(x \ e,z) * ((x \ e) \ (K(x,x \ e) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) = T(e / x,z) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) by Theorem 600. Then T(x \ e,z) * ((((x \ e) \ e) \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = T(e / x,z) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) by (L50). Then T(x \ e,z) * (T(x \ e,(x \ e) \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = T(e / x,z) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) by Proposition 49. Then (((x \ e) \ e) * T(x \ e,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) = T(e / x,z) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) by (L47). Then
(L56) (x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) = T(e / x,z) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))
by Theorem 1206. We have (e / x) * (T(e / x,z) * ((e / x) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) = T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 830. Then (e / x) * (T(x \ e,z) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) = T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 1210. Then (x \ e) * (T(e / x,z) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) = T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Theorem 1214. Then (T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / (T(e / x,z) * ((x \ e) \ ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) = x \ e by Proposition 1. Then
(L61) (T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = x \ e
by (L56). We have ((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * (((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) \ (((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * (x * (x \ ((e / x) \ y))))) = ((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * (x * (x \ ((e / x) \ y))) by Axiom 4. Then ((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) \ (((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * (x * (x \ ((e / x) \ y))))))) = ((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * (x * (x \ ((e / x) \ y))) by Axiom 4. Then
(L64) ((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = ((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * (x * (x \ ((e / x) \ y)))
by Axiom 3. We have ((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) = T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by Axiom 6. Then ((T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))) / ((x * T(e / x,z)) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))))) * (x * (x \ ((e / x) \ y))) = T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by (L64). Then
(L67) (x \ e) * (x * (x \ ((e / x) \ y))) = T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))))
by (L61). We have T(z,x \ e) * (z \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y)))))) = (T(x \ e,z) \ (x \ e)) * (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) by Theorem 1275. Then T(z,x \ e) * (z \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y)))))) = ((x \ e) \ T(x \ e,z)) \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) by Theorem 1281. Then
(L7) (T(z,x \ e) \ z) \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) = T(z,x \ e) * (z \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))))
by Theorem 1030. We have L(T(e / x,z) \ (x \ ((e / x) \ y)),T(x \ e,z),x) = (x * T(x \ e,z)) \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) by Definition 4. Then L(T(e / x,z) \ (x \ ((e / x) \ y)),T(x \ e,z),x) = (T(z,x \ e) \ z) \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) by Theorem 1028. Then
(L8) L(T(e / x,z) \ (x \ ((e / x) \ y)),T(x \ e,z),x) = T(z,x \ e) * (z \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))))
by (L7). We have (x * ((z * (x \ e)) / z)) * (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) = x * ((x * ((z * (x \ e)) / z)) * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) by Theorem 1128. Then K(z,(z / x) / z) * (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) = x * ((x * ((z * (x \ e)) / z)) * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) by Theorem 333. Then K(z,(z / x) / z) * (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) = x * ((z \ T(z,x \ e)) * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) by Theorem 980. Then (z \ T(z,x \ e)) * (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) = x * ((z \ T(z,x \ e)) * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) by Theorem 1004. Then T(z,x \ e) * (z \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y)))))) = x * ((z \ T(z,x \ e)) * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) by Theorem 1270. Then T(z,x \ e) * (z \ (x * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y)))))) = x * (T(z,x \ e) * (z \ (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y)))))) by Theorem 1270. Then
(L15) L(T(e / x,z) \ (x \ ((e / x) \ y)),T(x \ e,z),x) = x * (T(z,x \ e) * (z \ (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))))
by (L8). We have T(z,x \ e) * (z \ (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) = (T(x \ e,z) \ (x \ e)) * (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y)))) by Theorem 1275. Then T(z,x \ e) * (z \ (T(x \ e,z) * (T(e / x,z) \ (x \ ((e / x) \ y))))) = (x \ e) * (T(e / x,z) \ (x \ ((e / x) \ y))) by Theorem 1282. Then
(L16) x * ((x \ e) * (T(e / x,z) \ (x \ ((e / x) \ y)))) = L(T(e / x,z) \ (x \ ((e / x) \ y)),T(x \ e,z),x)
by (L15). We have x * ((x \ e) * (T(e / x,z) \ (x \ ((e / x) \ y)))) = L(T(e / x,z) \ (x \ ((e / x) \ y)),x \ e,x) by Proposition 56. Then
(L18) L(T(e / x,z) \ (x \ ((e / x) \ y)),T(x \ e,z),x) = L(T(e / x,z) \ (x \ ((e / x) \ y)),x \ e,x)
by (L16). We have L(T(e / x,z) \ (x \ ((e / x) \ y)),T(e / x,z),x) = (x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))) by Proposition 53. Then L(T(e / x,z) \ (x \ ((e / x) \ y)),T(x \ e,z),x) = (x * T(e / x,z)) \ (x * (x \ ((e / x) \ y))) by Theorem 1228. Then
(L21) L(T(e / x,z) \ (x \ ((e / x) \ y)),x \ e,x) = (x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))
by (L18). We have L(x \ L((e / x) \ ((z * (x \ ((e / x) \ y))) / z),x \ e,x),x,x \ e) = (e / x) * L((e / x) \ ((z * (x \ ((e / x) \ y))) / z),x \ e,x) by Theorem 434. Then
(L25) L((x \ e) * ((e / x) \ ((z * (x \ ((e / x) \ y))) / z)),x,x \ e) = (e / x) * L((e / x) \ ((z * (x \ ((e / x) \ y))) / z),x \ e,x)
by Proposition 57. We have T((e / x) * L((e / x) \ ((z * (x \ ((e / x) \ y))) / z),x \ e,x),z) = T(e / x,z) * L(T(e / x,z) \ (x \ ((e / x) \ y)),x \ e,x) by Theorem 879. Then
(L27) T(L((x \ e) * ((e / x) \ ((z * (x \ ((e / x) \ y))) / z)),x,x \ e),z) = T(e / x,z) * L(T(e / x,z) \ (x \ ((e / x) \ y)),x \ e,x)
by (L25). We have L(((z * (x \ ((e / x) \ y))) / z) * K(x \ e,x),x,x \ e) = (x \ e) * (x * ((z * (x \ ((e / x) \ y))) / z)) by Theorem 1195. Then L((x \ e) * ((e / x) \ ((z * (x \ ((e / x) \ y))) / z)),x,x \ e) = (x \ e) * (x * ((z * (x \ ((e / x) \ y))) / z)) by Theorem 1190. Then T((x \ e) * (x * ((z * (x \ ((e / x) \ y))) / z)),z) = T(e / x,z) * L(T(e / x,z) \ (x \ ((e / x) \ y)),x \ e,x) by (L27). Then T((x \ e) * (x * ((z * (x \ ((e / x) \ y))) / z)),z) = T(e / x,z) * ((x * T(e / x,z)) \ (x * (x \ ((e / x) \ y)))) by (L21). Then
(L68) T((x \ e) * (x * ((z * (x \ ((e / x) \ y))) / z)),z) = (x \ e) * (x * (x \ ((e / x) \ y)))
by (L67). We have (x \ e) * ((e / x) \ ((z * ((e / x) * ((e / x) \ y))) / z)) = K(x \ e,x) * ((z * ((e / x) * ((e / x) \ y))) / z) by Theorem 1161. Then
(L70) (x \ e) * (x * ((z * (x \ ((e / x) \ y))) / z)) = K(x \ e,x) * ((z * ((e / x) * ((e / x) \ y))) / z)
by Theorem 1104. We have T((K(x \ e,x) * (z * ((e / x) * ((e / x) \ y)))) / z,z) = z \ (K(x \ e,x) * (z * ((e / x) * ((e / x) \ y)))) by Proposition 47. Then T(K(x \ e,x) * ((z * ((e / x) * ((e / x) \ y))) / z),z) = z \ (K(x \ e,x) * (z * ((e / x) * ((e / x) \ y)))) by Theorem 969. Then
(L73) T((x \ e) * (x * ((z * (x \ ((e / x) \ y))) / z)),z) = z \ (K(x \ e,x) * (z * ((e / x) * ((e / x) \ y))))
by (L70). We have L((x \ e) * ((e / x) \ y),K(x,x \ e),z) = (z * K(x,x \ e)) \ (z * (K(x,x \ e) * ((x \ e) * ((e / x) \ y)))) by Definition 4. Then L((x \ e) * ((e / x) \ y),K(x,x \ e),z) = (z * K(x,x \ e)) \ (z * ((e / x) * ((e / x) \ y))) by Theorem 697. Then z \ (K(x \ e,x) * (z * ((e / x) * ((e / x) \ y)))) = L((x \ e) * ((e / x) \ y),K(x,x \ e),z) by Theorem 1223. Then T((x \ e) * (x * ((z * (x \ ((e / x) \ y))) / z)),z) = L((x \ e) * ((e / x) \ y),K(x,x \ e),z) by (L73). Then (x \ e) * (x * (x \ ((e / x) \ y))) = L((x \ e) * ((e / x) \ y),K(x,x \ e),z) by (L68). Then
(L79) L((x \ e) * ((e / x) \ y),K(x,x \ e),z) = (x \ e) * ((e / x) \ y)
by Axiom 4. We have (z * (K(x,x \ e) * ((x \ e) * ((e / x) \ y)))) / L((x \ e) * ((e / x) \ y),K(x,x \ e),z) = z * K(x,x \ e) by Theorem 453. Then (z * ((e / x) * ((e / x) \ y))) / L((x \ e) * ((e / x) \ y),K(x,x \ e),z) = z * K(x,x \ e) by Theorem 697. Then
(L82) (z * ((e / x) * ((e / x) \ y))) / ((x \ e) * ((e / x) \ y)) = z * K(x,x \ e)
by (L79). We have R(z,K(x \ e,x),(e / x) * ((e / x) \ y)) = ((z * K(x \ e,x)) * ((e / x) * ((e / x) \ y))) / (K(x \ e,x) * ((e / x) * ((e / x) \ y))) by Definition 5. Then R(z,K(x \ e,x),(e / x) * ((e / x) \ y)) = ((z * K(x \ e,x)) * ((e / x) * ((e / x) \ y))) / ((x \ e) * ((e / x) \ y)) by Theorem 682. Then
(L85) (K(x \ e,x) * (z * ((e / x) * ((e / x) \ y)))) / ((x \ e) * ((e / x) \ y)) = R(z,K(x \ e,x),(e / x) * ((e / x) \ y))
by Theorem 1222. We have (K(x \ e,x) * (z * ((e / x) * ((e / x) \ y)))) / ((x \ e) * ((e / x) \ y)) = K(x \ e,x) * ((z * ((e / x) * ((e / x) \ y))) / ((x \ e) * ((e / x) \ y))) by Theorem 969. Then R(z,K(x \ e,x),(e / x) * ((e / x) \ y)) = K(x \ e,x) * ((z * ((e / x) * ((e / x) \ y))) / ((x \ e) * ((e / x) \ y))) by (L85). Then
(L88) K(x \ e,x) * (z * K(x,x \ e)) = R(z,K(x \ e,x),(e / x) * ((e / x) \ y))
by (L82). We have K(x \ e,x) * (z * K(x,x \ e)) = z by Theorem 1182. Then R(z,K(x \ e,x),(e / x) * ((e / x) \ y)) = z by (L88). Hence we are done by Axiom 4.
Theorem 1293 (prov9_0a0a51f1b1): x = R(x,y,K(z \ e,z))
Proof. We have R(x,K(z \ e,(z \ e) \ e),y) = ((x * K(z \ e,(z \ e) \ e)) * y) / (K(z \ e,(z \ e) \ e) * y) by Definition 5. Then
(L7) R(x,K(z \ e,(z \ e) \ e),y) = (K(z \ e,(z \ e) \ e) * (x * y)) / (K(z \ e,(z \ e) \ e) * y)
by Theorem 1231. We have
(L1) (K(z \ e,(z \ e) \ e) * (x * y)) / (K(z \ e,(z \ e) \ e) * y) = K(z \ e,(z \ e) \ e) * ((x * y) / (K(z \ e,(z \ e) \ e) * y))
by Theorem 1226. Then
(L2) ((x * y) * K(z \ e,(z \ e) \ e)) / (K(z \ e,(z \ e) \ e) * y) = K(z \ e,(z \ e) \ e) * ((x * y) / (K(z \ e,(z \ e) \ e) * y))
by Theorem 1183. We have ((x * y) * T(K(z \ e,(z \ e) \ e),y)) / (K(z \ e,(z \ e) \ e) * y) = R(x,y,T(K(z \ e,(z \ e) \ e),y)) by Theorem 787. Then ((x * y) * K(z \ e,(z \ e) \ e)) / (K(z \ e,(z \ e) \ e) * y) = R(x,y,T(K(z \ e,(z \ e) \ e),y)) by Theorem 723. Then
(L5) K(z \ e,(z \ e) \ e) * ((x * y) / (K(z \ e,(z \ e) \ e) * y)) = R(x,y,T(K(z \ e,(z \ e) \ e),y))
by (L2). We have R(x,K(z \ e,(z \ e) \ e),y) = K(z \ e,(z \ e) \ e) * ((x * y) / (K(z \ e,(z \ e) \ e) * y)) by (L7) and (L1). Then R(x,y,T(K(z \ e,(z \ e) \ e),y)) = R(x,K(z \ e,(z \ e) \ e),y) by (L5). Then R(x,y,K(z \ e,(z \ e) \ e)) = R(x,K(z \ e,(z \ e) \ e),y) by Theorem 723. Then R(x,K(z \ e,z),y) = R(x,y,K(z \ e,(z \ e) \ e)) by Theorem 779. Then R(x,y,K(z \ e,z)) = R(x,K(z \ e,z),y) by Theorem 779. Hence we are done by Theorem 1292.
Theorem 1294 (prov9_df91bd9450): x \ (K(z \ e,z) * y) = (x \ y) * K(z \ e,z)
Proof. We have R(x,x \ y,K(z \ e,z)) \ (K(z \ e,z) * y) = (x \ y) * K(z \ e,z) by Theorem 1189. Hence we are done by Theorem 1293.
Theorem 1295 (prov9_e6be2f16fd): y * (K(x \ e,x) * z) = K(x \ e,x) * (y * z)
Proof. We have R(y,K(x \ e,x),z) * (K(x \ e,x) * z) = (y * K(x \ e,x)) * z by Proposition 54. Then
(L2) y * (K(x \ e,x) * z) = (y * K(x \ e,x)) * z
by Theorem 1292. We have (y * K(x \ e,x)) * z = K(x \ e,x) * (y * z) by Theorem 1222. Hence we are done by (L2).
Theorem 1296 (prov9_2eac628d37): L(y,z,w) * K(x \ e,x) = L(K(x \ e,x) * y,z,w)
Proof. We have (w * z) \ (K(x \ e,x) * (w * (z * y))) = ((w * z) \ (w * (z * y))) * K(x \ e,x) by Theorem 1294. Then (w * z) \ (w * (K(x \ e,x) * (z * y))) = ((w * z) \ (w * (z * y))) * K(x \ e,x) by Theorem 1295. Then
(L3) (w * z) \ (w * (K(x \ e,x) * (z * y))) = L(y,z,w) * K(x \ e,x)
by Definition 4. We have L(K(x \ e,x) * y,z,w) = (w * z) \ (w * (z * (K(x \ e,x) * y))) by Definition 4. Then L(K(x \ e,x) * y,z,w) = (w * z) \ (w * (K(x \ e,x) * (z * y))) by Theorem 1295. Hence we are done by (L3).
Theorem 1297 (prov9_01340bbfb0): K(x \ e,x) * L(y,z,w) = L(K(x \ e,x) * y,z,w)
Proof. We have K(x \ e,x) * L(y,z,w) = L(y,z,w) * K(x \ e,x) by Theorem 923. Hence we are done by Theorem 1296.
Theorem 1298 (prov9_1f156c68fd): (y \ L(y * x,z,w)) * y = L(x * y,z,w)
Proof. We have K(y \ e,y) * L(L(y \ ((y \ e) \ (((x * y) / y) * y)),z,w),y,y \ e) = (y \ e) * (y * L(y \ ((y \ e) \ (((x * y) / y) * y)),z,w)) by Theorem 59. Then K(y \ e,y) * L(((y \ e) * y) \ (((x * y) / y) * y),z,w) = (y \ e) * (y * L(y \ ((y \ e) \ (((x * y) / y) * y)),z,w)) by Theorem 1070. Then
(L5) K(y \ e,y) * L(K(y \ e,y) \ (((x * y) / y) * y),z,w) = (y \ e) * (y * L(y \ ((y \ e) \ (((x * y) / y) * y)),z,w))
by Proposition 76. We have (y \ e) * (y * L(y \ ((y \ e) \ (((x * y) / y) * y)),z,w)) = ((y \ e) * L((y \ e) \ ((x * y) / y),z,w)) * y by Theorem 1120. Then K(y \ e,y) * L(K(y \ e,y) \ (((x * y) / y) * y),z,w) = ((y \ e) * L((y \ e) \ ((x * y) / y),z,w)) * y by (L5). Then K(y \ e,y) * L(K(y \ e,y) \ (((x * y) / y) * y),z,w) = (y \ L(y * ((x * y) / y),z,w)) * y by Theorem 827. Then K(y \ e,y) * L(K(y,y \ e) * (((x * y) / y) * y),z,w) = (y \ L(y * ((x * y) / y),z,w)) * y by Theorem 696. Then K(y \ e,y) * L(((x * y) / y) * T(y,y \ e),z,w) = (y \ L(y * ((x * y) / y),z,w)) * y by Theorem 925. Then
(L11) K(y \ e,y) * L((x * y) * K(y,y \ e),z,w) = (y \ L(y * ((x * y) / y),z,w)) * y
by Theorem 1142. We have L(K(y \ e,y) * ((x * y) * K(y,y \ e)),z,w) = K(y \ e,y) * L((x * y) * K(y,y \ e),z,w) by Theorem 1297. Then L(x * y,z,w) = K(y \ e,y) * L((x * y) * K(y,y \ e),z,w) by Theorem 1182. Then L(x * y,z,w) = (y \ L(y * ((x * y) / y),z,w)) * y by (L11). Hence we are done by Axiom 5.
Theorem 1299 (prov9_8256262860): (y \ L(x,z,w)) * y = L((y \ x) * y,z,w)
Proof. We have (y \ L(y * (y \ x),z,w)) * y = L((y \ x) * y,z,w) by Theorem 1298. Hence we are done by Axiom 4.
Theorem 1300 (prov9_f8d0f6ab29): ((y \ e) * x) * (x / ((y \ e) * x)) = (y \ x) * y
Proof. We have ((((((e / y) * x) \ x) \ e) \ e) \ x) * (((((e / y) * x) \ x) \ e) \ e) = ((((e / y) * x) \ x) \ x) * (((e / y) * x) \ x) by Theorem 1187. Then ((((y \ e) * x) \ x) \ x) * (((((e / y) * x) \ x) \ e) \ e) = ((((e / y) * x) \ x) \ x) * (((e / y) * x) \ x) by Theorem 1212. Then
(L3) ((((y \ e) * x) \ x) \ x) * (((y \ e) * x) \ x) = ((((e / y) * x) \ x) \ x) * (((e / y) * x) \ x)
by Theorem 1212. We have x \ (((((e / y) * x) \ x) \ x) * (((e / y) * x) \ x)) = K((((e / y) * x) \ x) \ x,((e / y) * x) \ x) by Theorem 2. Then x \ (((((e / y) * x) \ x) \ x) * (((e / y) * x) \ x)) = K((((e / y) * x) \ x) \ x,((y \ e) * x) \ x) by Theorem 1219. Then
(L6) x \ (((((y \ e) * x) \ x) \ x) * (((y \ e) * x) \ x)) = K((((e / y) * x) \ x) \ x,((y \ e) * x) \ x)
by (L3). We have x \ (((((y \ e) * x) \ x) \ x) * (((y \ e) * x) \ x)) = K((((y \ e) * x) \ x) \ x,((y \ e) * x) \ x) by Theorem 2. Then
(L8) K((((e / y) * x) \ x) \ x,((y \ e) * x) \ x) = K((((y \ e) * x) \ x) \ x,((y \ e) * x) \ x)
by (L6). We have K((((e / y) * x) \ x) \ x,((e / y) * x) \ x) = K((((e / y) * x) \ x) \ x,((y \ e) * x) \ x) by Theorem 1219. Then K((e / y) * x,x / ((e / y) * x)) = K((((e / y) * x) \ x) \ x,((y \ e) * x) \ x) by Theorem 1251. Then
(L11) K((((y \ e) * x) \ x) \ x,((y \ e) * x) \ x) = K((e / y) * x,x / ((e / y) * x))
by (L8). We have K((((y \ e) * x) \ x) \ x,((y \ e) * x) \ x) = K((y \ e) * x,x / ((y \ e) * x)) by Theorem 1251. Then
(L13) K((e / y) * x,x / ((e / y) * x)) = K((y \ e) * x,x / ((y \ e) * x))
by (L11). We have x * K((e / y) * x,x / ((e / y) * x)) = ((e / y) * x) * (x / ((e / y) * x)) by Theorem 459. Then
(L15) x * K((y \ e) * x,x / ((y \ e) * x)) = ((e / y) * x) * (x / ((e / y) * x))
by (L13). We have x * K((y \ e) * x,x / ((y \ e) * x)) = ((y \ e) * x) * (x / ((y \ e) * x)) by Theorem 459. Then
(L17) ((e / y) * x) * (x / ((e / y) * x)) = ((y \ e) * x) * (x / ((y \ e) * x))
by (L15). We have (e / y) \ T((e / y) * x,(e / y) \ e) = y * T(y \ x,(e / y) \ e) by Theorem 1080. Then
(L19) (e / y) \ ((e / y) * (((e / y) * x) * (x / ((e / y) * x)))) = y * T(y \ x,(e / y) \ e)
by Theorem 1247. We have (e / y) \ ((e / y) * (((e / y) * x) * (x / ((e / y) * x)))) = ((e / y) * x) * (x / ((e / y) * x)) by Axiom 3. Then y * T(y \ x,(e / y) \ e) = ((e / y) * x) * (x / ((e / y) * x)) by (L19). Then ((y \ e) * x) * (x / ((y \ e) * x)) = y * T(y \ x,(e / y) \ e) by (L17). Then
(L23) ((y \ e) * x) * (x / ((y \ e) * x)) = y * T(y \ x,y)
by Proposition 25. We have y * T(y \ x,y) = (y \ x) * y by Proposition 46. Hence we are done by (L23).
Theorem 1301 (prov9_fe533093f6): ((y \ e) * x) * R(y,y \ e,x) = (y \ x) * y
Proof. We have ((y \ e) * x) * (x / ((y \ e) * x)) = (y \ x) * y by Theorem 1300. Hence we are done by Proposition 66.
Theorem 1302 (prov9_7b8e3d9d44): T(x,x \ y) = T(x,(x \ e) * y)
Proof. We have x * ((e / x) * ((x \ y) * x)) = L(x \ y,x,e / x) * x by Theorem 1074. Then
(L17) x * ((e / x) * ((x \ y) * x)) = L(x \ y,x,x \ e) * x
by Theorem 1224. We have (((x * ((e / x) * ((x \ y) * x))) / x) \ ((e / x) * ((x \ y) * x))) * ((((x * ((e / x) * ((x \ y) * x))) / x) \ ((e / x) * ((x \ y) * x))) \ ((e / x) * ((x \ y) * x))) = (e / x) * ((x \ y) * x) by Axiom 4. Then (((x * ((e / x) * ((x \ y) * x))) / x) \ ((e / x) * ((x \ y) * x))) * ((x * ((e / x) * ((x \ y) * x))) / x) = (e / x) * ((x \ y) * x) by Theorem 946. Then
(L20) (((L(x \ y,x,x \ e) * x) / x) \ ((e / x) * ((x \ y) * x))) * ((x * ((e / x) * ((x \ y) * x))) / x) = (e / x) * ((x \ y) * x)
by (L17). We have (T(x,(e / x) * ((x \ y) * x)) \ ((((e / x) * ((x \ y) * x)) * T(x,(e / x) * ((x \ y) * x))) / ((e / x) * ((x \ y) * x)))) \ ((e / x) * ((x \ y) * x)) = ((e / x) * ((x \ y) * x)) * (((((e / x) * ((x \ y) * x)) * T(x,(e / x) * ((x \ y) * x))) / ((e / x) * ((x \ y) * x))) \ T(x,(e / x) * ((x \ y) * x))) by Theorem 1216. Then (T(x,(e / x) * ((x \ y) * x)) \ x) \ ((e / x) * ((x \ y) * x)) = ((e / x) * ((x \ y) * x)) * (((((e / x) * ((x \ y) * x)) * T(x,(e / x) * ((x \ y) * x))) / ((e / x) * ((x \ y) * x))) \ T(x,(e / x) * ((x \ y) * x))) by Proposition 48. Then
(L12) ((e / x) * ((x \ y) * x)) * (x \ T(x,(e / x) * ((x \ y) * x))) = (T(x,(e / x) * ((x \ y) * x)) \ x) \ ((e / x) * ((x \ y) * x))
by Proposition 48. We have (T(x,(e / x) * ((x \ y) * x)) \ x) * ((T(x,(e / x) * ((x \ y) * x)) \ x) \ ((e / x) * ((x \ y) * x))) = (e / x) * ((x \ y) * x) by Axiom 4. Then
(L14) (T(x,(e / x) * ((x \ y) * x)) \ x) * (((e / x) * ((x \ y) * x)) * (x \ T(x,(e / x) * ((x \ y) * x)))) = (e / x) * ((x \ y) * x)
by (L12).
(L(x \ y,x,x \ e) \ ((e / x) * ((x \ y) * x))) * ((x * ((e / x) * ((x \ y) * x))) / x)
=(e / x) * ((x \ y) * x) by (L20), Axiom 5
=(T(x,(e / x) * ((x \ y) * x)) \ x) * ((x * ((e / x) * ((x \ y) * x))) / x) by (L14), Theorem 977

Then (L(x \ y,x,x \ e) \ ((e / x) * ((x \ y) * x))) * ((x * ((e / x) * ((x \ y) * x))) / x) = (T(x,(e / x) * ((x \ y) * x)) \ x) * ((x * ((e / x) * ((x \ y) * x))) / x). Then
(L23) L(x \ y,x,x \ e) \ ((e / x) * ((x \ y) * x)) = T(x,(e / x) * ((x \ y) * x)) \ x
by Proposition 10. We have ((e / x) * (x * (x \ y))) \ ((e / x) * ((x \ y) * x)) = L(K(x \ y,x),x * (x \ y),e / x) by Theorem 1058. Then L(x \ y,x,x \ e) \ ((e / x) * ((x \ y) * x)) = L(K(x \ y,x),x * (x \ y),e / x) by Theorem 1158. Then L(x \ y,x,x \ e) \ ((e / x) * ((x \ y) * x)) = L(K(x \ y,x),x * (x \ y),x \ e) by Theorem 1224. Then T(x,(e / x) * ((x \ y) * x)) \ x = L(K(x \ y,x),x * (x \ y),x \ e) by (L23). Then
(L28) L(K(x \ y,x),x * (x \ y),x \ e) = T(x,(x \ e) * ((x \ y) * x)) \ x
by Theorem 1159. We have (((x \ e) * (x * (x \ y))) * R(x,x \ e,x * (x \ y))) / (x * (x \ y)) = ((((x \ e) * (x * (x \ y))) * x) / ((x \ e) * (x * (x \ y)))) * (x \ e) by Theorem 1076. Then
(L2) ((x \ (x * (x \ y))) * x) / (x * (x \ y)) = ((((x \ e) * (x * (x \ y))) * x) / ((x \ e) * (x * (x \ y)))) * (x \ e)
by Theorem 1301. We have ((((x \ e) * (x * (x \ y))) * x) / ((x \ e) * (x * (x \ y)))) * (x \ e) = ((x \ e) * (x * (x \ y))) \ T((x \ e) * (x * (x \ y)),x) by Theorem 983. Then
(L4) ((x \ (x * (x \ y))) * x) / (x * (x \ y)) = ((x \ e) * (x * (x \ y))) \ T((x \ e) * (x * (x \ y)),x)
by (L2). We have ((x \ e) * (x * (x \ y))) \ ((x \ e) * ((x \ (x * (x \ y))) * x)) = L(K(x \ (x * (x \ y)),x),x * (x \ y),x \ e) by Theorem 1060. Then ((x \ e) * (x * (x \ y))) \ T((x \ e) * (x * (x \ y)),x) = L(K(x \ (x * (x \ y)),x),x * (x \ y),x \ e) by Theorem 891. Then ((x \ (x * (x \ y))) * x) / (x * (x \ y)) = L(K(x \ (x * (x \ y)),x),x * (x \ y),x \ e) by (L4). Then L(K(x \ y,x),x * (x \ y),x \ e) = ((x \ (x * (x \ y))) * x) / (x * (x \ y)) by Axiom 3. Then ((x \ y) * x) / (x * (x \ y)) = L(K(x \ y,x),x * (x \ y),x \ e) by Axiom 3. Then
(L29) ((x \ y) * x) / (x * (x \ y)) = T(x,(x \ e) * ((x \ y) * x)) \ x
by (L28). We have x * ((T(x,(x \ e) * ((x \ y) * x)) \ x) \ e) = T(x,(x \ e) * ((x \ y) * x)) by Theorem 1198. Then
(L31) x * ((((x \ y) * x) / (x * (x \ y))) \ e) = T(x,(x \ e) * ((x \ y) * x))
by (L29). We have x * ((T(x,x \ y) \ x) \ e) = T(x,x \ y) by Theorem 1198. Then x * ((((x \ y) * x) / (x * (x \ y))) \ e) = T(x,x \ y) by Theorem 1023. Then T(x,x \ y) = T(x,(x \ e) * ((x \ y) * x)) by (L31). Then
(L35) T(x,T((x \ e) * y,x)) = T(x,x \ y)
by Theorem 891. We have T(x,T((x \ e) * y,x)) = T(x,(x \ e) * y) by Theorem 1032. Hence we are done by (L35).
Theorem 1303 (prov9_b4cd7bd768): (((y \ e) * x) \ x) * ((y \ e) * x) = y * (x / y)
Proof. We have (K(y \ e,y) * ((e / y) * x)) / L(T(e / y,x),x,K(y \ e,y)) = K(y \ e,y) * x by Theorem 1066. Then ((y \ e) * x) / L(T(e / y,x),x,K(y \ e,y)) = K(y \ e,y) * x by Theorem 682. Then
(L5) ((y \ e) * x) / T(e / y,x) = K(y \ e,y) * x
by Theorem 967. We have ((e / y) * T((e / y) \ e,(y \ e) * x)) * x = x * K(((y \ e) * x) \ (((y \ e) * x) / T(e / y,x)),(y \ e) * x) by Theorem 1115. Then ((e / y) * T((e / y) \ e,(y \ e) * x)) * x = x * K(((y \ e) * x) \ (K(y \ e,y) * x),(y \ e) * x) by (L5). Then
(L8) x * K(((y \ e) * x) \ (K(y \ e,y) * x),(y \ e) * x) = (y \ T(y,(y \ e) * x)) * x
by Theorem 1199. We have (((y \ e) * x) * K(y,y \ e)) \ x = ((y \ e) * x) \ (K(y \ e,y) * x) by Theorem 1223. Then ((e / y) * x) \ x = ((y \ e) * x) \ (K(y \ e,y) * x) by Theorem 1186. Then x * K(((e / y) * x) \ x,(y \ e) * x) = (y \ T(y,(y \ e) * x)) * x by (L8). Then
(L10) (y \ T(y,(y \ e) * x)) * x = x * K(((y \ e) * x) \ x,(y \ e) * x)
by Theorem 1213. We have x * K(((y \ e) * x) \ x,(y \ e) * x) = (((y \ e) * x) \ x) * ((y \ e) * x) by Theorem 17. Then (y \ T(y,(y \ e) * x)) * x = (((y \ e) * x) \ x) * ((y \ e) * x) by (L10). Then
(L13) (((y \ e) * x) \ x) * ((y \ e) * x) = (y \ T(y,y \ x)) * x
by Theorem 1302. We have T(y,y \ x) * (y \ x) = y * (x / y) by Theorem 1245. Then (y \ T(y,y \ x)) * x = y * (x / y) by Theorem 1270. Hence we are done by (L13).
Theorem 1304 (prov9_8bed7636c1): y \ (x / (x / y)) = K(y \ x,y)
Proof. We have K(x \ e,R(x,y \ x,y)) = K(x \ e,x) by Theorem 966. Then
(L6) K(x \ e,R(x,y \ x,y)) = R(K(x \ e,x),y \ x,(y \ x) \ x)
by Theorem 1230. We have (x * R(K(x \ e,x),y \ x,(y \ x) \ x)) / ((y \ x) \ x) = ((x * K(x \ e,x)) / x) * (y \ x) by Theorem 828. Then (x * R(K(x \ e,x),y \ x,(y \ x) \ x)) / ((y \ x) \ x) = ((e / (e / x)) / x) * (y \ x) by Theorem 262. Then (x * K(x \ e,R(x,y \ x,y))) / ((y \ x) \ x) = ((e / (e / x)) / x) * (y \ x) by (L6). Then L(x,x \ e,R(x,y \ x,y)) / ((y \ x) \ x) = ((e / (e / x)) / x) * (y \ x) by Theorem 1220. Then
(L11) L(x,x \ e,R(x,y \ x,y)) / ((y \ x) \ x) = K(x \ e,x) * (y \ x)
by Theorem 1109. We have x * K(x \ e,R(x,y \ x,y)) = L(x,x \ e,R(x,y \ x,y)) by Theorem 1220. Then
(L2) x * K(x \ e,x) = L(x,x \ e,R(x,y \ x,y))
by Theorem 966. We have x * K(x \ e,x) = e / (e / x) by Theorem 262. Then L(x,x \ e,R(x,y \ x,y)) = e / (e / x) by (L2). Then (e / (e / x)) / ((y \ x) \ x) = K(x \ e,x) * (y \ x) by (L11). Then (e / (e / x)) / T(y,y \ x) = K(x \ e,x) * (y \ x) by Proposition 49. Then
(L14) y \ (e / (e / x)) = K(x \ e,x) * (y \ x)
by Theorem 1266.
K(x \ e,x) * (K(x,x \ e) * ((K(x \ e,x) * (y \ x)) * y))
=(K(x \ e,x) * (y \ x)) * y by Theorem 693
=K(x \ e,x) * ((y \ x) * y) by Theorem 968

Then K(x \ e,x) * (K(x,x \ e) * ((K(x \ e,x) * (y \ x)) * y)) = K(x \ e,x) * ((y \ x) * y). Then K(x,x \ e) * ((K(x \ e,x) * (y \ x)) * y) = (y \ x) * y by Proposition 9. Then
(L19) K(x,x \ e) * ((y \ (e / (e / x))) * y) = (y \ x) * y
by (L14). We have K(x,x \ e) * ((y \ (e / (e / x))) * y) = x * K(y \ (e / (e / x)),y) by Theorem 1164. Then
(L21) (y \ x) * y = x * K(y \ (e / (e / x)),y)
by (L19). We have
(L22) (K(x,x \ e) * ((y \ (e / (e / x))) * y)) * (x \ e) = ((y \ (e / (e / x))) * y) * (e / x)
by Theorem 1191.
R((y \ (e / (e / x))) * y,e / x,(e / x) \ e) / ((e / x) \ e)
=((y \ (e / (e / x))) * y) * (e / x) by Theorem 73
=(x * K(y \ (e / (e / x)),y)) * (x \ e) by (L22), Theorem 1164

Then R((y \ (e / (e / x))) * y,e / x,(e / x) \ e) / ((e / x) \ e) = (x * K(y \ (e / (e / x)),y)) * (x \ e). Then R((y \ (e / (e / x))) * y,e / x,x) / ((e / x) \ e) = (x * K(y \ (e / (e / x)),y)) * (x \ e) by Proposition 25. Then R((y \ (e / (e / x))) * y,x \ e,x) / ((e / x) \ e) = (x * K(y \ (e / (e / x)),y)) * (x \ e) by Theorem 1151. Then ((y \ (e * x)) * y) / ((e / x) \ e) = (x * K(y \ (e / (e / x)),y)) * (x \ e) by Theorem 974. Then (x * K(y \ (e / (e / x)),y)) * (x \ e) = ((y \ x) * y) / ((e / x) \ e) by Axiom 1. Then (x * K(y \ (e / (e / x)),y)) * (x \ e) = ((y \ x) * y) / x by Proposition 25. Then
(L31) ((y \ x) * y) * (x \ e) = ((y \ x) * y) / x
by (L21). We have ((y \ x) * y) * (x \ e) = (e / x) * ((y \ x) * y) by Theorem 1268. Then
(L33) ((y \ x) * y) / x = (e / x) * ((y \ x) * y)
by (L31). We have (((y \ e) * x) \ T((y \ e) * x,((y \ e) * x) \ x)) \ x = ((y \ e) * x) * (T((y \ e) * x,((y \ e) * x) \ x) \ x) by Theorem 1272. Then
(L35) (((y \ e) * x) \ T((y \ e) * x,((y \ e) * x) \ x)) \ x = ((y \ e) * x) * T(((y \ e) * x) \ x,(y \ e) * x)
by Theorem 1276. We have ((y \ e) * x) * T(((y \ e) * x) \ x,(y \ e) * x) = (((y \ e) * x) \ x) * ((y \ e) * x) by Proposition 46. Then (((y \ e) * x) \ T((y \ e) * x,((y \ e) * x) \ x)) \ x = (((y \ e) * x) \ x) * ((y \ e) * x) by (L35). Then
(L38) (((y \ e) * x) \ ((((y \ e) * x) \ x) \ x)) \ x = (((y \ e) * x) \ x) * ((y \ e) * x)
by Proposition 49. We have x / ((((y \ e) * x) \ ((((y \ e) * x) \ x) \ x)) \ x) = ((y \ e) * x) \ ((((y \ e) * x) \ x) \ x) by Proposition 24. Then
(L40) x / ((((y \ e) * x) \ x) * ((y \ e) * x)) = ((y \ e) * x) \ ((((y \ e) * x) \ x) \ x)
by (L38). We have (((y \ e) * x) * (x / ((y \ e) * x))) / x = ((y \ e) * x) \ ((((y \ e) * x) \ x) \ x) by Theorem 1024. Then (((y \ e) * x) * (x / ((y \ e) * x))) / x = x / ((((y \ e) * x) \ x) * ((y \ e) * x)) by (L40). Then T(((y \ e) * x) * (x / ((y \ e) * x)),x) / x = x / ((((y \ e) * x) \ x) * ((y \ e) * x)) by Theorem 1291. Then T((y \ x) * y,x) / x = x / ((((y \ e) * x) \ x) * ((y \ e) * x)) by Theorem 1300. Then x / (y * (x / y)) = T((y \ x) * y,x) / x by Theorem 1303. Then
(L46) ((y \ x) * y) / x = x / (y * (x / y))
by Theorem 1290. We have (x / y) \ T(x / y,y) = y \ (x / (x / y)) by Theorem 987. Then
(L48) T(y,x / y) \ y = y \ (x / (x / y))
by Theorem 1030. We have ((x / y) * y) / (y * (x / y)) = T(y,x / y) \ y by Theorem 1023. Then x / (y * (x / y)) = T(y,x / y) \ y by Axiom 6. Then y \ (x / (x / y)) = x / (y * (x / y)) by (L48). Then ((y \ x) * y) / x = y \ (x / (x / y)) by (L46). Then
(L53) (e / x) * ((y \ x) * y) = y \ (x / (x / y))
by (L33). We have L((x \ e) \ ((e / x) * ((y \ x) * y)),x \ e,x) = x * ((e / x) * ((y \ x) * y)) by Theorem 62. Then L(K(x,x \ e) * ((y \ x) * y),x \ e,x) = x * ((e / x) * ((y \ x) * y)) by Theorem 1166. Then L((y \ ((x \ e) \ e)) * y,x \ e,x) = x * ((e / x) * ((y \ x) * y)) by Theorem 1265. Then
(L57) L((y \ ((x \ e) \ e)) * y,x \ e,x) = x * (y \ (x / (x / y)))
by (L53). We have L((y \ ((x \ e) \ e)) * y,x \ e,x) = (y \ L((x \ e) \ e,x \ e,x)) * y by Theorem 1299. Then L((y \ ((x \ e) \ e)) * y,x \ e,x) = (y \ (e \ (x * e))) * y by Theorem 60. Then x * (y \ (x / (x / y))) = (y \ (e \ (x * e))) * y by (L57). Then (y \ (e \ x)) * y = x * (y \ (x / (x / y))) by Axiom 2. Then
(L62) x * (y \ (x / (x / y))) = (y \ x) * y
by Proposition 26. We have x * K(y \ x,y) = (y \ x) * y by Theorem 17. Hence we are done by (L62) and Proposition 7.
Theorem 1305 (prov9_e7159097f0): T(K(y \ x,y),x) = K(y \ x,y)
Proof. We have K(y \ x,y) = T(T(y,y \ x) \ y,y * (y \ x)) by Theorem 1273. Then K(y \ x,y) = T(T(y,y \ x) \ y,x) by Axiom 4. Then T(y \ (x / (x / y)),x) = K(y \ x,y) by Theorem 1010. Hence we are done by Theorem 1304.
Theorem 1306 (prov9_42cd59acf4): T(x * y,K(y,x)) = x * y
Proof. We have T(K(x \ (x * y),x),x * y) = K(x \ (x * y),x) by Theorem 1305. Then T(x * y,K(x \ (x * y),x)) = x * y by Proposition 21. Hence we are done by Axiom 3.
Theorem 1307 (prov9_96a83a9995): T(y,x) * K(x,y) = y
Proof. We have x * (x \ (K(x,y) * (y * x))) = K(x,y) * (y * x) by Axiom 4. Then x * (T(y,x) * K(x,y)) = K(x,y) * (y * x) by Theorem 1274. Then x * (T(y,x) * K(x,y)) = y * (x * K(x,y)) by Theorem 1280. Then
(L5) x * (T(y,x) * K(x,y)) = T(x * y,K(y,x))
by Theorem 1284. We have T(x * y,K(y,x)) = x * y by Theorem 1306. Then x * (T(y,x) * K(x,y)) = x * y by (L5). Hence we are done by Proposition 9.
Theorem 1308 (KT0): x \ T(x,y) = K(x,y)
Proof.
T(y,x) * (x \ T(x,y))
=y by Theorem 1031
=T(y,x) * K(x,y) by Theorem 1307

Then T(y,x) * (x \ T(x,y)) = T(y,x) * K(x,y). Hence we are done by Proposition 9.
Theorem 1309 (prov9_e11ceba1): T(x,y) = x * K(x,y)
Proof. We have x \ T(x,y) = K(x,y) by Theorem 1308. Hence we are done by Proposition 5.
Theorem 1310 (prov9_22d34f9e57): (x * K(x,y)) / x = K(x,y)
Proof. We have (x * T(K(x,y),x)) / x = K(x,y) by Proposition 48. Hence we are done by Theorem 734.
Theorem 1311 (prov9_6e80e5d5ed): K(y,z) * T(y,x) = T(y,x) * K(y,z)
Proof. We have K(y,z) * T(T(y,K(y,z)),x) = T(y,x) * K(y,z) by Proposition 50. Hence we are done by Theorem 736.
Theorem 1312 (prov9_051cfe5e80): T(x,((y * x) \ x) \ e) = x * K(x,y)
Proof. We have T(x,((y * x) \ x) \ e) = x * K(x,(y * x) / x) by Theorem 1001. Hence we are done by Axiom 5.
Theorem 1313 (prov9_ebef0c3bf6): K(T(x,T(y,x)),T(y,x)) = K(x,y)
Proof. We have K(T(x,T(y,x)),T(y,x)) = K(x,(x * T(y,x)) / x) by Theorem 1008. Hence we are done by Proposition 48.
Theorem 1314 (prov9_395b28220d): x * ((T(y,x) \ y) * z) = T(x,y) * z
Proof. We have x * ((x \ T(x,y)) * z) = T(x,y) * z by Theorem 1017. Hence we are done by Theorem 1030.
Theorem 1315 (prov9_5e94a02e93): R(T(y,x),x,K(x,y)) = K(x,y) \ y
Proof. We have T(x,((y * x) \ x) \ e) \ (x * y) = x * (T(x,((y * x) \ x) \ e) \ y) by Theorem 174. Then (x * K(x,y)) \ (x * y) = x * (T(x,((y * x) \ x) \ e) \ y) by Theorem 1312. Then
(L10) x * ((x * K(x,y)) \ y) = (x * K(x,y)) \ (x * y)
by Theorem 1312. We have (x * K(x,y)) \ ((y * x) * K(x,y)) = R(T(y,x * K(x,y)),x,K(x,y)) by Proposition 81. Then (x * K(x,y)) \ (x * y) = R(T(y,x * K(x,y)),x,K(x,y)) by Proposition 82. Then
(L13) R(T(y,x * K(x,y)),x,K(x,y)) = x * ((x * K(x,y)) \ y)
by (L10). We have y * ((T(x,y) \ x) * (T(x,y) * K(x,y))) = T(y,x) * (T(x,y) * K(x,y)) by Theorem 1314. Then y * (x * K(x,y)) = T(y,x) * (T(x,y) * K(x,y)) by Theorem 1282. Then
(L3) T(y,x) * (K(x,y) * T(x,y)) = y * (x * K(x,y))
by Theorem 1311.
T(y,x * K(x,y)) * T(x * K(x,y),y)
=y * (x * K(x,y)) by Theorem 1033
=T(y,x) * T(x * K(x,y),y) by (L3), Theorem 1049

Then T(y,x * K(x,y)) * T(x * K(x,y),y) = T(y,x) * T(x * K(x,y),y). Then T(y,x * K(x,y)) = T(y,x) by Proposition 10. Then
(L14) R(T(y,x),x,K(x,y)) = x * ((x * K(x,y)) \ y)
by (L13). We have R((x * K(x,y)) / x,x,(x * K(x,y)) \ y) \ y = x * ((x * K(x,y)) \ y) by Theorem 1065. Then R(K(x,y),x,(x * K(x,y)) \ y) \ y = x * ((x * K(x,y)) \ y) by Theorem 1310. Then K(x,y) \ y = x * ((x * K(x,y)) \ y) by Theorem 1025. Hence we are done by (L14).
Theorem 1316 (prov9_14e7931f15): K(x,T(y,x)) = K(x,y)
Proof. We have K(T(x,T(y,x)),T(y,x)) = K(x,y) by Theorem 1313. Hence we are done by Theorem 1053.
Theorem 1317 (prov9_9616049a61): R(T(x,y),y,K(y,x)) = x * K(x,y)
Proof. We have R(T(x,y),y,K(y,x)) = K(y,x) \ x by Theorem 1315. Hence we are done by Theorem 1283.
Theorem 1318 (prov9_79803736b1): R(y,x,K(x,y)) = y
Proof. We have x * R(T(y,x),x,K(x,y)) = R(y,x,K(x,y)) * x by Proposition 59. Then
(L3) x * (y * K(y,x)) = R(y,x,K(x,y)) * x
by Theorem 1317. We have T(y * x,K(x,y)) = x * (y * K(y,x)) by Theorem 1284. Then
(L5) T(y * x,K(x,y)) = R(y,x,K(x,y)) * x
by (L3). We have T(y * x,K(x,y)) = y * x by Theorem 1306. Then R(y,x,K(x,y)) * x = y * x by (L5). Hence we are done by Proposition 10.
Theorem 1319 (prov9_5ceb61b7a4): T(y,x) = K(x,y) \ y
Proof. We have R(T(y,x),x,K(x,y)) = K(x,y) \ y by Theorem 1315. Then R(T(y,x),x,K(x,T(y,x))) = K(x,y) \ y by Theorem 1316. Hence we are done by Theorem 1318.
Theorem 1320 (KT1): y / T(y,x) = K(x,y)
Proof. We have y / (K(x,y) \ y) = K(x,y) by Proposition 24. Hence we are done by Theorem 1319.
Theorem 1321 (prov9_7a31aa90cf): L(z,T(x,y),y) = (x * y) \ (y * (T(x,y) * z))
Proof. We have L(z,T(x,y),y) = (y * T(x,y)) \ (y * (T(x,y) * z)) by Definition 4. Hence we are done by Proposition 46.
Theorem 1322 (prov9_ad6683bb74): y * T(y,x) = T(y,x) * y
Proof. We have y * T(T(y,x),y) = T(y,x) * y by Proposition 46. Hence we are done by Theorem 451.
Theorem 1323 (prov9_10fa87c455): R(z \ e,x,y) = R(R(e / z,x,y),z,z \ e)
Proof. We have R(R(e / z,z,z \ e),x,y) = R(R(e / z,x,y),z,z \ e) by Axiom 12. Hence we are done by Theorem 41.
Theorem 1324 (prov9_610e44e177): x / L(z \ y,z,x / y) = (x / y) * z
Proof. We have x / (((x / y) * z) \ ((x / y) * y)) = (x / y) * z by Theorem 24. Hence we are done by Proposition 53.
Theorem 1325 (prov9_499e912d1c): R(R(e / x,x,y),x * y,z) = (y * z) / ((x * y) * z)
Proof. We have R(y / (x * y),x * y,z) = (y * z) / ((x * y) * z) by Proposition 55. Hence we are done by Proposition 79.
Theorem 1326 (prov9_a4c195405b): T(x * y,y \ e) = (y \ e) \ R(x,y,y \ e)
Proof. We have T(x * y,y \ e) = (y \ e) \ ((x * y) * (y \ e)) by Definition 3. Hence we are done by Proposition 68.
Theorem 1327 (prov9_db45cfb8f8): L((e / w) * x,y,z) = L(L(w \ x,y,z),w,e / w)
Proof. We have L(L(w \ x,w,e / w),y,z) = L(L(w \ x,y,z),w,e / w) by Axiom 11. Hence we are done by Theorem 63.
Theorem 1328 (prov9_b3398b3220): L(x,z,y) * (y * z) = y * (z * T(x,y * z))
Proof. We have (y * z) * L(T(x,y * z),z,y) = y * (z * T(x,y * z)) by Proposition 52. Hence we are done by Proposition 58.
Theorem 1329 (prov9_2859a1d7d2): T(x * (y \ e),y) = R(y \ x,y,y \ e)
Proof. We have T(R(x / y,y,y \ e),y) = R(y \ x,y,y \ e) by Proposition 60. Hence we are done by Proposition 71.
Theorem 1330 (prov9_1a98f7db6d): (x * (((x \ y) \ z) \ z)) / x = T(y / x,(x \ y) \ z)
Proof. We have (x * T(x \ y,(x \ y) \ z)) / x = T(y / x,(x \ y) \ z) by Theorem 49. Hence we are done by Proposition 49.
Theorem 1331 (prov9_d88664d929): R(x,y,y \ e) = T((y * x) * (y \ e),y)
Proof. We have R(y \ (y * x),y,y \ e) = T((y * x) * (y \ e),y) by Theorem 1329. Hence we are done by Axiom 3.
Theorem 1332 (prov9_2f3112254a): R((z * (x / y)) / z,y,y \ e) = (z * (x * (y \ e))) / z
Proof. We have R((z * (x / y)) / z,y,y \ e) = (z * R(x / y,y,y \ e)) / z by Theorem 123. Hence we are done by Proposition 71.
Theorem 1333 (prov9_49b4545847): T((e / y) * x,y) = (e / y) * ((y \ x) * y)
Proof. We have L(T(y \ x,y),y,e / y) = (e / y) * ((y \ x) * y) by Theorem 72. Hence we are done by Theorem 1072.
Theorem 1334 (prov9_d76c32fb3b): ((x * y) / x) / T(y / x,T(x,y)) = T(x,y)
Proof. We have ((y / x) * T(x,y)) / T(y / x,T(x,y)) = T(x,y) by Theorem 5. Hence we are done by Theorem 818.
Theorem 1335 (prov9_4177d56588): (e / x) \ T(y,z) = x * T(x \ ((e / x) \ y),z)
Proof. We have (e / x) \ T((e / x) * ((e / x) \ y),z) = x * T(x \ ((e / x) \ y),z) by Theorem 1080. Hence we are done by Axiom 4.
Theorem 1336 (prov9_f79b8f3e16): (e / y) \ T(x,y) = (y \ ((e / y) \ x)) * y
Proof. We have y * T(y \ ((e / y) \ x),y) = (y \ ((e / y) \ x)) * y by Proposition 46. Hence we are done by Theorem 1335.
Theorem 1337 (prov9_d66faf6b94): (y * T(x,z)) * (y \ e) = T((y * x) * (y \ e),z)
Proof. We have (y * T(y \ (y * x),z)) * (y \ e) = T((y * x) * (y \ e),z) by Theorem 506. Hence we are done by Axiom 3.
Theorem 1338 (prov9_71ac573df0): (e / x) \ L((e / x) * y,z,w) = x * L(x \ y,z,w)
Proof. We have (e / x) \ L(L(x \ y,z,w),x,e / x) = x * L(x \ y,z,w) by Theorem 71. Hence we are done by Theorem 1327.
Theorem 1339 (prov9_53e5770239): x \ (T(x,y) \ x) = T(x,y) \ e
Proof. We have x * (T(x,y) \ e) = T(x,y) \ x by Theorem 175. Hence we are done by Proposition 2.
Theorem 1340 (prov9_d9d66c49d1): ((y * x) / y) \ T(y,x) = x \ y
Proof. We have ((y * x) / y) * (x \ y) = T(y,x) by Theorem 179. Hence we are done by Proposition 2.
Theorem 1341 (prov9_7ae9d4a8f9): L(y * (x \ e),z,w) / (x \ e) = x * L(x \ y,z,w)
Proof. We have R(L(y / x,z,w),x,x \ e) = L(R(y / x,x,x \ e),z,w) by Axiom 10. Then
(L2) R(L(y / x,z,w),x,x \ e) = L(y * (x \ e),z,w)
by Proposition 71. We have R(L(y / x,z,w),x,x \ e) / (x \ e) = L(y / x,z,w) * x by Theorem 73. Then
(L4) L(y * (x \ e),z,w) / (x \ e) = L(y / x,z,w) * x
by (L2). We have L(y / x,z,w) * x = x * L(x \ y,z,w) by Theorem 85. Hence we are done by (L4).
Theorem 1342 (prov9_ed458327dd): K(x,T(y,y \ z)) = K(x,(y \ z) \ z)
Proof. We have K(x,T(y,y \ z)) = (T(y,y \ z) * x) \ (x * T(y,y \ z)) by Definition 2. Then
(L2) K(x,T(y,y \ z)) = (T(y,y \ z) * x) \ (x * ((y \ z) \ z))
by Theorem 806. We have (T(y,y \ z) * x) \ (x * ((y \ z) \ z)) = K(x,(y \ z) \ z) by Theorem 478. Hence we are done by (L2).
Theorem 1343 (prov9_d8f6c26b64): (x / z) * T(z,y) = (x * T(z,y)) / z
Proof. We have (x * T(z,y)) / T(z,y) = x by Axiom 5. Then
(L3) (((x * T(z,y)) / z) * z) / T(z,y) = x
by Axiom 6. We have ((((x * T(z,y)) / z) * z) / T(z,y)) * T(z,y) = ((x * T(z,y)) / z) * z by Axiom 6. Then ((((((x * T(z,y)) / z) * z) / T(z,y)) / z) * z) * T(z,y) = ((x * T(z,y)) / z) * z by Axiom 6. Then
(L6) ((x / z) * z) * T(z,y) = ((x * T(z,y)) / z) * z
by (L3). We have ((x / z) * T(z,y)) * z = ((x / z) * z) * T(z,y) by Theorem 238. Hence we are done by (L6) and Proposition 8.
Theorem 1344 (prov9_2101ea6c00): ((x / y) * T(y,z)) * y = x * T(y,z)
Proof. We have ((x / y) * T(y,z)) * y = ((x / y) * y) * T(y,z) by Theorem 238. Hence we are done by Axiom 6.
Theorem 1345 (prov9_3b5b8ea54c): (e / T(y,x)) \ (y / T(y,x)) = y
Proof. We have (e / T(y,x)) * y = y / T(y,x) by Theorem 550. Hence we are done by Proposition 2.
Theorem 1346 (prov9_e02ba562d9): ((x / z) * T(z,y)) \ (x * T(z,y)) = z
Proof. We have ((x / z) * T(z,y)) \ (((x / z) * z) * T(z,y)) = z by Theorem 1105. Hence we are done by Axiom 6.
Theorem 1347 (prov9_5f6457f4fa): x / y = ((x / T(y,z)) / y) * T(y,z)
Proof. We have ((x / T(y,z)) * T(y,z)) / y = ((x / T(y,z)) / y) * T(y,z) by Theorem 1343. Hence we are done by Axiom 6.
Theorem 1348 (prov9_788edfcf22): (x / (y \ z)) * (z / y) = (x * (z / y)) / (y \ z)
Proof. We have (x * (z / y)) / (z / y) = x by Axiom 5. Then
(L3) (((x * (z / y)) / (y \ z)) * (y \ z)) / (z / y) = x
by Axiom 6. We have ((((x * (z / y)) / (y \ z)) * (y \ z)) / (z / y)) * (z / y) = ((x * (z / y)) / (y \ z)) * (y \ z) by Axiom 6. Then ((((((x * (z / y)) / (y \ z)) * (y \ z)) / (z / y)) / (y \ z)) * (y \ z)) * (z / y) = ((x * (z / y)) / (y \ z)) * (y \ z) by Axiom 6. Then
(L6) ((x / (y \ z)) * (y \ z)) * (z / y) = ((x * (z / y)) / (y \ z)) * (y \ z)
by (L3). We have ((x / (y \ z)) * (z / y)) * (y \ z) = ((x / (y \ z)) * (y \ z)) * (z / y) by Theorem 241. Hence we are done by (L6) and Proposition 8.
Theorem 1349 (prov9_e939b6c585): T(y,x) * K(y,y \ e) = y * (T(y,x) * (e / y))
Proof. We have T(y,x) * K(y,e / y) = y * (T(y,x) * (e / y)) by Theorem 176. Hence we are done by Theorem 780.
Theorem 1350 (prov9_325887979f): T(y,T(y,x) \ e) / y = T(y,x) * ((y * (T(y,x) \ e)) / y)
Proof. We have (x * T(T(y,x),e / T(y,x))) / x = T((x * T(y,x)) / x,e / T(y,x)) by Theorem 50. Then
(L2) (x * T(T(y,x),T(y,x) \ e)) / x = T((x * T(y,x)) / x,e / T(y,x))
by Theorem 211. We have (x * T(T(y,x),T(y,x) \ e)) / x = T((x * T(y,x)) / x,T(y,x) \ e) by Theorem 50. Then T((x * T(y,x)) / x,e / T(y,x)) = T((x * T(y,x)) / x,T(y,x) \ e) by (L2). Then T(y,e / T(y,x)) = T((x * T(y,x)) / x,T(y,x) \ e) by Proposition 48. Then
(L6) T(y,T(y,x) \ e) = T(y,e / T(y,x))
by Proposition 48. We have ((e / T(y,x)) \ (y / T(y,x))) * (e / T(y,x)) = T(y,e / T(y,x)) / T(y,x) by Theorem 152. Then y * (e / T(y,x)) = T(y,e / T(y,x)) / T(y,x) by Theorem 1345. Then
(L9) T(y,T(y,x) \ e) / T(y,x) = y * (e / T(y,x))
by (L6). We have ((T(y,T(y,x) \ e) / T(y,x)) / y) * T(y,x) = T(y,T(y,x) \ e) / y by Theorem 1347. Then
(L11) ((y * (e / T(y,x))) / y) * T(y,x) = T(y,T(y,x) \ e) / y
by (L9). We have ((y * (e / T(y,x))) / y) * T(y,x) = T(y,x) * ((y * (T(y,x) \ e)) / y) by Theorem 125. Hence we are done by (L11).
Theorem 1351 (prov9_c07f187d3e): T(y,x) * ((T(y,x) \ y) / y) = T(y,T(y,x) \ e) / y
Proof. We have T(y,x) * ((y * (T(y,x) \ e)) / y) = T(y,T(y,x) \ e) / y by Theorem 1350. Hence we are done by Theorem 175.
Theorem 1352 (prov9_9d145d9547): K(e / x,y) = L(x * T(x \ e,y),e / x,y)
Proof. We have
(L1) L((e / x) \ T(e / x,y),e / x,y) = (y * (e / x)) \ ((e / x) * y)
by Proposition 73.
K(e / x,y)
=(y * (e / x)) \ ((e / x) * y) by Definition 2
=L(x * T(x \ e,y),e / x,y) by (L1), Theorem 102

Hence we are done.
Theorem 1353 (prov9_aef9adbde4): T(x / y,y / x) = ((x / y) \ e) \ e
Proof. We have T(x / y,(x / y) \ e) = ((x / y) \ e) \ e by Proposition 49. Then T(x / y,y / ((x / y) * y)) = ((x / y) \ e) \ e by Theorem 362. Hence we are done by Axiom 6.
Theorem 1354 (prov9_f7d1938391): K(x,y) * (e / y) = (e / y) * K(x,y)
Proof. We have (((e / y) * K(x,y)) / (e / y)) * (e / y) = (e / y) * K(x,y) by Axiom 6. Hence we are done by Theorem 365.
Theorem 1355 (prov9_af66ae8b95): L(x,y \ e,y) / y = (y \ e) * (y * (x / y))
Proof. We have L((x / y) * y,y \ e,y) / y = (y \ e) * (y * (x / y)) by Theorem 894. Hence we are done by Axiom 6.
Theorem 1356 (prov9_ff94fae6dc): K(y,z) * K(z,x) = K(z,x) * K(y,z)
Proof. We have K(y,z) * T(K(z,x),K(y,z)) = K(z,x) * K(y,z) by Proposition 46. Hence we are done by Theorem 899.
Theorem 1357 (prov9_9351e3c64d): K(z,w) * T(K(w,x),y) = T(K(w,x),y) * K(z,w)
Proof. We have K(z,w) * T(T(K(w,x),K(z,w)),y) = T(K(w,x),y) * K(z,w) by Proposition 50. Hence we are done by Theorem 899.
Theorem 1358 (prov9_9faa80e511): L(x \ T(x,w),y,z) = K(w \ (w / L(x \ e,y,z)),w)
Proof. We have L((x \ e) * T((x \ e) \ e,w),y,z) = K(w \ (w / L(x \ e,y,z)),w) by Theorem 635. Hence we are done by Theorem 145.
Theorem 1359 (prov9_bbae943f9a): L(K(w \ (w / x),w),y,z) = K(w \ (w / L(x,y,z)),w)
Proof. We have L(x * T(x \ e,w),y,z) = K(w \ (w / L(x,y,z)),w) by Theorem 635. Hence we are done by Theorem 196.
Theorem 1360 (prov9_35c79a4e37): ((w * x) / w) * T(((w * x) / w) \ y,z) = (w * (x * T(x \ T(y,w),z))) / w
Proof. We have ((w * x) * T((w * x) \ (y * w),z)) / w = ((w * x) / w) * T(((w * x) / w) \ y,z) by Theorem 1116. Then ((w * x) * T((w * x) \ (y * w),z)) / w = ((x / w) * T(w,x)) * T(((w * x) / w) \ y,z) by Theorem 818. Then ((x / w) * T(w,x)) * T(((w * x) / w) \ y,z) = (w * (x * T(x \ (w \ (y * w)),z))) / w by Theorem 136. Then ((x / w) * T(w,x)) * T(((w * x) / w) \ y,z) = (w * (x * T(x \ T(y,w),z))) / w by Definition 3. Hence we are done by Theorem 818.
Theorem 1361 (prov9_6a7e760342): (x / T(y,y \ e)) * K(y,y \ e) = x / y
Proof. We have ((x / T(y,y \ e)) / y) * T(y,y \ e) = x / y by Theorem 1347. Hence we are done by Theorem 1142.
Theorem 1362 (prov9_7428a7062b): x / (e / y) = (x / (y \ e)) * K(y \ e,y)
Proof. We have x / L(K(y \ e,y) \ (y \ e),K(y \ e,y),x / (y \ e)) = (x / (y \ e)) * K(y \ e,y) by Theorem 1324. Then x / L(e / y,K(y \ e,y),x / (y \ e)) = (x / (y \ e)) * K(y \ e,y) by Theorem 256. Hence we are done by Theorem 644.
Theorem 1363 (prov9_339282ac0c): (x * (y \ e)) / K(y \ e,y) = x * (e / y)
Proof. We have (x * (e / y)) * K(y \ e,y) = x * (y \ e) by Theorem 412. Hence we are done by Proposition 1.
Theorem 1364 (prov9_2dab4f67ce): (x * (y \ e)) * K(y,y \ e) = x * (e / y)
Proof. We have (x * (y \ e)) / K(y \ e,y) = x * (e / y) by Theorem 1363. Hence we are done by Theorem 656.
Theorem 1365 (prov9_fb0f9a7977): (x * y) * K(y \ e,y) = x * (e / (e / y))
Proof. We have
(L1) ((x * y) * K(y \ e,y)) * K(y,y \ e) = x * y
by Theorem 426. We have (x * (e / (e / y))) * K(y,y \ e) = x * y by Theorem 415. Hence we are done by (L1) and Proposition 8.
Theorem 1366 (prov9_e760f59e0b): T(y,x * (y \ e)) = T(y,x * (e / y))
Proof. We have T(y,(x * (e / y)) * K(y \ e,y)) = T(y,x * (e / y)) by Theorem 660. Hence we are done by Theorem 412.
Theorem 1367 (prov9_28ef746f21): (x * y) * (y \ e) = R(x,y,e / y)
Proof. We have R(x,y,e / y) * (y * (e / y)) = (x * y) * (e / y) by Proposition 54. Then
(L3) R(x,y,e / y) * K(y,y \ e) = (x * y) * (e / y)
by Theorem 250. We have ((x * y) * (y \ e)) * K(y,y \ e) = (x * y) * (e / y) by Theorem 1364. Hence we are done by (L3) and Proposition 8.
Theorem 1368 (prov9_7096a7e94a): T(x * (y \ e),K(y,y \ e)) = K(y,y \ e) \ (x * (e / y))
Proof. We have T(x * (y \ e),K(y,y \ e)) = K(y,y \ e) \ ((x * (y \ e)) * K(y,y \ e)) by Definition 3. Hence we are done by Theorem 1364.
Theorem 1369 (prov9_08be62222e): R(x,y,e / y) = R(x,y,y \ e)
Proof. We have (x * y) * (y \ e) = R(x,y,y \ e) by Proposition 68. Hence we are done by Theorem 1367.
Theorem 1370 (prov9_4e9955cca0): R(x,y \ e,y) = R(x,y \ e,(y \ e) \ e)
Proof. We have R(x,y \ e,e / (y \ e)) = R(x,y \ e,(y \ e) \ e) by Theorem 1369. Hence we are done by Proposition 24.
Theorem 1371 (prov9_cf6a7fc421): K(x \ (x / y),y) = K(y \ e,y)
Proof. We have K(((x / y) * y) \ (x / y),y) = K(y \ e,y) by Theorem 665. Hence we are done by Axiom 6.
Theorem 1372 (prov9_59883f3ba4): L(x \ T(x,y),(x \ e) \ e,y) = K(x,y)
Proof. We have L(x \ T(x,y),(x \ e) \ e,y) = K(y \ (y / L(x \ e,(x \ e) \ e,y)),y) by Theorem 1358. Then L(x \ T(x,y),(x \ e) \ e,y) = K(e / (x \ e),y) by Theorem 650. Hence we are done by Proposition 24.
Theorem 1373 (prov9_7040c50612): ((e / (e / x)) \ y) / y = e / (e / ((x \ y) / y))
Proof. We have ((K(y / (x \ y),(x \ y) / y) * (x \ y)) / y) * y = K(y / (x \ y),(x \ y) / y) * (x \ y) by Axiom 6. Then (e / (e / ((x \ y) / y))) * y = K(y / (x \ y),(x \ y) / y) * (x \ y) by Theorem 640. Then K(x,x \ e) * (x \ y) = (e / (e / ((x \ y) / y))) * y by Theorem 403. Then (e / (e / x)) \ y = (e / (e / ((x \ y) / y))) * y by Theorem 690. Hence we are done by Proposition 1.
Theorem 1374 (prov9_84e74dd6e3): ((x \ e) \ e) \ ((x \ e) \ y) = x \ ((e / x) \ y)
Proof. We have ((x \ e) \ e) \ (K(x,x \ e) * ((e / x) \ y)) = x \ ((e / x) \ y) by Theorem 1152. Hence we are done by Theorem 440.
Theorem 1375 (prov9_3cdb9ae9be): (e / x) * ((x \ e) \ y) = K(x,x \ e) * y
Proof. We have (e / x) * ((e / x) \ (K(x,x \ e) * y)) = K(x,x \ e) * y by Axiom 4. Hence we are done by Theorem 692.
Theorem 1376 (prov9_d595beb3c1): K(x \ e,x) * (((x \ e) \ e) * y) = x * y
Proof. We have
(L1) K(x,x \ e) * (K(x \ e,x) * (((x \ e) \ e) * y)) = ((x \ e) \ e) * y
by Theorem 442. We have K(x,x \ e) * (x * y) = ((x \ e) \ e) * y by Theorem 679. Hence we are done by (L1) and Proposition 7.
Theorem 1377 (prov9_bdada501ca): K(y,y \ e) * (x * K(y \ e,y)) = T(x,K(y \ e,y))
Proof. We have K(y,y \ e) * (K(y \ e,y) * T(x,K(y \ e,y))) = T(x,K(y \ e,y)) by Theorem 442. Hence we are done by Proposition 46.
Theorem 1378 (prov9_a2507d28cc): K(y \ e,y) * (x * ((y \ e) \ e)) = y * T(x,(y \ e) \ e)
Proof. We have K(y \ e,y) * (((y \ e) \ e) * T(x,(y \ e) \ e)) = y * T(x,(y \ e) \ e) by Theorem 1376. Hence we are done by Proposition 46.
Theorem 1379 (prov9_ef2ce3f00a): (((e / x) \ y) / y) \ e = e / (((x \ e) \ y) / y)
Proof. We have (e / (e / (((x \ e) \ y) / y))) \ e = e / (((x \ e) \ y) / y) by Proposition 25. Then (((e / (e / (x \ e))) \ y) / y) \ e = e / (((x \ e) \ y) / y) by Theorem 1373. Hence we are done by Proposition 24.
Theorem 1380 (prov9_730be0814f): ((((e / x) \ y) / y) \ e) \ e = ((x \ e) \ y) / y
Proof. We have (e / (((x \ e) \ y) / y)) \ e = ((x \ e) \ y) / y by Proposition 25. Hence we are done by Theorem 1379.
Theorem 1381 (prov9_3aef88027a): R(x \ e,x,y) = y / (T(x,x \ e) * y)
Proof. We have R(x \ e,x,y) = y / (((x \ e) \ e) * y) by Theorem 703. Hence we are done by Proposition 49.
Theorem 1382 (prov9_669d50bee9): R(x,e / x,y) \ y = (x \ e) * y
Proof. We have R((e / x) \ e,e / x,y) \ y = (((e / x) \ e) \ e) * y by Theorem 702. Then R(x,e / x,y) \ y = (((e / x) \ e) \ e) * y by Proposition 25. Hence we are done by Proposition 25.
Theorem 1383 (prov9_df8a49c68f): R(x,x \ e,y) = R(x,e / x,y)
Proof. We have y / (R(x,e / x,y) \ y) = R(x,e / x,y) by Proposition 24. Then
(L2) y / ((x \ e) * y) = R(x,e / x,y)
by Theorem 1382. We have R(x,x \ e,y) = y / ((x \ e) * y) by Proposition 66. Hence we are done by (L2).
Theorem 1384 (prov9_c60c9a20fd): R(e / (e / x),e / x,y) = R(e / (e / x),x \ e,y)
Proof. We have L(R(x,e / x,y),x \ e,x) = R(e / (e / x),e / x,y) by Theorem 851. Then
(L2) L(R(x,x \ e,y),x \ e,x) = R(e / (e / x),e / x,y)
by Theorem 1383. We have L(R(x,x \ e,y),x \ e,x) = R(e / (e / x),x \ e,y) by Theorem 851. Hence we are done by (L2).
Theorem 1385 (prov9_c486fdfc73): T((y \ e) * x,K(y \ e,y)) = (e / y) * (x * K(y \ e,y))
Proof. We have T(K(y \ e,y) * ((e / y) * x),K(y \ e,y)) = ((e / y) * x) * K(y \ e,y) by Theorem 903. Then
(L2) T((y \ e) * x,K(y \ e,y)) = ((e / y) * x) * K(y \ e,y)
by Theorem 682. We have ((e / y) * x) * K(y \ e,y) = (e / y) * (x * K(y \ e,y)) by Theorem 1138. Hence we are done by (L2).
Theorem 1386 (prov9_14b95bb943): x \ K(e / x,y) = (x \ e) * K(x \ e,y)
Proof. We have
(L10) L((e / (e / x)) * T((e / (e / x)) \ e,y),e / (e / (e / x)),y) = K(e / (e / (e / x)),y)
by Theorem 1352. Then
(L11) L((e / (e / x)) * T(e / x,y),e / (e / (e / x)),y) = K(e / (e / (e / x)),y)
by Proposition 25. We have (e / (e / x)) \ L((e / (e / x)) * T(e / x,y),e / (e / (e / x)),y) = (e / x) * L((e / x) \ T(e / x,y),e / (e / (e / x)),y) by Theorem 1338. Then
(L13) (e / (e / x)) \ K(e / (e / (e / x)),y) = (e / x) * L((e / x) \ T(e / x,y),e / (e / (e / x)),y)
by (L11). We have
(L14) L(T(e / x,y) / (e / x),e / (e / (e / x)),y) = K(e / (e / (e / x)),y)
by (L10) and Theorem 149. We have L(T(e / x,y) / (e / x),e / (e / (e / x)),y) * (e / x) = (e / x) * L((e / x) \ T(e / x,y),e / (e / (e / x)),y) by Theorem 85. Then K(e / (e / (e / x)),y) * (e / x) = (e / x) * L((e / x) \ T(e / x,y),e / (e / (e / x)),y) by (L14). Then
(L17) (e / (e / x)) \ K(e / (e / (e / x)),y) = K(e / (e / (e / x)),y) * (e / x)
by (L13). We have (((e / x) * K(e / x,y)) / (e / x)) * (e / x) = (e / x) * K(e / x,y) by Axiom 6. Then K(e / (e / (e / x)),y) * (e / x) = (e / x) * K(e / x,y) by Theorem 672. Then
(L18) (e / (e / x)) \ K(e / (e / (e / x)),y) = (e / x) * K(e / x,y)
by (L17). We have ((e / x) \ ((e / (e / x)) \ K(e / (e / (e / x)),y))) * (e / x) = (e / (e / x)) \ T(K(e / (e / (e / x)),y),e / x) by Theorem 1336. Then ((e / x) \ ((e / (e / x)) \ K(e / (e / (e / x)),y))) * (e / x) = (e / (e / x)) \ K(e / x,y) by Theorem 673. Then ((e / x) \ ((e / x) * K(e / x,y))) * (e / x) = (e / (e / x)) \ K(e / x,y) by (L18). Then
(L22) K(e / x,y) * (e / x) = (e / (e / x)) \ K(e / x,y)
by Axiom 3. We have K(x \ e,x) * ((e / (e / x)) \ K(e / x,y)) = x \ K(e / x,y) by Theorem 695. Then
(L24) K(x \ e,x) * (K(e / x,y) * (e / x)) = x \ K(e / x,y)
by (L22). We have T(K(e / x,y),((e / x) \ e) \ e) = T(K(e / x,y),e / x) by Theorem 721. Then
(L4) T(K(e / x,y),x \ e) = T(K(e / x,y),e / x)
by Proposition 25. We have (e / x) * T(K(e / x,y),e / x) = K(e / x,y) * (e / x) by Proposition 46. Then
(L6) (e / x) * T(K(e / x,y),x \ e) = K(e / x,y) * (e / x)
by (L4). We have T(K(e / (e / (x \ e)),y),x \ e) = K(x \ e,y) by Theorem 673. Then T(K(e / x,y),x \ e) = K(x \ e,y) by Proposition 24. Then (e / x) * K(x \ e,y) = K(e / x,y) * (e / x) by (L6). Then
(L25) K(x \ e,x) * ((e / x) * K(x \ e,y)) = x \ K(e / x,y)
by (L24). We have K(x \ e,x) * ((e / x) * K(x \ e,y)) = (x \ e) * K(x \ e,y) by Theorem 682. Hence we are done by (L25).
Theorem 1387 (prov9_73be46f003): (x \ K(e / x,y)) * ((x \ e) \ e) = K(x \ e,y)
Proof. We have K(x \ e,y) = L((x \ e) \ T(x \ e,y),((x \ e) \ e) \ e,y) by Theorem 1372. Then
(L2) K(x \ e,y) = L(T(x \ e,y) * ((x \ e) \ e),((x \ e) \ e) \ e,y)
by Theorem 146. We have L(T(x \ e,y) * ((x \ e) \ e),((x \ e) \ e) \ e,y) / ((x \ e) \ e) = (x \ e) * L((x \ e) \ T(x \ e,y),((x \ e) \ e) \ e,y) by Theorem 1341. Then L(T(x \ e,y) * ((x \ e) \ e),((x \ e) \ e) \ e,y) / ((x \ e) \ e) = (x \ e) * K(x \ e,y) by Theorem 1372. Then
(L5) K(x \ e,y) / ((x \ e) \ e) = (x \ e) * K(x \ e,y)
by (L2). We have (K(x \ e,y) / ((x \ e) \ e)) * ((x \ e) \ e) = K(x \ e,y) by Axiom 6. Then ((x \ e) * K(x \ e,y)) * ((x \ e) \ e) = K(x \ e,y) by (L5). Hence we are done by Theorem 1386.
Theorem 1388 (prov9_2dea57197a): T(y * x,K(y \ e,y)) = y * T(x,K(y \ e,y))
Proof. We have K(y,y \ e) * ((y * x) * K(y \ e,y)) = T(y * x,K(y \ e,y)) by Theorem 1377. Then
(L5) K(y,y \ e) * (((y * x) / ((y \ e) \ e)) * y) = T(y * x,K(y \ e,y))
by Theorem 424. We have (K(y,y \ e) * ((y * x) / ((y \ e) \ e))) * y = K(y,y \ e) * (((y * x) / ((y \ e) \ e)) * y) by Theorem 1225. Then (K(y,y \ e) * ((y * x) / ((y \ e) \ e))) * y = T(y * x,K(y \ e,y)) by (L5). Then (K(y \ e,y) \ ((y * x) / ((y \ e) \ e))) * y = T(y * x,K(y \ e,y)) by Theorem 696. Then
(L9) (K(y \ e,y) \ ((y * x) / T(y,y \ e))) * y = T(y * x,K(y \ e,y))
by Proposition 49. We have T((y * x) / y,K(y \ e,y)) = K(y \ e,y) \ (((y * x) / y) * K(y \ e,y)) by Definition 3. Then T((y * x) / y,K(y \ e,y)) = K(y \ e,y) \ ((y * x) * T((y * x) \ ((y * x) / y),y)) by Theorem 154. Then K(y \ e,y) \ ((y * x) / T(y,y \ e)) = T((y * x) / y,K(y \ e,y)) by Theorem 1146. Then
(L10) T((y * x) / y,K(y \ e,y)) * y = T(y * x,K(y \ e,y))
by (L9). We have T((y * x) / y,K(y \ e,y)) * y = y * T(x,K(y \ e,y)) by Theorem 47. Hence we are done by (L10).
Theorem 1389 (prov9_0b0addf2a8): (y * (((y \ x) \ e) \ e)) / y = ((x / y) \ e) \ e
Proof. We have (y * T(((x / y) \ e) \ e,y)) / y = ((x / y) \ e) \ e by Proposition 48. Hence we are done by Theorem 920.
Theorem 1390 (prov9_2305a31443): T(x / y,(y \ x) \ e) = ((x / y) \ e) \ e
Proof. We have T(x / y,(y \ x) \ e) = (y * (((y \ x) \ e) \ e)) / y by Theorem 1330. Hence we are done by Theorem 1389.
Theorem 1391 (prov9_585493754c): ((x / y) \ e) \ e = T(x / y,x \ y)
Proof. We have T(x / y,(y \ x) \ e) = (y * (((y \ x) \ e) \ e)) / y by Theorem 1330. Then
(L2) T(x / y,(y \ x) \ e) = (y * T(y \ x,x \ y)) / y
by Theorem 387. We have (y * T(y \ x,x \ y)) / y = T(x / y,x \ y) by Theorem 49. Then T(x / y,(y \ x) \ e) = T(x / y,x \ y) by (L2). Hence we are done by Theorem 1390.
Theorem 1392 (prov9_4fab0816ae): T(y,((x * y) / x) \ e) = (y \ e) \ e
Proof. We have T(T((x * y) / x,x),((x * y) / x) \ e) = T((((x * y) / x) \ e) \ e,x) by Theorem 18. Then T(y,((x * y) / x) \ e) = T((((x * y) / x) \ e) \ e,x) by Theorem 7. Then ((x \ (x * y)) \ e) \ e = T(y,((x * y) / x) \ e) by Theorem 920. Hence we are done by Axiom 3.
Theorem 1393 (prov9_eb1addbd8b): ((x / (x * y)) \ e) \ e = x / (x * ((y \ e) \ e))
Proof. We have
(L7) (((x / (x * y)) \ e) \ e) * (K((x / (x * y)) \ e,x / (x * y)) * (x * y)) = (x / (x * y)) * (x * y)
by Theorem 1169. We have T(x / (x * y),(x / (x * y)) \ e) * (K((x / (x * y)) \ e,x / (x * y)) * (x * y)) = (((x / (x * y)) \ e) \ e) * (K((x / (x * y)) \ e,x / (x * y)) * (x * y)) by Theorem 464. Then T(x / (x * y),(x / (x * y)) \ e) * (K((x / (x * y)) \ e,x / (x * y)) * (x * y)) = (x / (x * y)) * (x * y) by (L7). Then
(L10) T(x / (x * y),(x / (x * y)) \ e) \ ((x / (x * y)) * (x * y)) = K((x / (x * y)) \ e,x / (x * y)) * (x * y)
by Proposition 2. We have x / (T(x / (x * y),(x / (x * y)) \ e) \ ((x / (x * y)) * (x * y))) = T(x / (x * y),(x / (x * y)) \ e) by Theorem 24. Then x / (K((x / (x * y)) \ e,x / (x * y)) * (x * y)) = T(x / (x * y),(x / (x * y)) \ e) by (L10). Then
(L13) x / (K((x * y) / x,x / (x * y)) * (x * y)) = T(x / (x * y),(x / (x * y)) \ e)
by Theorem 429. We have T(x / (x * y),(x / (x * y)) \ e) = ((x / (x * y)) \ e) \ e by Proposition 49. Then
(L15) x / (K((x * y) / x,x / (x * y)) * (x * y)) = ((x / (x * y)) \ e) \ e
by (L13). We have K((x * y) / x,x / (x * y)) * (x * y) = x * T(x \ (x * y),x / (x * y)) by Theorem 430. Then K((x * y) / x,x / (x * y)) * (x * y) = x * T(y,x / (x * y)) by Axiom 3. Then
(L16) x / (x * T(y,x / (x * y))) = ((x / (x * y)) \ e) \ e
by (L15). We have T(y,y \ e) = (y \ e) \ e by Proposition 49. Then
(L2) T(y,y \ e) = T(y,((x * y) / x) \ e)
by Theorem 1392. We have T(y,((x * y) / x) \ e) = T(y,x / (x * y)) by Theorem 1034. Then T(y,y \ e) = T(y,x / (x * y)) by (L2). Then x / (x * T(y,y \ e)) = ((x / (x * y)) \ e) \ e by (L16). Hence we are done by Proposition 49.
Theorem 1394 (prov9_2034587b87): x / (x * (y \ e)) = ((x / (x * (e / y))) \ e) \ e
Proof. We have x / (x * (((e / y) \ e) \ e)) = ((x / (x * (e / y))) \ e) \ e by Theorem 1393. Hence we are done by Proposition 25.
Theorem 1395 (prov9_2ba4d031e6): y * T(x,(y \ e) \ e) = x * y
Proof. We have T(T(x * y,K(y,y \ e)),K(y \ e,y)) = x * y by Theorem 701. Then T(y * T(x,(y \ e) \ e),K(y \ e,y)) = x * y by Theorem 705. Hence we are done by Theorem 717.
Theorem 1396 (prov9_e09ca7dfc6): (e / x) * (y * K(x \ e,x)) = (x \ e) * y
Proof. We have (e / x) * (y * K(x \ e,x)) = (x \ e) * T(y,K(x \ e,x)) by Theorem 1162. Hence we are done by Theorem 717.
Theorem 1397 (prov9_1e256f1074): T(x,y \ e) = T(x,e / y)
Proof. We have T(T(x,((e / y) \ e) \ e),K((e / y) \ e,e / y)) = T(x,e / y) by Theorem 708. Then T(T(x,y \ e),K((e / y) \ e,e / y)) = T(x,e / y) by Proposition 25. Hence we are done by Theorem 717.
Theorem 1398 (prov9_e511b83eff): x * (e / y) = (e / y) * T(x,y \ e)
Proof. We have K(y,y \ e) * ((x * (e / y)) * K(y \ e,y)) = T(x * (e / y),K(y \ e,y)) by Theorem 1377. Then
(L2) K(y,y \ e) * (x * (y \ e)) = T(x * (e / y),K(y \ e,y))
by Theorem 412. We have (e / y) * ((y \ e) \ (x * (y \ e))) = K(y,y \ e) * (x * (y \ e)) by Theorem 1375. Then (e / y) * T(x,y \ e) = K(y,y \ e) * (x * (y \ e)) by Definition 3. Then T(x * (e / y),K(y \ e,y)) = (e / y) * T(x,y \ e) by (L2). Hence we are done by Theorem 717.
Theorem 1399 (prov9_e0492260c4): (y * K(x \ e,x)) / (x * y) = R(x \ e,x,y)
Proof. We have K(((y / K(x,x \ e)) / (x * (y / K(x,x \ e)))) \ e,(y / K(x,x \ e)) / (x * (y / K(x,x \ e)))) = K(x,x \ e) by Theorem 687. Then K(R(e / x,x,y / K(x,x \ e)) \ e,(y / K(x,x \ e)) / (x * (y / K(x,x \ e)))) = K(x,x \ e) by Proposition 79. Then
(L13) K(R(e / x,x,y / K(x,x \ e)) \ e,R(e / x,x,y / K(x,x \ e))) = K(x,x \ e)
by Proposition 79. We have R(R(e / x,x,y / K(x,x \ e)),x * (y / K(x,x \ e)),K(R(e / x,x,y / K(x,x \ e)) \ e,R(e / x,x,y / K(x,x \ e)))) = R(e / x,x,y / K(x,x \ e)) by Theorem 1293. Then
(L15) R(R(e / x,x,y / K(x,x \ e)),x * (y / K(x,x \ e)),K(x,x \ e)) = R(e / x,x,y / K(x,x \ e))
by (L13). We have R(R(e / x,x,y / K(x,x \ e)),x * (y / K(x,x \ e)),K(x,x \ e)) = ((y / K(x,x \ e)) * K(x,x \ e)) / ((x * (y / K(x,x \ e))) * K(x,x \ e)) by Theorem 1325. Then
(L17) R(e / x,x,y / K(x,x \ e)) = ((y / K(x,x \ e)) * K(x,x \ e)) / ((x * (y / K(x,x \ e))) * K(x,x \ e))
by (L15). We have R(e / x,x,(y / K(x,x \ e)) * K(x,x \ e)) = ((y / K(x,x \ e)) * K(x,x \ e)) / (x * ((y / K(x,x \ e)) * K(x,x \ e))) by Proposition 79. Then R(e / x,x,(y / K(x,x \ e)) * K(x,x \ e)) = ((y / K(x,x \ e)) * K(x,x \ e)) / ((x * (y / K(x,x \ e))) * K(x,x \ e)) by Theorem 623. Then
(L20) R(e / x,x,y / K(x,x \ e)) = R(e / x,x,(y / K(x,x \ e)) * K(x,x \ e))
by (L17). We have R(R(e / x,x,(y / K(x,x \ e)) * K(x,x \ e)),x,x \ e) = R(x \ e,x,(y / K(x,x \ e)) * K(x,x \ e)) by Theorem 1323. Then
(L22) R(R(e / x,x,y / K(x,x \ e)),x,x \ e) = R(x \ e,x,(y / K(x,x \ e)) * K(x,x \ e))
by (L20). We have R(R(e / x,x,y / K(x,x \ e)),x,x \ e) = R(x \ e,x,y / K(x,x \ e)) by Theorem 1323. Then
(L24) R(x \ e,x,(y / K(x,x \ e)) * K(x,x \ e)) = R(x \ e,x,y / K(x,x \ e))
by (L22). We have (K(x \ e,x) * (K(x,x \ e) * (y * K(x \ e,x)))) / (x * (K(x,x \ e) * (y * K(x \ e,x)))) = R(x \ e,x,K(x,x \ e) * (y * K(x \ e,x))) by Theorem 791. Then (y * K(x \ e,x)) / (x * (K(x,x \ e) * (y * K(x \ e,x)))) = R(x \ e,x,K(x,x \ e) * (y * K(x \ e,x))) by Theorem 693. Then
(L6) R(x \ e,x,K(x,x \ e) * (y * K(x \ e,x))) = (y * K(x \ e,x)) / (T(x,x \ e) * (y * K(x \ e,x)))
by Theorem 1153. We have (y * K(x \ e,x)) / (T(x,x \ e) * (y * K(x \ e,x))) = R(x \ e,x,y * K(x \ e,x)) by Theorem 1381. Then R(x \ e,x,K(x,x \ e) * (y * K(x \ e,x))) = R(x \ e,x,y * K(x \ e,x)) by (L6). Then
(L9) R(x \ e,x,T(y,K(x \ e,x))) = R(x \ e,x,y * K(x \ e,x))
by Theorem 1377. We have (K(x \ e,x) * T(y,K(x \ e,x))) / (x * T(y,K(x \ e,x))) = R(x \ e,x,T(y,K(x \ e,x))) by Theorem 791. Then
(L2) (y * K(x \ e,x)) / (x * T(y,K(x \ e,x))) = R(x \ e,x,T(y,K(x \ e,x)))
by Proposition 46.
(y * K(x \ e,x)) / T(x * y,K(x \ e,x))
=R(x \ e,x,T(y,K(x \ e,x))) by (L2), Theorem 1388
=R(x \ e,x,y / K(x,x \ e)) by (L9), Theorem 425
=R(x \ e,x,y) by (L24), Axiom 6

Then (y * K(x \ e,x)) / T(x * y,K(x \ e,x)) = R(x \ e,x,y). Hence we are done by Theorem 717.
Theorem 1400 (prov9_8b8a2faef5): x * (y \ e) = K(y \ e,y) * (x * (e / y))
Proof. We have K(y,y \ e) \ (x * (e / y)) = T(x * (y \ e),K(y,y \ e)) by Theorem 1368. Then K(y \ e,y) * (x * (e / y)) = T(x * (y \ e),K(y,y \ e)) by Theorem 694. Hence we are done by Theorem 718.
Theorem 1401 (prov9_bf12809a18): T(x,T(y \ e,y)) = T(x,y \ e)
Proof. We have T(x,((y \ e) \ e) \ e) = T(x,y \ e) by Theorem 721. Hence we are done by Theorem 209.
Theorem 1402 (prov9_b70540a166): T(x,T(y / z,y \ z)) = T(x,y / z)
Proof. We have T(x,((y / z) \ e) \ e) = T(x,y / z) by Theorem 721. Hence we are done by Theorem 1391.
Theorem 1403 (prov9_cf50311f15): ((y \ e) \ e) * T(x,y) = x * T(y,y \ e)
Proof. We have T(y,y \ e) * T(x,T(y,y \ e)) = x * T(y,y \ e) by Proposition 46. Then ((y \ e) \ e) * T(x,T(y,y \ e)) = x * T(y,y \ e) by Theorem 464. Hence we are done by Theorem 722.
Theorem 1404 (prov9_87b2eaef72): T(((e / y) * x) * y,y \ e) = R(x,e / y,y)
Proof. We have (((e / y) * x) * y) / e = ((e / y) * x) * y by Proposition 27. Then
(L2) (((e / y) * x) * y) / ((e / y) * y) = ((e / y) * x) * y
by Axiom 6. We have R(x,e / y,y) = T((((e / y) * x) * y) / ((e / y) * y),e / y) by Theorem 110. Then R(x,e / y,y) = T(((e / y) * x) * y,e / y) by (L2). Hence we are done by Theorem 1397.
Theorem 1405 (prov9_e8ea33f228): x * K(x,x \ e) = T(x,T(x,y) \ e)
Proof. We have T(x,y) * ((x * (T(x,y) \ e)) / x) = T(x * ((x * (x \ e)) / x),y) by Theorem 345. Then T(x,y) * ((x * (T(x,y) \ e)) / x) = T(x * (e / x),y) by Axiom 4. Then T(x,y) * ((T(x,y) \ x) / x) = T(x * (e / x),y) by Theorem 175. Then
(L4) T(x,y) * ((T(x,y) \ x) / x) = T(K(x,x \ e),y)
by Theorem 250. We have x * (T(x,T(x,y) \ e) / x) = T(x,T(x,y) \ e) by Theorem 757. Then x * (T(x,y) * ((T(x,y) \ x) / x)) = T(x,T(x,y) \ e) by Theorem 1351. Then x * T(K(x,x \ e),y) = T(x,T(x,y) \ e) by (L4). Hence we are done by Theorem 723.
Theorem 1406 (prov9_93f7381319): (K(y \ e,y) * x) * ((y \ e) \ e) = x * y
Proof. We have (x * K(y \ e,y)) * ((y \ e) \ e) = x * y by Theorem 908. Hence we are done by Theorem 923.
Theorem 1407 (prov9_4784d63248): K(y \ e,y) * (x * y) = x * (e / (e / y))
Proof. We have (x * y) * K(y \ e,y) = x * (e / (e / y)) by Theorem 1365. Hence we are done by Theorem 923.
Theorem 1408 (prov9_8597dfd0a8): x * ((y \ e) \ e) = K(y,y \ e) * (x * y)
Proof.
(x * ((y \ e) \ e)) * K(y \ e,y)
=x * y by Theorem 655
=(K(y,y \ e) * (x * y)) * K(y \ e,y) by Theorem 924

Then (x * ((y \ e) \ e)) * K(y \ e,y) = (K(y,y \ e) * (x * y)) * K(y \ e,y). Hence we are done by Proposition 10.
Theorem 1409 (prov9_7d6d329c2d): ((y \ e) \ e) * x = (y * x) * K(y,y \ e)
Proof.
K(y \ e,y) * (((y \ e) \ e) * x)
=y * x by Theorem 1376
=K(y \ e,y) * ((y * x) * K(y,y \ e)) by Theorem 1182

Then K(y \ e,y) * (((y \ e) \ e) * x) = K(y \ e,y) * ((y * x) * K(y,y \ e)). Hence we are done by Proposition 9.
Theorem 1410 (prov9_18276855bc): T(x \ T(x,y),y \ e) = x \ T(x,y)
Proof. We have T(e / y,K(y \ (y / (x \ e)),y)) = e / y by Theorem 367. Then
(L3) T(e / y,x \ T(x,y)) = e / y
by Theorem 198. We have (x \ T(x,y)) * T(e / y,x \ T(x,y)) = (e / y) * (x \ T(x,y)) by Proposition 46. Then
(L5) (x \ T(x,y)) * (e / y) = (e / y) * (x \ T(x,y))
by (L3). We have (e / y) * T(x \ T(x,y),y \ e) = (x \ T(x,y)) * (e / y) by Theorem 1398. Hence we are done by (L5) and Proposition 7.
Theorem 1411 (prov9_77119aba5e): T(y,T(y,x) \ e) = (y \ e) \ e
Proof. We have y * (y \ ((y \ e) \ e)) = (y \ e) \ e by Axiom 4. Then y * K(y,e / y) = (y \ e) \ e by Theorem 559. Then y * K(y,y \ e) = (y \ e) \ e by Theorem 780. Hence we are done by Theorem 1405.
Theorem 1412 (prov9_7a5d6c449a): T(y,T(y,x) \ e) = T(y,y \ e)
Proof. We have y * K(y,y \ e) = T(y,y \ e) by Theorem 562. Hence we are done by Theorem 1405.
Theorem 1413 (prov9_c8c2d5aad2): (K(y,y \ e) * x) * y = x * T(y,y \ e)
Proof. We have (K(y,y \ e) * x) * y = K(y,y \ e) * (x * y) by Theorem 1225. Hence we are done by Theorem 925.
Theorem 1414 (prov9_78331cf990): K(x \ e,((x \ e) \ e) * y) = K(x \ e,x * y)
Proof. We have T(x \ e,(((x * y) * (e / x)) / (x \ e)) * K(x \ e,(x \ e) \ e)) = T(x \ e,((x * y) * (e / x)) / (x \ e)) by Theorem 626. Then
(L6) T(x \ e,(((x * y) * (e / x)) / (x \ e)) * K(x \ e,x)) = T(x \ e,((x * y) * (e / x)) / (x \ e))
by Theorem 779. We have (((x * y) * (e / x)) / (x \ e)) * (K(x \ e,x) * T(e / x,(((x * y) * (e / x)) / (x \ e)) * K(x \ e,x))) = L(e / x,K(x \ e,x),((x * y) * (e / x)) / (x \ e)) * ((((x * y) * (e / x)) / (x \ e)) * K(x \ e,x)) by Theorem 1328. Then (((x * y) * (e / x)) / (x \ e)) * ((x \ e) * (x * T(x \ e,(((x * y) * (e / x)) / (x \ e)) * K(x \ e,x)))) = L(e / x,K(x \ e,x),((x * y) * (e / x)) / (x \ e)) * ((((x * y) * (e / x)) / (x \ e)) * K(x \ e,x)) by Theorem 567. Then (((x * y) * (e / x)) / (x \ e)) * ((x \ e) * (x * T(x \ e,((x * y) * (e / x)) / (x \ e)))) = L(e / x,K(x \ e,x),((x * y) * (e / x)) / (x \ e)) * ((((x * y) * (e / x)) / (x \ e)) * K(x \ e,x)) by (L6). Then (e / x) * ((((x * y) * (e / x)) / (x \ e)) * K(x \ e,x)) = (((x * y) * (e / x)) / (x \ e)) * ((x \ e) * (x * T(x \ e,((x * y) * (e / x)) / (x \ e)))) by Theorem 644. Then
(L11) (((x * y) * (e / x)) / (x \ e)) * ((x \ e) * (x * T(x \ e,((x * y) * (e / x)) / (x \ e)))) = T((x \ e) * (((x * y) * (e / x)) / (x \ e)),K(x \ e,x))
by Theorem 1385. We have ((x * y) * (e / x)) \ ((((x * y) * (e / x)) / (x \ e)) * ((x \ e) * (x * T(x \ e,((x * y) * (e / x)) / (x \ e))))) = L(x * T(x \ e,((x * y) * (e / x)) / (x \ e)),x \ e,((x * y) * (e / x)) / (x \ e)) by Theorem 4. Then ((x * y) * (e / x)) \ ((((x * y) * (e / x)) / (x \ e)) * ((x \ e) * (x * T(x \ e,((x * y) * (e / x)) / (x \ e))))) = K(e / x,((x * y) * (e / x)) / (x \ e)) by Theorem 654. Then
(L14) ((x * y) * (e / x)) \ T((x \ e) * (((x * y) * (e / x)) / (x \ e)),K(x \ e,x)) = K(e / x,((x * y) * (e / x)) / (x \ e))
by (L11). We have T((x \ e) * (((x * y) * (e / x)) / (x \ e)),K(x \ e,x)) = (e / x) * ((((x * y) * (e / x)) / (x \ e)) * K(x \ e,x)) by Theorem 1385. Then T((x \ e) * (((x * y) * (e / x)) / (x \ e)),K(x \ e,x)) = (e / x) * (((x * y) * (e / x)) / (e / x)) by Theorem 1362. Then
(L15) ((x * y) * (e / x)) \ ((e / x) * (((x * y) * (e / x)) / (e / x))) = K(e / x,((x * y) * (e / x)) / (x \ e))
by (L14). We have (x \ K(e / x,((x * y) * (e / x)) / (e / x))) * ((x \ e) \ e) = K(x \ e,((x * y) * (e / x)) / (e / x)) by Theorem 1387. Then (x \ (((x * y) * (e / x)) \ ((e / x) * (((x * y) * (e / x)) / (e / x))))) * ((x \ e) \ e) = K(x \ e,((x * y) * (e / x)) / (e / x)) by Theorem 3. Then
(L18) (x \ K(e / x,((x * y) * (e / x)) / (x \ e))) * ((x \ e) \ e) = K(x \ e,((x * y) * (e / x)) / (e / x))
by (L15). We have (x \ K(e / x,((x * y) * (e / x)) / (x \ e))) * ((x \ e) \ e) = K(x \ e,((x * y) * (e / x)) / (x \ e)) by Theorem 1387. Then K(x \ e,((x * y) * (e / x)) / (e / x)) = K(x \ e,((x * y) * (e / x)) / (x \ e)) by (L18). Then K(x \ e,x * y) = K(x \ e,((x * y) * (e / x)) / (x \ e)) by Axiom 5. Then
(L22) K(x \ e,((x * y) / (x \ e)) * (e / x)) = K(x \ e,x * y)
by Theorem 1348. We have (((x * y) / (x \ e)) * (x \ e)) * K(x,x \ e) = ((x * y) / (x \ e)) * (e / x) by Theorem 1364. Then (x * y) * K(x,x \ e) = ((x * y) / (x \ e)) * (e / x) by Axiom 6. Then K(x \ e,(x * y) * K(x,x \ e)) = K(x \ e,x * y) by (L22). Hence we are done by Theorem 1409.
Theorem 1415 (prov9_f38c595de6): K(T(e / x,y) \ e,z) = K(T(x \ e,y) \ e,z)
Proof. We have K(e / T(x \ e,y),z) = K(T(x \ e,y) \ e,z) by Theorem 732. Hence we are done by Theorem 917.
Theorem 1416 (prov9_ce349c69b7): K(y,y \ e) = K(T(y,x),y \ e)
Proof. We have K(T(y,x),T(y,x) \ e) = K(T(y,x),y \ e) by Theorem 711. Hence we are done by Theorem 744.
Theorem 1417 (prov9_9f74cb1155): (e / T(x,y)) * K(x \ e,x) = T(x,y) \ e
Proof.
((e / T(x,y)) * K(x \ e,x)) * K(x,x \ e)
=e / T(x,y) by Theorem 426
=(T(x,y) \ e) * K(x,x \ e) by Theorem 746

Then ((e / T(x,y)) * K(x \ e,x)) * K(x,x \ e) = (T(x,y) \ e) * K(x,x \ e). Hence we are done by Proposition 10.
Theorem 1418 (prov9_e3e5a6f8b1): (y \ e) \ (T(y,x) \ e) = T(y,x) \ y
Proof. We have (y \ e) \ (T(y,x) \ e) = (y \ T(y,x)) \ e by Theorem 638. Hence we are done by Theorem 749.
Theorem 1419 (prov9_22b8c723dc): T(y,x * T(x \ e,y)) = y
Proof. We have T(y,K(y \ (y / x),y)) = y by Theorem 755. Hence we are done by Theorem 196.
Theorem 1420 (prov9_5dbada98f6): R(x,K(y,z),z) = ((x * K(y,z)) * z) / (z * K(y,z))
Proof. We have R(x,K(y,z),z) = ((x * K(y,z)) * z) / (K(y,z) * z) by Definition 5. Hence we are done by Theorem 933.
Theorem 1421 (prov9_21fcc39301): K((y / x) \ e,y / x) = K(y \ x,x \ y)
Proof. We have K(y \ x,(((x * y) / x) * (y \ x)) \ ((x * y) / x)) = K(y \ x,(y \ x) \ e) by Theorem 392. Then K(y \ x,T(x,y) \ ((x * y) / x)) = K(y \ x,(y \ x) \ e) by Theorem 179. Then
(L6) K(y \ x,T(y / x,T(x,y))) = K(y \ x,(y \ x) \ e)
by Theorem 1083. We have K(y \ x,(y \ x) \ e) = K(y \ x,x \ y) by Theorem 612. Then
(L8) K(y \ x,T(y / x,T(x,y))) = K(y \ x,x \ y)
by (L6). We have K(((x * y) / x) \ (((x * y) / x) / T(y / x,T(x,y))),T(y / x,T(x,y))) = K(T(y / x,T(x,y)) \ e,T(y / x,T(x,y))) by Theorem 1371. Then K(((x * y) / x) \ T(x,y),T(y / x,T(x,y))) = K(T(y / x,T(x,y)) \ e,T(y / x,T(x,y))) by Theorem 1334. Then K(T(y / x,T(x,y)) \ e,T(y / x,T(x,y))) = K(y \ x,T(y / x,T(x,y))) by Theorem 1340. Then K(T(y / x,T(x,y)) \ e,T(y / x,T(x,y))) = K(y \ x,x \ y) by (L8). Hence we are done by Theorem 929.
Theorem 1422 (prov9_fa31b65495): y * (x * T(x \ e,y)) = x * ((y \ (x \ y)) * y)
Proof. We have T(y,x * T(x \ e,y)) = y by Theorem 1419. Then
(L2) T(x * T(x \ e,y),y) = x * T(x \ e,y)
by Proposition 21. We have y * T(x * T(x \ e,y),y) = (x * T(x \ e,y)) * y by Proposition 46. Then y * T(x * T(x \ e,y),y) = x * ((y \ (x \ y)) * y) by Theorem 236. Hence we are done by (L2).
Theorem 1423 (prov9_87277ea181): K(y \ x,x \ y) = K(x / y,(x / y) \ e)
Proof. We have K(x / y,(x / y) \ e) = K(x / y,y / x) by Theorem 402. Then K(x / y,(x / y) \ e) = K((y / x) \ e,y / x) by Theorem 429. Hence we are done by Theorem 1421.
Theorem 1424 (prov9_0560b9c5a7): T(x \ T(x,y),x) = x \ T(x,y)
Proof. We have T(T(x,y) / x,x) = x \ T(x,y) by Proposition 47. Hence we are done by Theorem 758.
Theorem 1425 (prov9_4cfdd6233e): K(y \ (y / (x \ e)),y) * x = T(x,y)
Proof. We have (x \ T(x,y)) * x = T(x,y) by Theorem 759. Hence we are done by Theorem 198.
Theorem 1426 (prov9_62cea11bcf): x \ T(x,y) = K(y \ (y / T(x \ e,x)),y)
Proof. We have T(x \ T(x,y),x) = K(y \ (y / T(x \ e,x)),y) by Theorem 575. Hence we are done by Theorem 1424.
Theorem 1427 (prov9_ac396a6d0c): K(z \ y,y \ z) * (x * (y / z)) = x * (((y / z) \ e) \ e)
Proof. We have K(y / z,(y / z) \ e) * (x * (y / z)) = x * (((y / z) \ e) \ e) by Theorem 1408. Hence we are done by Theorem 1423.
Theorem 1428 (prov9_5e68441418): T(y \ e,T(y,x) \ y) = y \ e
Proof. We have T(y \ e,K(y \ (y / ((x \ y) / T(y,x))),y)) = y \ e by Theorem 607. Hence we are done by Theorem 932.
Theorem 1429 (prov9_5fe0c95d9f): y \ (((y * x) / y) \ x) = T(y,R(x,x \ e,y)) \ e
Proof. We have y \ (T(y,R(x,x \ e,y)) \ y) = T(y,R(x,x \ e,y)) \ e by Theorem 1339. Hence we are done by Theorem 769.
Theorem 1430 (prov9_e07ea8db90): x \ (y \ T(y,x)) = (x * (T(y,x) \ y)) \ e
Proof. We have x \ (T(x,R(T(y,x),T(y,x) \ e,x)) \ x) = T(x,R(T(y,x),T(y,x) \ e,x)) \ e by Theorem 1339. Then x \ (((T(y,x) \ y) * x) \ x) = T(x,R(T(y,x),T(y,x) \ e,x)) \ e by Theorem 591. Then ((T(y,x) \ y) * x) \ e = x \ (((T(y,x) \ y) * x) \ x) by Theorem 591. Then x \ ((x * (T(y,x) \ y)) \ x) = ((T(y,x) \ y) * x) \ e by Theorem 740. Then x \ ((x * (T(y,x) \ y)) \ x) = (x * (T(y,x) \ y)) \ e by Theorem 740. Hence we are done by Theorem 771.
Theorem 1431 (prov9_0bbda83c22): (x * K(x,y)) \ x = K(x,y) \ e
Proof. We have T(x,R(y,x,(y * x) \ x)) \ x = K(x,y) \ e by Theorem 949. Hence we are done by Theorem 871.
Theorem 1432 (prov9_9f9ff06116): (x \ T(x,y)) * (y \ e) = y \ (x \ T(x,y))
Proof. We have (T(y,R(T(x,y),T(x,y) \ e,y)) \ y) * T(y \ e,T(y,R(T(x,y),T(x,y) \ e,y)) \ y) = (y \ e) * (T(y,R(T(x,y),T(x,y) \ e,y)) \ y) by Proposition 46. Then
(L4) (T(y,R(T(x,y),T(x,y) \ e,y)) \ y) * (y \ e) = (y \ e) * (T(y,R(T(x,y),T(x,y) \ e,y)) \ y)
by Theorem 1428. We have (y \ e) * ((y \ e) \ (T(y,R(T(x,y),T(x,y) \ e,y)) \ e)) = T(y,R(T(x,y),T(x,y) \ e,y)) \ e by Axiom 4. Then (y \ e) * (T(y,R(T(x,y),T(x,y) \ e,y)) \ y) = T(y,R(T(x,y),T(x,y) \ e,y)) \ e by Theorem 1418. Then (T(y,R(T(x,y),T(x,y) \ e,y)) \ y) * (y \ e) = T(y,R(T(x,y),T(x,y) \ e,y)) \ e by (L4). Then (T(y,R(T(x,y),T(x,y) \ e,y)) \ y) \ (T(y,R(T(x,y),T(x,y) \ e,y)) \ e) = y \ e by Proposition 2. Then
(L9) (x \ T(x,y)) \ (T(y,R(T(x,y),T(x,y) \ e,y)) \ e) = y \ e
by Theorem 770. We have y \ (T(y,R(T(x,y),T(x,y) \ e,y)) \ y) = T(y,R(T(x,y),T(x,y) \ e,y)) \ e by Theorem 1339. Then y \ (x \ T(x,y)) = T(y,R(T(x,y),T(x,y) \ e,y)) \ e by Theorem 770. Then
(L10) (x \ T(x,y)) \ (y \ (x \ T(x,y))) = y \ e
by (L9). We have (x \ T(x,y)) * ((x \ T(x,y)) \ (y \ (x \ T(x,y)))) = y \ (x \ T(x,y)) by Axiom 4. Hence we are done by (L10).
Theorem 1433 (prov9_199343d374): T(y,y \ K(x,y)) = T(y,y \ e)
Proof. We have T(y,T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ e) = T(y,y \ e) by Theorem 1412. Then T(y,y \ (((y * T(((y * x) \ y) \ e,y)) / y) \ T(((y * x) \ y) \ e,y))) = T(y,y \ e) by Theorem 1429. Then T(y,y \ ((((y * x) \ y) \ e) \ T(((y * x) \ y) \ e,y))) = T(y,y \ e) by Proposition 48. Then T(y,y \ ((((y * x) \ y) \ e) \ (((y * x) \ y) \ K(x,y)))) = T(y,y \ e) by Theorem 538. Hence we are done by Theorem 735.
Theorem 1434 (prov9_4b9c5a842e): K(x,y) / T(y,y \ e) = y \ K(x,y)
Proof. We have K(x,y) / T(y,y \ K(x,y)) = y \ K(x,y) by Theorem 15. Hence we are done by Theorem 1433.
Theorem 1435 (prov9_179fa87fca): (y \ K(x,y)) * K(y,y \ e) = K(x,y) / y
Proof. We have (K(x,y) / T(y,y \ e)) * K(y,y \ e) = K(x,y) / y by Theorem 1361. Hence we are done by Theorem 1434.
Theorem 1436 (prov9_40925b8813): (y \ K(x,y)) * y = K(x,y) * K(y \ e,y)
Proof. We have (K(x,y) / T(y,y \ e)) * y = K(x,y) * K(y \ e,y) by Theorem 1148. Hence we are done by Theorem 1434.
Theorem 1437 (prov9_52f12e955c): y * (K(x,y) / y) = K(x,y) * K(y,y \ e)
Proof. We have y * ((y \ K(x,y)) * K(y,y \ e)) = K(x,y) * K(y,y \ e) by Theorem 622. Hence we are done by Theorem 1435.
Theorem 1438 (prov9_48b76cd717): K(y,y \ e) * K(x,y) = y * (K(x,y) / y)
Proof. We have K(y,y \ e) * K(x,y) = K(x,y) * K(y,y \ e) by Theorem 1356. Hence we are done by Theorem 1437.
Theorem 1439 (prov9_18ab2bfb51): (y \ e) * (y * (K(x,y) / y)) = (e / y) * K(x,y)
Proof. We have (y \ e) * (K(x,y) * K(y,y \ e)) = (e / y) * K(x,y) by Theorem 1185. Hence we are done by Theorem 1437.
Theorem 1440 (prov9_236d5d609d): R(T(x,y * K(z \ e,z)),y,K(z \ e,z)) = L(T(x,y),y,K(z \ e,z))
Proof. We have (y * K(z \ e,z)) \ ((x * y) * K(z \ e,z)) = R(T(x,y * K(z \ e,z)),y,K(z \ e,z)) by Proposition 81. Then
(L2) (K(z \ e,z) * y) \ ((x * y) * K(z \ e,z)) = R(T(x,y * K(z \ e,z)),y,K(z \ e,z))
by Theorem 923. We have (K(z \ e,z) * y) \ (K(z \ e,z) * (x * y)) = L(T(x,y),y,K(z \ e,z)) by Proposition 62. Then (K(z \ e,z) * y) \ ((x * y) * K(z \ e,z)) = L(T(x,y),y,K(z \ e,z)) by Theorem 923. Hence we are done by (L2).
Theorem 1441 (prov9_7549fc56a8): y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e)) = K(x,y) \ y
Proof. We have y / K(x,y) = K(x,y) \ y by Theorem 766. Hence we are done by Theorem 776.
Theorem 1442 (prov9_5f2e7a556c): K(x,y) / y = e / (K(x,y) \ y)
Proof. We have R((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * ((T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) / (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y),e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y),(e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e) / ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e) = ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * ((T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) / (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) * (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) by Theorem 73. Then R((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * ((T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) / (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y),e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y),(e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e) / ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e) = (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ (T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) by Theorem 125. Then ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * ((T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e) = (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ (T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) by Theorem 1332. Then ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * ((T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e) = (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) by Theorem 1345. Then ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * ((T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e) = (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) by Proposition 25. Then ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / ((e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) \ e) = (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) by Axiom 6. Then ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y) = (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) by Proposition 25. Then (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * y = ((R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y) * T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y))) / R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y) by Proposition 48. Then y / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y) = (e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) * y by Proposition 48. Then (y / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y)) / y = e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y) by Proposition 1. Then (T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y) / y = e / T(T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)),T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y) by Theorem 15. Then e / ((T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y) \ y) = (T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y) / y by Proposition 49. Then e / T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) = (T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) \ y) / y by Theorem 930. Then ((((y * x) \ y) \ e) \ T(((y * x) \ y) \ e,y)) / y = e / T(y,R(T(((y * x) \ y) \ e,y),T(((y * x) \ y) \ e,y) \ e,y)) by Theorem 770. Then e / (y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e))) = ((((y * x) \ y) \ e) \ T(((y * x) \ y) \ e,y)) / y by Theorem 742. Then e / (y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e))) = (((y * x) \ y) * T(((y * x) \ y) \ e,y)) / y by Theorem 729. Then e / (K(x,y) \ y) = (((y * x) \ y) * T(((y * x) \ y) \ e,y)) / y by Theorem 1441. Hence we are done by Theorem 539.
Theorem 1443 (prov9_ebc6c71e2f): (K(x,y) / y) \ e = K(x,y) \ y
Proof. We have (e / (K(x,y) \ y)) \ e = K(x,y) \ y by Proposition 25. Hence we are done by Theorem 1442.
Theorem 1444 (prov9_3647aba64e): y / (e / (K(x,y) / y)) = y * (e / (e / (y \ K(x,y))))
Proof. We have y / (e / (e / (K(x,y) \ y))) = y * (e / (e / (y \ K(x,y)))) by Theorem 647. Hence we are done by Theorem 1442.
Theorem 1445 (prov9_5b09abfbc6): K(K(y,y \ e) * x,y) = K(x,y)
Proof. We have K(K(y \ e,y) * (K(y,y \ e) * x),(y \ e) \ e) = (((y \ e) \ e) * (K(y \ e,y) * (K(y,y \ e) * x))) \ ((K(y \ e,y) * (K(y,y \ e) * x)) * ((y \ e) \ e)) by Definition 2. Then K(K(y \ e,y) * (K(y,y \ e) * x),(y \ e) \ e) = (y * (K(y,y \ e) * x)) \ ((K(y \ e,y) * (K(y,y \ e) * x)) * ((y \ e) \ e)) by Theorem 1169. Then
(L3) (y * (K(y,y \ e) * x)) \ ((K(y,y \ e) * x) * y) = K(K(y \ e,y) * (K(y,y \ e) * x),(y \ e) \ e)
by Theorem 1406. We have K(K(y,y \ e) * x,y) = (y * (K(y,y \ e) * x)) \ ((K(y,y \ e) * x) * y) by Definition 2. Then K(K(y,y \ e) * x,y) = K(K(y \ e,y) * (K(y,y \ e) * x),(y \ e) \ e) by (L3). Then K(x,(y \ e) \ e) = K(K(y,y \ e) * x,y) by Theorem 693. Hence we are done by Theorem 779.
Theorem 1446 (prov9_a454635335): K(y \ x,y) = K(((y \ e) \ e) \ x,y)
Proof. We have K(K(y,y \ e) * (((y \ e) \ e) \ x),y) = K(((y \ e) \ e) \ x,y) by Theorem 1445. Hence we are done by Theorem 680.
Theorem 1447 (prov9_1a62d1e356): K((e / y) * x,y) = K((y \ e) * x,y)
Proof. We have K(K(y,y \ e) * ((y \ e) * x),y) = K((y \ e) * x,y) by Theorem 1445. Hence we are done by Theorem 697.
Theorem 1448 (prov9_d1641e15b9): y \ K(x,y) = (K(x,y) \ y) \ e
Proof. We have (y * (T(((y * x) \ y) \ e,y) \ (((y * x) \ y) \ e))) \ e = y \ ((((y * x) \ y) \ e) \ T(((y * x) \ y) \ e,y)) by Theorem 1430. Then (K(x,y) \ y) \ e = y \ ((((y * x) \ y) \ e) \ T(((y * x) \ y) \ e,y)) by Theorem 1441. Then y \ (((y * x) \ y) * T(((y * x) \ y) \ e,y)) = (K(x,y) \ y) \ e by Theorem 729. Hence we are done by Theorem 539.
Theorem 1449 (prov9_f1769cb5f6): e / (y \ K(x,y)) = K(x,y) \ y
Proof. We have e / ((K(x,y) \ y) \ e) = K(x,y) \ y by Proposition 24. Hence we are done by Theorem 1448.
Theorem 1450 (prov9_5a0559dbe3): T(x,z \ K(y,z)) = T(x,K(y,z) / z)
Proof. We have T(e / (K(y,z) \ z),K(y,z) \ z) = (K(y,z) \ z) \ e by Proposition 47. Then T(K(y,z) / z,K(y,z) \ z) = (K(y,z) \ z) \ e by Theorem 1442. Then
(L3) T(K(y,z) / z,K(y,z) \ z) = z \ K(y,z)
by Theorem 1448. We have T(x,T(K(y,z) / z,K(y,z) \ z)) = T(x,K(y,z) / z) by Theorem 1402. Hence we are done by (L3).
Theorem 1451 (prov9_2b464efc89): (K(y,z) / z) * T(x,z \ K(y,z)) = x * (K(y,z) / z)
Proof. We have (K(y,z) / z) * T(x,K(y,z) / z) = x * (K(y,z) / z) by Proposition 46. Hence we are done by Theorem 1450.
Theorem 1452 (prov9_2356dd5fc8): K(K(x,y) \ y,y \ K(x,y)) = K(y,y \ e)
Proof. We have
(L1) y * (e / (e / (y \ K(x,y)))) = y / (e / (K(x,y) / y))
by Theorem 1444. Then y * (e / (K(x,y) \ y)) = y / (e / (K(x,y) / y)) by Theorem 1449. Then y / (e / (K(x,y) / y)) = y * (K(x,y) / y) by Theorem 1442. Then
(L4) y * (e / (e / (y \ K(x,y)))) = y * (K(x,y) / y)
by (L1). We have K(y,y \ e) * K(x,y) = y * (K(x,y) / y) by Theorem 1438. Then
(L6) K(y,y \ e) * K(x,y) = y * (e / (e / (y \ K(x,y))))
by (L4). We have K(K(x,y) \ y,y \ K(x,y)) * K(x,y) = y * (e / (e / (y \ K(x,y)))) by Theorem 731. Then K(K(x,y) \ y,y \ K(x,y)) * K(x,y) = K(y,y \ e) * K(x,y) by (L6). Hence we are done by Proposition 10.
Theorem 1453 (prov9_639deede91): K(x,z \ K(y,z)) = K(x,K(y,z) / z)
Proof. We have
(L4) (K(y,z) / z) * T(z,z \ K(y,z)) = z * (K(y,z) / z)
by Theorem 1451. Then (K(y,z) / z) * T(z,z \ K(y,z)) = z * (e / (K(y,z) \ z)) by Theorem 1442. Then z * (e / (K(y,z) \ z)) = z * (K(y,z) / z) by (L4). Then
(L7) z * (e / (e / (z \ K(y,z)))) = z * (K(y,z) / z)
by Theorem 1449. We have K((z \ K(y,z)) \ e,z \ K(y,z)) * (z * (z \ K(y,z))) = z * (e / (e / (z \ K(y,z)))) by Theorem 1407. Then K(K(y,z) \ z,z \ K(y,z)) * (z * (z \ K(y,z))) = z * (e / (e / (z \ K(y,z)))) by Theorem 666. Then K(z,z \ e) * (z * (z \ K(y,z))) = z * (e / (e / (z \ K(y,z)))) by Theorem 1452. Then
(L8) K(z,z \ e) * (z * (z \ K(y,z))) = z * (K(y,z) / z)
by (L7). We have K(z,z \ e) * (K(z \ e,z) * (z * (K(y,z) / z))) = z * (K(y,z) / z) by Theorem 442. Then K(z,z \ e) * (K(z \ e,z) * (z * (K(y,z) / z))) = K(z,z \ e) * (z * (z \ K(y,z))) by (L8). Then K(z \ e,z) * (z * (K(y,z) / z)) = z * (z \ K(y,z)) by Proposition 9. Then K(z \ e,z) * (z * (K(y,z) / z)) = z * ((K(y,z) \ z) \ e) by Theorem 1448. Then
(L13) K(z \ e,z) * (z * (K(y,z) / z)) = z * (((K(y,z) / z) \ e) \ e)
by Theorem 1443. We have K(z \ K(y,z),K(y,z) \ z) * (z * (K(y,z) / z)) = z * (((K(y,z) / z) \ e) \ e) by Theorem 1427. Then
(L15) K(z \ e,z) = K(z \ K(y,z),K(y,z) \ z)
by (L13) and Proposition 8. We have (K(y,z) * K((K(y,z) \ z) \ e,K(y,z) \ z)) / ((K(y,z) \ z) * K(y,z)) = R((K(y,z) \ z) \ e,K(y,z) \ z,K(y,z)) by Theorem 1399. Then (K(y,z) * K((K(y,z) \ z) \ e,K(y,z) \ z)) / z = R((K(y,z) \ z) \ e,K(y,z) \ z,K(y,z)) by Theorem 765. Then (K(y,z) * K(z \ K(y,z),K(y,z) \ z)) / z = R((K(y,z) \ z) \ e,K(y,z) \ z,K(y,z)) by Theorem 666. Then (K(y,z) * K(z \ e,z)) / z = R((K(y,z) \ z) \ e,K(y,z) \ z,K(y,z)) by (L15). Then
(L20) R((K(y,z) \ z) \ e,K(y,z) \ z,K(y,z)) = ((z \ K(y,z)) * z) / z
by Theorem 1436. We have ((z \ K(y,z)) * z) / z = z \ K(y,z) by Axiom 5. Then
(L22) R((K(y,z) \ z) \ e,K(y,z) \ z,K(y,z)) = z \ K(y,z)
by (L20). We have K(x,T(K(y,z) / ((K(y,z) \ z) * K(y,z)),(K(y,z) / ((K(y,z) \ z) * K(y,z))) \ e)) = K(x,((K(y,z) / ((K(y,z) \ z) * K(y,z))) \ e) \ e) by Theorem 1342. Then K(x,T(K(y,z) / ((K(y,z) \ z) * K(y,z)),((K(y,z) \ z) * K(y,z)) / K(y,z))) = K(x,((K(y,z) / ((K(y,z) \ z) * K(y,z))) \ e) \ e) by Theorem 1034. Then K(x,T(K(y,z) / ((K(y,z) \ z) * K(y,z)),K(y,z) \ z)) = K(x,((K(y,z) / ((K(y,z) \ z) * K(y,z))) \ e) \ e) by Axiom 5. Then K(x,R((K(y,z) \ z) \ e,K(y,z) \ z,K(y,z))) = K(x,((K(y,z) / ((K(y,z) \ z) * K(y,z))) \ e) \ e) by Theorem 89. Then K(x,R((K(y,z) \ z) \ e,K(y,z) \ z,K(y,z))) = K(x,K(y,z) / ((K(y,z) \ z) * K(y,z))) by Theorem 779. Then K(x,R((K(y,z) \ z) \ e,K(y,z) \ z,K(y,z))) = K(x,K(y,z) / z) by Theorem 765. Hence we are done by (L22).
Theorem 1454 (prov9_e75cf2bd1a): (x * (e / z)) \ (y * (e / z)) = (x * (z \ e)) \ (y * (z \ e))
Proof. We have L((x * (e / z)) \ (K(z,z \ e) * (y * (z \ e))),x * (e / z),K(z \ e,z)) = (K(z \ e,z) * (x * (e / z))) \ (K(z \ e,z) * (K(z,z \ e) * (y * (z \ e)))) by Proposition 53. Then L((x * (e / z)) \ (K(z,z \ e) * (y * (z \ e))),x * (e / z),K(z \ e,z)) = (K(z \ e,z) * (x * (e / z))) \ (y * (z \ e)) by Theorem 693. Then (x * (e / z)) \ (K(z,z \ e) * (y * (z \ e))) = (K(z \ e,z) * (x * (e / z))) \ (y * (z \ e)) by Theorem 967. Then (K(z \ e,z) * (x * (e / z))) \ (y * (z \ e)) = (x * (e / z)) \ ((y * (z \ e)) * K(z,z \ e)) by Theorem 1183. Then (x * (z \ e)) \ (y * (z \ e)) = (x * (e / z)) \ ((y * (z \ e)) * K(z,z \ e)) by Theorem 1400. Hence we are done by Theorem 1364.
Theorem 1455 (prov9_d903a7f954): (x * y) \ ((x * y) / x) = L(x \ e,T(x,y),y)
Proof. We have (x * y) \ (y * (T(x,y) * (x \ e))) = L(x \ e,T(x,y),y) by Theorem 1321. Then (x * y) \ (y * (x \ T(x,y))) = L(x \ e,T(x,y),y) by Theorem 146. Hence we are done by Theorem 977.
Theorem 1456 (prov9_f0160d1cdc): K(y \ (y / (x \ e)),y) = T(y,T(x,y)) \ y
Proof. We have K(y \ (y / (x \ e)),y) = x \ T(x,y) by Theorem 198. Hence we are done by Theorem 986.
Theorem 1457 (prov9_bbd99d3b70): x * K(x,(x / y) / x) = T(x,y \ e)
Proof. We have x * K(x,(x / y) / x) = T(x,x / (x * y)) by Theorem 870. Hence we are done by Theorem 1003.
Theorem 1458 (prov9_fdb918d8fb): x \ T(y,x \ e) = y * R(e / x,x,x \ y)
Proof. We have x \ T(y,x \ e) = y * ((x \ y) / y) by Theorem 1248. Hence we are done by Theorem 37.
Theorem 1459 (prov9_3743ef0ac3): ((y * x) * (y \ e)) \ R(x,y,y \ e) = T(y,R(x,y,y \ e)) \ y
Proof. We have ((y * R(x,y,y \ e)) / y) \ R(x,y,y \ e) = T(y,R(x,y,y \ e)) \ y by Theorem 984. Then R((y * x) / y,y,y \ e) \ R(x,y,y \ e) = T(y,R(x,y,y \ e)) \ y by Theorem 123. Hence we are done by Proposition 71.
Theorem 1460 (prov9_4a3dce9ca8): L(z,x,x \ T(x,y)) = z
Proof. We have
(L1) L(x * z,x \ T(x,y),x) = x * z
by Theorem 1269. We have L(x * z,x \ T(x,y),x) = x * L(z,x,x \ T(x,y)) by Theorem 950. Then x * L(z,x,x \ T(x,y)) = x * z by (L1). Hence we are done by Proposition 9.
Theorem 1461 (prov9_fd671b8f9d): T(T(z,x \ e) \ z,y) = K(z \ (z / T(x,y)),z)
Proof. We have T(x * T(x \ e,z),y) = K(z \ (z / T(x,y)),z) by Theorem 296. Hence we are done by Theorem 1028.
Theorem 1462 (prov9_c4ae895d39): (T(z,y / x) \ z) * y = x * T(x \ y,z)
Proof. We have
(L1) (y / x) * ((T(z,y / x) \ z) * y) = T(y / x,z) * y
by Theorem 1314. We have (y / x) * (x * T(x \ y,z)) = T(y / x,z) * y by Theorem 1094. Hence we are done by (L1) and Proposition 7.
Theorem 1463 (prov9_f44e89543d): (T(x,y) \ x) * z = x * (T(x,y) \ z)
Proof. We have (x \ T(x,y)) \ z = x * (T(x,y) \ z) by Theorem 1272. Hence we are done by Theorem 1281.
Theorem 1464 (prov9_5ae878a360): K(y \ x,y) = K(x / y,y)
Proof. We have K(T(x / y,y),y) = K(x / y,y) by Theorem 1053. Hence we are done by Proposition 47.
Theorem 1465 (prov9_969b812910): K(y,y \ x) = K(y,x / y)
Proof. We have K(T(y,y \ x),y \ x) = K(y,x / y) by Theorem 1250. Hence we are done by Theorem 1053.
Theorem 1466 (prov9_3996d75717): K(y,x) \ e = K(x,y)
Proof. We have K(T(y,x),x) \ e = K(x,y) by Theorem 1036. Hence we are done by Theorem 1053.
Theorem 1467 (prov9_ae1035360c): K(y,x) \ z = K(x,y) * z
Proof. We have K(T(y,x),x) * (K(T(y,x),x) \ z) = z by Axiom 4. Then
(L3) K(y,(y * x) / y) * (K(T(y,x),x) \ z) = z
by Theorem 1016.
(K(x,y) \ e) * (K(T(y,x),x) \ z)
=z by (L3), Theorem 1035
=(K(x,y) \ e) * (K(x,y) * z) by Theorem 1051

Then (K(x,y) \ e) * (K(T(y,x),x) \ z) = (K(x,y) \ e) * (K(x,y) * z). Then K(T(y,x),x) \ z = K(x,y) * z by Proposition 9. Hence we are done by Theorem 1053.
Theorem 1468 (prov9_a67cdef714): (y * (x / y)) \ x = K(y \ x,y)
Proof. We have (y * (x / y)) \ x = K(x / y,y) by Theorem 448. Hence we are done by Theorem 1464.
Theorem 1469 (prov9_51ef323d2b): (x * (y / x)) * K(x \ y,x) = y
Proof. We have (x * (y / x)) * ((x * (y / x)) \ y) = y by Axiom 4. Hence we are done by Theorem 1468.
Theorem 1470 (prov9_b23b48936f): T(x,y / (e / z)) = T(x,y / (z \ e))
Proof. We have R(T(x,(y / (z \ e)) * K(z \ e,z)),y / (z \ e),K(z \ e,z)) = L(T(x,y / (z \ e)),y / (z \ e),K(z \ e,z)) by Theorem 1440. Then R(T(x,(y / (z \ e)) * K(z \ e,z)),y / (z \ e),K(z \ e,z)) = T(x,y / (z \ e)) by Theorem 967. Then R(T(x,y / (e / z)),y / (z \ e),K(z \ e,z)) = T(x,y / (z \ e)) by Theorem 1362. Hence we are done by Theorem 1293.
Theorem 1471 (prov9_7d4e2c445b): z \ T(z * (x / z),y) = T(x,y) / z
Proof. We have (e / z) * T((e / z) \ (x / z),y) = T((x / z) * z,y) / z by Theorem 498. Then
(L4) (x / z) * K(y \ (y / ((x / z) \ (e / z))),y) = T((x / z) * z,y) / z
by Theorem 284. We have
(L1) (e / z) * T((e / z) \ (x / z),y) = (x / z) * K(y \ (y / ((x / z) \ (e / z))),y)
by Theorem 284.
(T(y,(x / z) / (e / z)) \ y) * (x / z)
=(x / z) * K(y \ (y / ((x / z) \ (e / z))),y) by (L1), Theorem 1462
=T(x,y) / z by (L4), Axiom 6

Then (T(y,(x / z) / (e / z)) \ y) * (x / z) = T(x,y) / z. Then y * (T(y,(x / z) / (e / z)) \ (x / z)) = T(x,y) / z by Theorem 1463. Then
(L8) y * (T(y,(x / z) / (z \ e)) \ (x / z)) = T(x,y) / z
by Theorem 1470. We have (T(y,(x / z) / (z \ e)) \ y) * (x / z) = (z \ e) * T((z \ e) \ (x / z),y) by Theorem 1462. Then
(L10) y * (T(y,(x / z) / (z \ e)) \ (x / z)) = (z \ e) * T((z \ e) \ (x / z),y)
by Theorem 1463. We have (z \ e) * T((z \ e) \ (x / z),y) = z \ T(z * (x / z),y) by Theorem 140. Then y * (T(y,(x / z) / (z \ e)) \ (x / z)) = z \ T(z * (x / z),y) by (L10). Hence we are done by (L8).
Theorem 1472 (prov9_3ad947c998): z \ T(z * x,y) = T(x * z,y) / z
Proof. We have z \ T(z * ((x * z) / z),y) = T(x * z,y) / z by Theorem 1471. Hence we are done by Axiom 5.
Theorem 1473 (prov9_f27839b52d): (y \ T(x,z)) * y = T((y \ x) * y,z)
Proof. We have T((y \ x) * y,z) / y = y \ T(y * (y \ x),z) by Theorem 1472. Then
(L2) T((y \ x) * y,z) / y = y \ T(x,z)
by Axiom 4. We have (T((y \ x) * y,z) / y) * y = T((y \ x) * y,z) by Axiom 6. Hence we are done by (L2).
Theorem 1474 (prov9_55b0ab11b3): K(y \ ((y \ e) \ x),y) = K(x,y)
Proof. We have R(e / (e / y),e / y,(e / y) \ x) = ((e / y) \ x) / x by Theorem 37. Then
(L18) R(e / (e / y),y \ e,(e / y) \ x) = ((e / y) \ x) / x
by Theorem 1384. We have ((e / y) \ x) \ (x * (((e / y) \ x) / x)) = K(x,((e / y) \ x) / x) by Theorem 3. Then
(L20) ((e / y) \ x) \ (x * R(e / (e / y),y \ e,(e / y) \ x)) = K(x,((e / y) \ x) / x)
by (L18). We have K(x,((((e / y) \ x) / x) \ e) \ e) = K(x,((e / y) \ x) / x) by Theorem 779. Then K(x,((y \ e) \ x) / x) = K(x,((e / y) \ x) / x) by Theorem 1380. Then
(L21) ((e / y) \ x) \ (x * R(e / (e / y),y \ e,(e / y) \ x)) = K(x,((y \ e) \ x) / x)
by (L20). We have x * R(e / (e / y),e / y,(e / y) \ x) = (e / y) \ T(x,(e / y) \ e) by Theorem 1458. Then x * R(e / (e / y),e / y,(e / y) \ x) = (e / y) \ T(x,y) by Proposition 25. Then x * R(e / (e / y),y \ e,(e / y) \ x) = (e / y) \ T(x,y) by Theorem 1384. Then
(L22) ((e / y) \ x) \ ((e / y) \ T(x,y)) = K(x,((y \ e) \ x) / x)
by (L21). We have K(((y \ e) \ e) \ ((y \ e) \ x),y) = K(y \ ((y \ e) \ x),y) by Theorem 1446. Then
(L24) K(y \ ((e / y) \ x),y) = K(y \ ((y \ e) \ x),y)
by Theorem 1374. We have ((e / y) \ x) \ (y * T(y \ ((e / y) \ x),y)) = K(y \ ((e / y) \ x),y) by Theorem 46. Then ((e / y) \ x) \ ((e / y) \ T(x,y)) = K(y \ ((e / y) \ x),y) by Theorem 1335. Then ((e / y) \ x) \ ((e / y) \ T(x,y)) = K(y \ ((y \ e) \ x),y) by (L24). Then
(L28) K(x,((y \ e) \ x) / x) = K(y \ ((y \ e) \ x),y)
by (L22). We have
(L1) K(x,(y \ e) \ e) = T(T((y \ e) \ e,x) \ ((y \ e) \ e),((y \ e) \ e) * x)
by Theorem 1273.
x \ T(x,T(y \ e,((y \ e) \ e) * x) \ e)
=T(x \ T(x,(y \ e) \ e),((y \ e) \ e) * x) by Theorem 1014
=K(x,(y \ e) \ e) by (L1), Theorem 1030

Then x \ T(x,T(y \ e,((y \ e) \ e) * x) \ e) = K(x,(y \ e) \ e). Then
(L5) x \ T(x,T(y \ e,(y \ e) \ x) \ e) = K(x,(y \ e) \ e)
by Theorem 1302. We have x * (x \ T(x,T(y \ e,(y \ e) \ x) \ e)) = T(x,T(y \ e,(y \ e) \ x) \ e) by Axiom 4. Then
(L7) x * K(x,(y \ e) \ e) = T(x,T(y \ e,(y \ e) \ x) \ e)
by (L5). We have T(x,(((y \ e) \ x) \ x) \ e) = x * K(x,((y \ e) \ x) / x) by Theorem 1001. Then T(x,T(y \ e,(y \ e) \ x) \ e) = x * K(x,((y \ e) \ x) / x) by Proposition 49. Then x * K(x,((y \ e) \ x) / x) = x * K(x,(y \ e) \ e) by (L7). Then K(x,((y \ e) \ x) / x) = K(x,(y \ e) \ e) by Proposition 9. Then
(L29) K(y \ ((y \ e) \ x),y) = K(x,(y \ e) \ e)
by (L28). We have K(x,(y \ e) \ e) = K(x,y) by Theorem 779. Hence we are done by (L29).
Theorem 1475 (prov9_035a91c279): K(y \ x,y) = K((y \ e) * x,y)
Proof. We have K(y \ ((y \ e) \ ((y \ e) * x)),y) = K((y \ e) * x,y) by Theorem 1474. Hence we are done by Axiom 3.
Theorem 1476 (prov9_7ff84cdb02): K((e / y) * x,y) = K(y \ x,y)
Proof. We have K((e / y) * x,y) = K((y \ e) * x,y) by Theorem 1447. Hence we are done by Theorem 1475.
Theorem 1477 (prov9_083a0b3c85): K(x,(x \ e) * y) = K(x,x \ y)
Proof. We have K((e / x) * ((x \ y) * x),x) = K(x \ ((x \ y) * x),x) by Theorem 1476. Then K((e / x) * ((x \ y) * x),x) = K(T(x \ y,x),x) by Definition 3. Then
(L3) K(T((e / x) * y,x),x) = K(T(x \ y,x),x)
by Theorem 1333. We have K(K(x,x \ e) * T((x \ e) * y,x),x) = K(T((x \ e) * y,x),x) by Theorem 1445. Then K(T(K(x,x \ e) * ((x \ e) * y),x),x) = K(T((x \ e) * y,x),x) by Theorem 1135. Then K(T((e / x) * y,x),x) = K(T((x \ e) * y,x),x) by Theorem 697. Then
(L7) K(T(x \ y,x),x) = K(T((x \ e) * y,x),x)
by (L3). We have K(T((x \ e) * y,x),x) \ e = K(x,(x \ e) * y) by Theorem 1036. Then
(L9) K(T(x \ y,x),x) \ e = K(x,(x \ e) * y)
by (L7). We have K(T(x \ y,x),x) \ e = K(x,x \ y) by Theorem 1036. Hence we are done by (L9).
Theorem 1478 (prov9_02e1ae3bfa): K(x \ e,(x \ e) \ y) = K(x \ e,x * y)
Proof. We have K(x \ e,((x \ e) \ e) * y) = K(x \ e,x * y) by Theorem 1414. Hence we are done by Theorem 1477.
Theorem 1479 (prov9_d43d0b0430): T(x * T(y,y \ e),T(y,y \ e) \ e) = T(y \ e,y) \ R(x,y,y \ e)
Proof. We have R(x,(y \ e) \ e,((y \ e) \ e) \ e) = R(x,(y \ e) \ e,y \ e) by Theorem 1370. Then R(x,(y \ e) \ e,((y \ e) \ e) \ e) = R(x,e / (y \ e),y \ e) by Theorem 1151. Then R(x,(y \ e) \ e,T(y \ e,y)) = R(x,e / (y \ e),y \ e) by Theorem 209. Then R(x,(y \ e) \ e,T(y \ e,y)) = R(x,y,y \ e) by Proposition 24. Then
(L5) R(x,T(y,y \ e),T(y \ e,y)) = R(x,y,y \ e)
by Proposition 49. We have (T(y,y \ e) \ e) \ R(x,T(y,y \ e),T(y,y \ e) \ e) = T(x * T(y,y \ e),T(y,y \ e) \ e) by Theorem 1326. Then (T(y,y \ e) \ e) \ R(x,T(y,y \ e),T(y,y \ e) \ e) = T(((y \ e) \ e) * T(x,y),T(y,y \ e) \ e) by Theorem 1403. Then T(y \ e,y) \ R(x,T(y,y \ e),T(y,y \ e) \ e) = T(((y \ e) \ e) * T(x,y),T(y,y \ e) \ e) by Theorem 1276. Then T(y \ e,y) \ R(x,T(y,y \ e),T(y \ e,y)) = T(((y \ e) \ e) * T(x,y),T(y,y \ e) \ e) by Theorem 1276. Then T(y \ e,y) \ R(x,y,y \ e) = T(((y \ e) \ e) * T(x,y),T(y,y \ e) \ e) by (L5). Hence we are done by Theorem 1403.
Theorem 1480 (prov9_22be3c9257): K(x,y) = L(K(x,y),y \ e,y)
Proof. We have ((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * (((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) \ (((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * T(y,y \ e))) = ((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * T(y,y \ e) by Axiom 4. Then ((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * (y * (y \ (((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) \ (((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * T(y,y \ e))))) = ((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * T(y,y \ e) by Axiom 4. Then
(L6) ((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * (y * (y \ T(y,y \ e))) = ((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * T(y,y \ e)
by Axiom 3. We have ((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * (y * (y \ T(y,y \ e))) = T(y,x) \ y by Axiom 6. Then ((T(y,x) \ y) / (y * (y \ T(y,y \ e)))) * T(y,y \ e) = T(y,x) \ y by (L6). Then
(L9) ((T(y,x) \ y) / ((y \ e) \ e)) * T(y,y \ e) = T(y,x) \ y
by Theorem 795. We have (y * (T(y,x) \ e)) / T(y,T(y,x) \ e) = T(y,x) \ e by Theorem 5. Then (T(y,x) \ y) / T(y,T(y,x) \ e) = T(y,x) \ e by Theorem 175. Then (T(y,x) \ y) / ((y \ e) \ e) = T(y,x) \ e by Theorem 1411. Then
(L10) (T(y,x) \ e) * T(y,y \ e) = T(y,x) \ y
by (L9). We have
(L11) T((T(y,x) \ e) * T(y,y \ e),T(y,y \ e) \ e) = T(y \ e,y) \ R(T(y,x) \ e,y,y \ e)
by Theorem 1479. Then
(L12) T(T(y,x) \ y,T(y,y \ e) \ e) = T(y \ e,y) \ R(T(y,x) \ e,y,y \ e)
by (L10). We have
(L13) T((T(y,x) \ e) * T(y,y \ e),T(y \ e,y)) = T(y \ e,y) \ R(T(y,x) \ e,y,y \ e)
by (L11) and Theorem 1276. We have T((T(y,x) \ e) * T(y,y \ e),T(y \ e,y)) = T((T(y,x) \ e) * T(y,y \ e),y \ e) by Theorem 1401. Then
(L15) T(y \ e,y) \ R(T(y,x) \ e,y,y \ e) = T((T(y,x) \ e) * T(y,y \ e),y \ e)
by (L13). We have T(((e / y) * ((y \ e) \ (T(y,x) \ e))) * y,y \ e) = R((y \ e) \ (T(y,x) \ e),e / y,y) by Theorem 1404. Then T((K(y,y \ e) * (T(y,x) \ e)) * y,y \ e) = R((y \ e) \ (T(y,x) \ e),e / y,y) by Theorem 1375. Then T((T(y,x) \ e) * T(y,y \ e),y \ e) = R((y \ e) \ (T(y,x) \ e),e / y,y) by Theorem 1413. Then
(L19) T(y \ e,y) \ R(T(y,x) \ e,y,y \ e) = R((y \ e) \ (T(y,x) \ e),e / y,y)
by (L15). We have R((y \ e) \ (T(y,x) \ e),e / y,y) = R((y \ e) \ (T(y,x) \ e),y \ e,y) by Theorem 1151. Then T(y \ e,y) \ R(T(y,x) \ e,y,y \ e) = R((y \ e) \ (T(y,x) \ e),y \ e,y) by (L19). Then T(T(y,x) \ y,T(y,y \ e) \ e) = R((y \ e) \ (T(y,x) \ e),y \ e,y) by (L12). Then
(L23) R((y \ e) \ (T(y,x) \ e),y \ e,y) = T(T(y,x) \ y,T(y \ e,y))
by Theorem 1276. We have T(T(y,x) \ y,T(y \ e,y)) = T(T(y,x) \ y,y \ e) by Theorem 1401. Then
(L25) R((y \ e) \ (T(y,x) \ e),y \ e,y) = T(T(y,x) \ y,y \ e)
by (L23). We have T(y \ e,T(y,x) \ y) = y \ e by Theorem 1428. Then T(T(y,x) \ y,y \ e) = T(y,x) \ y by Proposition 21. Then R((y \ e) \ (T(y,x) \ e),y \ e,y) = T(y,x) \ y by (L25). Then
(L29) R(T(y,x) \ y,y \ e,y) = T(y,x) \ y
by Theorem 1418. We have y * (T(y,x) \ e) = T(y,x) \ y by Theorem 175. Then
(L36) y * ((e / T(y,x)) * K(y \ e,y)) = T(y,x) \ y
by Theorem 1417. We have (y * (e / T(y,x))) * K(y \ e,y) = y * ((e / T(y,x)) * K(y \ e,y)) by Theorem 668. Then
(L38) (y * (e / T(y,x))) * K(y \ e,y) = T(y,x) \ y
by (L36). We have (e / y) * ((y * (e / T(y,x))) * K(y \ e,y)) = (y \ e) * (y * (e / T(y,x))) by Theorem 1396. Then
(L40) (e / y) * (T(y,x) \ y) = (y \ e) * (y * (e / T(y,x)))
by (L38). We have ((y \ e) * (y * (e / T(y,x)))) * y = L((e / T(y,x)) * y,y \ e,y) by Theorem 893. Then
(L42) ((e / y) * (T(y,x) \ y)) * y = L((e / T(y,x)) * y,y \ e,y)
by (L40). We have (e / T(y,x)) * y = y / T(y,x) by Theorem 550. Then (e / T(y,x)) * y = T(y,x) \ y by Theorem 753. Then
(L43) L(T(y,x) \ y,y \ e,y) = ((e / y) * (T(y,x) \ y)) * y
by (L42). We have (K(y \ (y / ((x \ y) / T(y,x))),y) * (e / y)) * y = R(K(y \ (y / ((x \ y) / T(y,x))),y),y \ e,y) by Theorem 911. Then ((e / y) * K(y \ (y / ((x \ y) / T(y,x))),y)) * y = R(K(y \ (y / ((x \ y) / T(y,x))),y),y \ e,y) by Theorem 1354. Then ((e / y) * (T(y,x) \ y)) * y = R(K(y \ (y / ((x \ y) / T(y,x))),y),y \ e,y) by Theorem 932. Then R(K(y \ (y / ((x \ y) / T(y,x))),y),y \ e,y) = L(T(y,x) \ y,y \ e,y) by (L43). Then R(T(y,x) \ y,y \ e,y) = L(T(y,x) \ y,y \ e,y) by Theorem 932. Then
(L46) T(y,x) \ y = L(T(y,x) \ y,y \ e,y)
by (L29). We have L(L(T(y,x) \ y,T(y,x),x),y \ e,y) = L(L(T(y,x) \ y,y \ e,y),T(y,x),x) by Axiom 11. Then L((y * x) \ (x * y),y \ e,y) = L(L(T(y,x) \ y,y \ e,y),T(y,x),x) by Theorem 798. Then
(L49) L(T(y,x) \ y,T(y,x),x) = L((y * x) \ (x * y),y \ e,y)
by (L46). We have L(T(y,x) \ y,T(y,x),x) = (y * x) \ (x * y) by Theorem 798. Then
(L51) L((y * x) \ (x * y),y \ e,y) = (y * x) \ (x * y)
by (L49). We have K(x,y) = (y * x) \ (x * y) by Definition 2. Then K(x,y) = L((y * x) \ (x * y),y \ e,y) by (L51). Hence we are done by Definition 2.
Theorem 1481 (prov9_08105bcdf3): K(x,y) / y = (e / y) * K(x,y)
Proof. We have (y \ e) * (y * (K(x,y) / y)) = L(K(x,y),y \ e,y) / y by Theorem 1355. Then (e / y) * K(x,y) = L(K(x,y),y \ e,y) / y by Theorem 1439. Hence we are done by Theorem 1480.
Theorem 1482 (prov9_1263a9b13e): T(y,x) \ y = K(x,y)
Proof. We have T(y,x) * K(x,y) = y by Theorem 1307. Hence we are done by Proposition 2.
Theorem 1483 (prov9_a97005dd44): L(K(x,z),T(z,x),y) = (y * T(z,x)) \ (y * z)
Proof. We have L(K(x,z),T(z,x),y) = (y * T(z,x)) \ (y * (T(z,x) * K(x,z))) by Definition 4. Hence we are done by Theorem 1307.
Theorem 1484 (prov9_39041a18f4): (y * x) / (K(z,y) * x) = T(y,z)
Proof. We have (y * x) / (K(z,y) * x) = K(z,y) \ y by Theorem 1050. Hence we are done by Theorem 1319.
Theorem 1485 (prov9_34f9cc50af): K(x,x \ (x / y)) = K(x,y \ e)
Proof. We have x * K(x,(x / y) / x) = T(x,y \ e) by Theorem 1457. Then
(L3) x * K(x,x \ (x / y)) = T(x,y \ e)
by Theorem 1465. We have T(x,y \ e) = x * K(x,y \ e) by Theorem 1309. Then x * K(x,x \ (x / y)) = x * K(x,y \ e) by (L3). Hence we are done by Proposition 9.
Theorem 1486 (prov9_7bc864a235): x * T(x \ e,y) = K(x \ e,y)
Proof. We have
(L1) T(x \ e,y) = (x \ e) * K(x \ e,y)
by Theorem 1309. We have (x \ e) * (x * T(x \ e,y)) = T(x \ e,y) by Theorem 728. Then (x \ e) * (x * T(x \ e,y)) = (x \ e) * K(x \ e,y) by (L1). Hence we are done by Proposition 9.
Theorem 1487 (prov9_e8c57a686b): L(K(y,z),y,x) = (x * y) \ (x * T(y,z))
Proof. We have L(K(y,z),y,x) = (x * y) \ (x * (y * K(y,z))) by Definition 4. Hence we are done by Theorem 1309.
Theorem 1488 (prov9_a9c08ff7f5): y \ ((y / z) * T(z,x)) = L(K(z,x),z,y / z)
Proof. We have y \ ((y / z) * (z * K(z,x))) = L(K(z,x),z,y / z) by Theorem 4. Hence we are done by Theorem 1309.
Theorem 1489 (prov9_210a98bc78): (z * K(x,y)) \ (z * T(x,y)) = L(x,K(x,y),z)
Proof. We have (z * K(x,y)) \ (z * (x * K(x,y))) = L(T(x,K(x,y)),K(x,y),z) by Proposition 62. Then (z * K(x,y)) \ (z * (x * K(x,y))) = L(x,K(x,y),z) by Theorem 736. Hence we are done by Theorem 1309.
Theorem 1490 (prov9_93ad643b4d): y * (x * T(K(x,y),z)) = (y * x) * T(K(x,y),z)
Proof. We have y * (x * T(x \ T(x,y),z)) = (y * x) * T(K(x,y),z) by Theorem 1113. Hence we are done by Theorem 1308.
Theorem 1491 (prov9_ff53e1a36a): K(y \ (y / x),y) = K(x \ e,y)
Proof. We have K(y \ (y / x),y) = (x \ e) \ T(x \ e,y) by Theorem 730. Hence we are done by Theorem 1308.
Theorem 1492 (prov9_dfbb312205): K(y,x) * z = T(y,x) * (y \ z)
Proof. We have (y \ T(y,x)) * z = T(y,x) * (y \ z) by Theorem 1270. Hence we are done by Theorem 1308.
Theorem 1493 (prov9_383a30061b): K(K(x,y),z) = K(y,x) * T(K(x,y),z)
Proof. We have K(x,y) \ T(K(x,y),z) = K(y,x) * T(K(x,y),z) by Theorem 1467. Hence we are done by Theorem 1308.
Theorem 1494 (prov9_4ad4dfb116): y * K(x \ e,y) = x * ((y \ (x \ y)) * y)
Proof. We have x * ((y \ (x \ y)) * y) = y * (x * T(x \ e,y)) by Theorem 1422. Then x * ((y \ (x \ y)) * y) = y * (T(y,x \ e) \ y) by Theorem 1028. Hence we are done by Theorem 1482.
Theorem 1495 (prov9_e296567ee7): T(K(x,y \ e),z) = K(x,T(y,z) \ e)
Proof. We have x \ T(x,T(y,z) \ e) = T(x \ T(x,y \ e),z) by Theorem 1014. Then K(x,T(y,z) \ e) = T(x \ T(x,y \ e),z) by Theorem 1308. Hence we are done by Theorem 1308.
Theorem 1496 (prov9_a8a8d9fae2): K(x,T(y \ e,z) \ e) = T(K(x,y),z)
Proof. We have K(x,T(y \ e,z) \ e) = T(K(x,(y \ e) \ e),z) by Theorem 1495. Hence we are done by Theorem 779.
Theorem 1497 (prov9_7b5fdf7cc4): T(K(x \ e,z),y) = K(T(x,y) \ e,z)
Proof. We have T(T(z,x \ e) \ z,y) = K(z \ (z / T(x,y)),z) by Theorem 1461. Then
(L2) T(K(x \ e,z),y) = K(z \ (z / T(x,y)),z)
by Theorem 1482. We have K(z \ (z / T(x,y)),z) = K(T(x,y) \ e,z) by Theorem 1491. Hence we are done by (L2).
Theorem 1498 (prov9_f6260867f1): K(T(x \ e,z) \ e,y) = T(K(x,y),z)
Proof. We have K(T(x \ e,z) \ e,y) = T(K((x \ e) \ e,y),z) by Theorem 1497. Hence we are done by Theorem 733.
Theorem 1499 (prov9_7cf7bddf4b): K(x * K(z,x \ e),y) = T(K(x,y),z)
Proof. We have K(x \ e,x) * ((z \ T(z,x \ e)) * ((x \ e) \ e)) = x * T(z \ T(z,x \ e),(x \ e) \ e) by Theorem 1378. Then
(L2) K(x \ e,x) * ((x \ e) \ (z \ T(z,x \ e))) = x * T(z \ T(z,x \ e),(x \ e) \ e)
by Theorem 1432. We have K(x \ e,x) * ((x \ e) \ (z \ T(z,x \ e))) = (e / x) \ (z \ T(z,x \ e)) by Theorem 436. Then x * T(z \ T(z,x \ e),(x \ e) \ e) = (e / x) \ (z \ T(z,x \ e)) by (L2). Then
(L5) x * (z \ T(z,x \ e)) = (e / x) \ (z \ T(z,x \ e))
by Theorem 1410. We have (e / x) \ (T(e / x,T(z,e / x)) \ (e / x)) = T(e / x,T(z,e / x)) \ e by Theorem 1339. Then
(L7) (e / x) \ (z \ T(z,e / x)) = T(e / x,T(z,e / x)) \ e
by Theorem 986. We have K(T(e / x,T(z,e / x)) \ e,y) = K(T(x \ e,T(z,e / x)) \ e,y) by Theorem 1415. Then K((e / x) \ (z \ T(z,e / x)),y) = K(T(x \ e,T(z,e / x)) \ e,y) by (L7). Then K(T(x \ e,T(z,e / x)) \ e,y) = K((e / x) \ (z \ T(z,x \ e)),y) by Theorem 1397. Then K(T(x \ e,T(z,e / x)) \ e,y) = K(x * (z \ T(z,x \ e)),y) by (L5). Then K(T(x \ e,T(z,x \ e)) \ e,y) = K(x * (z \ T(z,x \ e)),y) by Theorem 1397. Then K(T(x \ e,z) \ e,y) = K(x * (z \ T(z,x \ e)),y) by Theorem 1032. Then K(x * (z \ T(z,x \ e)),y) = T(K(x,y),z) by Theorem 1498. Hence we are done by Theorem 1308.
Theorem 1500 (prov9_a42723e25d): K(x,(y \ z) \ e) = L(K(x,z \ y),z \ y,z)
Proof. We have
(L3) T(x,R((y \ z) \ e,((y \ z) \ e) \ e,x)) = (x \ T(x,(y \ z) \ e)) * x
by Theorem 989. We have T(x,R((y \ z) \ e,((y \ z) \ e) \ e,x)) = (y \ (z * ((x * (z \ y)) / x))) * x by Theorem 953. Then (y \ (z * ((x * (z \ y)) / x))) * x = (x \ T(x,(y \ z) \ e)) * x by (L3). Then
(L6) y \ (z * ((x * (z \ y)) / x)) = x \ T(x,(y \ z) \ e)
by Proposition 10. We have y \ (z * ((z \ y) * (x \ T(x,z \ y)))) = L(x \ T(x,z \ y),z \ y,z) by Theorem 449. Then y \ (z * ((x * (z \ y)) / x)) = L(x \ T(x,z \ y),z \ y,z) by Theorem 977. Then
(L7) L(x \ T(x,z \ y),z \ y,z) = x \ T(x,(y \ z) \ e)
by (L6).
K(x,(y \ z) \ e)
=x \ T(x,(y \ z) \ e) by Theorem 1308
=L(K(x,z \ y),z \ y,z) by (L7), Theorem 1308

Hence we are done.
Theorem 1501 (prov9_af8794c78f): ((z \ x) * z) * K(z,y) = (z \ (x * K(z,y))) * z
Proof. We have (z \ (x * (((y * z) / y) \ z))) * z = (((y * z) / y) * T(((y * z) / y) \ x,z)) * (((y * z) / y) \ z) by Theorem 1096. Then (z \ (x * (T(y,z) \ y))) * z = (((y * z) / y) * T(((y * z) / y) \ x,z)) * (((y * z) / y) \ z) by Theorem 984. Then ((y * (z * T(z \ T(x,y),z))) / y) * (((y * z) / y) \ z) = (z \ (x * (T(y,z) \ y))) * z by Theorem 1360. Then ((y * ((z \ T(x,y)) * z)) / y) * (((y * z) / y) \ z) = (z \ (x * (T(y,z) \ y))) * z by Proposition 46. Then ((y * ((z \ T(x,y)) * z)) / y) * (T(y,z) \ y) = (z \ (x * (T(y,z) \ y))) * z by Theorem 984. Then ((y * T((z \ x) * z,y)) / y) * (T(y,z) \ y) = (z \ (x * (T(y,z) \ y))) * z by Theorem 1473. Then ((z \ x) * z) * (T(y,z) \ y) = (z \ (x * (T(y,z) \ y))) * z by Proposition 48. Then (z \ (x * K(z,y))) * z = ((z \ x) * z) * (T(y,z) \ y) by Theorem 1482. Hence we are done by Theorem 1482.
Theorem 1502 (prov9_68c646006a): x * (z * K(y \ e,z)) = (x * K(y \ e,z)) * z
Proof. We have (x * y) * (z * T(z \ L(y \ z,y,x),z)) = (x * (y * T(y \ e,z))) * z by Theorem 884. Then (x * y) * ((z \ L(y \ z,y,x)) * z) = (x * (y * T(y \ e,z))) * z by Proposition 46. Then (x * y) * ((z \ L(y \ z,y,x)) * z) = (x * K(y \ e,z)) * z by Theorem 1486. Then
(L4) (x * y) * L((z \ (y \ z)) * z,y,x) = (x * K(y \ e,z)) * z
by Theorem 1299. We have (x * y) * L((z \ (y \ z)) * z,y,x) = x * (y * ((z \ (y \ z)) * z)) by Proposition 52. Then (x * K(y \ e,z)) * z = x * (y * ((z \ (y \ z)) * z)) by (L4). Hence we are done by Theorem 1494.
Theorem 1503 (prov9_1242fdb307): R(z,K(x \ e,y),y) = z
Proof. We have
(L1) (z * K(x \ e,y)) * y = z * (y * K(x \ e,y))
by Theorem 1502. We have K(x \ e,y) * y = y * K(x \ e,y) by Theorem 933. Then (z * K(x \ e,y)) * y = z * (K(x \ e,y) * y) by (L1). Hence we are done by Theorem 64.
Theorem 1504 (prov9_8fad0370e0): R(z,K(x,y),y) = z
Proof. We have R(z,K((e / x) \ e,y),y) = z by Theorem 1503. Hence we are done by Proposition 25.
Theorem 1505 (prov9_cd1e617642): K(y,x) = y \ T(y,R(x,y,y \ e))
Proof. We have K((((y * x) / ((y * x) * (e / y))) \ e) \ e,((y * x) / ((y * x) * (y \ e))) \ e) = K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (y \ e))) \ e) by Theorem 733. Then
(L16) K((y * x) / ((y * x) * (y \ e)),((y * x) / ((y * x) * (y \ e))) \ e) = K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (y \ e))) \ e)
by Theorem 1394. We have K((y * x) / ((y * x) * (y \ e)),((y * x) / ((y * x) * (y \ e))) \ e) = K(((y * x) * (y \ e)) \ (y * x),(y * x) \ ((y * x) * (y \ e))) by Theorem 1423. Then
(L18) K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (y \ e))) \ e) = K(((y * x) * (y \ e)) \ (y * x),(y * x) \ ((y * x) * (y \ e)))
by (L16). We have K(((y * x) * (y \ e)) \ (y * x),(y * x) \ ((y * x) * (y \ e))) = K((y \ e) \ e,y \ e) by Theorem 664. Then
(L20) K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (y \ e))) \ e) = K((y \ e) \ e,y \ e)
by (L18). We have K((y \ e) \ e,y \ e) = K(y,y \ e) by Theorem 733. Then
(L22) K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (y \ e))) \ e) = K(y,y \ e)
by (L20). We have K((y * x) / ((y * x) * (e / y)),((((y * x) / ((y * x) * (e / y))) \ e) \ e) \ e) = K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (e / y))) \ e) by Theorem 779. Then
(L24) K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (y \ e))) \ e) = K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (e / y))) \ e)
by Theorem 1394. We have (((y * x) * (e / y)) * ((y * x) / ((y * x) * (e / y)))) * K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (e / y))) \ e) = ((y * x) * (e / y)) * ((((y * x) / ((y * x) * (e / y))) \ e) \ e) by Theorem 407. Then (((y * x) * (e / y)) * ((y * x) / ((y * x) * (e / y)))) * K((y * x) / ((y * x) * (e / y)),((y * x) / ((y * x) * (y \ e))) \ e) = ((y * x) * (e / y)) * ((((y * x) / ((y * x) * (e / y))) \ e) \ e) by (L24). Then (((y * x) * (e / y)) * ((y * x) / ((y * x) * (e / y)))) * K(y,y \ e) = ((y * x) * (e / y)) * ((((y * x) / ((y * x) * (e / y))) \ e) \ e) by (L22). Then
(L28) (((y * x) * (e / y)) * ((y * x) / ((y * x) * (e / y)))) * K(y,y \ e) = ((y * x) * (e / y)) * ((y * x) / ((y * x) * (y \ e)))
by Theorem 1394. We have (K(y \ e,y) * ((y * x) * (e / y))) * ((y * x) / ((y * x) * (y \ e))) = K(y \ e,y) * (((y * x) * (e / y)) * ((y * x) / ((y * x) * (y \ e)))) by Theorem 968. Then
(L30) ((y * x) * (y \ e)) * ((y * x) / ((y * x) * (y \ e))) = K(y \ e,y) * (((y * x) * (e / y)) * ((y * x) / ((y * x) * (y \ e))))
by Theorem 1400. We have K(y \ e,y) * ((((y * x) * (y \ e)) * ((y * x) / ((y * x) * (y \ e)))) * K(y,y \ e)) = ((y * x) * (y \ e)) * ((y * x) / ((y * x) * (y \ e))) by Theorem 1182. Then K(y \ e,y) * ((((y * x) * (y \ e)) * ((y * x) / ((y * x) * (y \ e)))) * K(y,y \ e)) = K(y \ e,y) * (((y * x) * (e / y)) * ((y * x) / ((y * x) * (y \ e)))) by (L30). Then (((y * x) * (y \ e)) * ((y * x) / ((y * x) * (y \ e)))) * K(y,y \ e) = ((y * x) * (e / y)) * ((y * x) / ((y * x) * (y \ e))) by Proposition 9. Then
(L34) ((y * x) * (e / y)) * ((y * x) / ((y * x) * (e / y))) = ((y * x) * (y \ e)) * ((y * x) / ((y * x) * (y \ e)))
by (L28) and Proposition 8. We have (((y * x) * (e / y)) * ((y * x) / ((y * x) * (e / y)))) * K(((y * x) * (e / y)) \ (y * x),(y * x) * (e / y)) = y * x by Theorem 1469. Then (((y * x) * (e / y)) * ((y * x) / ((y * x) * (e / y)))) * K(L(y,y \ e,y * x),(y * x) * (e / y)) = y * x by Theorem 649. Then
(L37) (((y * x) * (y \ e)) * ((y * x) / ((y * x) * (y \ e)))) * K(L(y,y \ e,y * x),(y * x) * (e / y)) = y * x
by (L34). We have
(L9) (((y * x) * (y \ e)) * ((((y * x) * (y \ e)) / (y \ e)) / ((y * x) * (y \ e)))) * (y \ e) = T((y * x) * (y \ e),(y \ e) \ e)
by Theorem 1005. We have (y * T(x,(y \ e) \ e)) * (y \ e) = T((y * x) * (y \ e),(y \ e) \ e) by Theorem 1337. Then ((y * x) * (y \ e)) * ((((y * x) * (y \ e)) / (y \ e)) / ((y * x) * (y \ e))) = y * T(x,(y \ e) \ e) by (L9) and Proposition 8. Then
(L12) ((y * x) * (y \ e)) * ((y * x) / ((y * x) * (y \ e))) = y * T(x,(y \ e) \ e)
by Axiom 5. We have y * T(x,(y \ e) \ e) = x * y by Theorem 1395. Then ((y * x) * (y \ e)) * ((y * x) / ((y * x) * (y \ e))) = x * y by (L12). Then
(L38) (x * y) * K(L(y,y \ e,y * x),(y * x) * (e / y)) = y * x
by (L37). We have T(y,(y * x) * (e / y)) = K(((y * x) * (e / y)) \ (((y * x) * (e / y)) / (e / y)),(y * x) * (e / y)) * y by Theorem 201. Then T(y,(y * x) * (e / y)) = K(((y * x) * (e / y)) \ (y * x),(y * x) * (e / y)) * y by Axiom 5. Then
(L7) K(L(y,y \ e,y * x),(y * x) * (e / y)) * y = T(y,(y * x) * (e / y))
by Theorem 649. We have T(y,(y * x) * (e / y)) = T(y,(y * x) * (y \ e)) by Theorem 1366. Then T(y,(y * x) * (e / y)) = (y \ T(y,(y * x) * (y \ e))) * y by Theorem 759. Then y \ T(y,(y * x) * (y \ e)) = K(L(y,y \ e,y * x),(y * x) * (e / y)) by (L7) and Proposition 8. Then
(L39) (x * y) * (y \ T(y,(y * x) * (y \ e))) = y * x
by (L38). We have T(y,T((y * x) * (y \ e),y)) = T(y,(y * x) * (y \ e)) by Theorem 1032. Then T(y,R(x,y,y \ e)) = T(y,(y * x) * (y \ e)) by Theorem 1331. Then (x * y) * (y \ T(y,R(x,y,y \ e))) = y * x by (L39). Hence we are done by Theorem 460.
Theorem 1506 (prov9_9b2346c800): T(K(y,x),z) = T(K(x,y),z) \ e
Proof. We have T((x \ e) * ((((x * y) * (x \ e)) * ((x \ e) \ e)) / ((x * y) * (x \ e))),z) = K((x * y) * (x \ e),(((x * y) * (x \ e)) / T(x \ e,z)) / ((x * y) * (x \ e))) by Theorem 344. Then T(x \ ((((x * y) * (x \ e)) * x) / ((x * y) * (x \ e))),z) = K((x * y) * (x \ e),(((x * y) * (x \ e)) / T(x \ e,z)) / ((x * y) * (x \ e))) by Theorem 546. Then
(L25) T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z) = K((x * y) * (x \ e),(((x * y) * (x \ e)) / T(x \ e,z)) / ((x * y) * (x \ e)))
by Theorem 979. We have (((x * y) * (x \ e)) \ T((x * y) * (x \ e),T(x \ e,z) \ e)) \ e = T((x * y) * (x \ e),T(x \ e,z) \ e) \ ((x * y) * (x \ e)) by Theorem 749. Then K((x * y) * (x \ e),(((x * y) * (x \ e)) / T(x \ e,z)) / ((x * y) * (x \ e))) \ e = T((x * y) * (x \ e),T(x \ e,z) \ e) \ ((x * y) * (x \ e)) by Theorem 1004. Then
(L28) T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z) \ e = T((x * y) * (x \ e),T(x \ e,z) \ e) \ ((x * y) * (x \ e))
by (L25). We have K(((x * y) * (x \ e)) \ (((x * y) * (x \ e)) / T(x \ e,z)),(x * y) * (x \ e)) = T((x * y) * (x \ e),T(x \ e,z) \ e) \ ((x * y) * (x \ e)) by Theorem 1029. Then T(T((x * y) * (x \ e),(x \ e) \ e) \ ((x * y) * (x \ e)),z) = T((x * y) * (x \ e),T(x \ e,z) \ e) \ ((x * y) * (x \ e)) by Theorem 1461. Then T(T((x * y) * (x \ e),(x \ e) \ e) \ ((x * y) * (x \ e)),z) = T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z) \ e by (L28). Then
(L30) T(T((x * y) * (x \ e),x) \ ((x * y) * (x \ e)),z) = T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z) \ e
by Theorem 721. We have e / (T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z) \ e) = T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z) by Proposition 24. Then
(L32) e / T(T((x * y) * (x \ e),x) \ ((x * y) * (x \ e)),z) = T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z)
by (L30). We have ((T((x * y) * (x \ e),x) \ ((x * y) * (x \ e))) \ e) \ K(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),T((x * y) * (x \ e),x) \ ((x * y) * (x \ e))) = T((x * y) * (x \ e),x) \ ((x * y) * (x \ e)) by Theorem 1150. Then
(L34) ((T((x * y) * (x \ e),x) \ ((x * y) * (x \ e))) \ e) \ e = T((x * y) * (x \ e),x) \ ((x * y) * (x \ e))
by Theorem 741. We have e / T(((T((x * y) * (x \ e),x) \ ((x * y) * (x \ e))) \ e) \ e,z) = T(T((x * y) * (x \ e),x) \ ((x * y) * (x \ e)),z) \ e by Theorem 918. Then e / T(T((x * y) * (x \ e),x) \ ((x * y) * (x \ e)),z) = T(T((x * y) * (x \ e),x) \ ((x * y) * (x \ e)),z) \ e by (L34). Then T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z) = T(T((x * y) * (x \ e),x) \ ((x * y) * (x \ e)),z) \ e by (L32). Then
(L38) T(R(y,x,x \ e) \ ((x * y) * (x \ e)),z) \ e = T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z)
by Theorem 1331. We have (((x * y) * (e / x)) \ T((x * y) * (e / x),x)) \ e = T((x * y) * (e / x),x) \ ((x * y) * (e / x)) by Theorem 749. Then (((x * y) * (e / x)) \ ((y * x) * (e / x))) \ e = T((x * y) * (e / x),x) \ ((x * y) * (e / x)) by Theorem 611. Then ((y * x) * (e / x)) \ ((x * y) * (e / x)) = (((x * y) * (e / x)) \ ((y * x) * (e / x))) \ e by Theorem 611. Then (((x * y) * (x \ e)) \ ((y * x) * (x \ e))) \ e = ((y * x) * (e / x)) \ ((x * y) * (e / x)) by Theorem 1454. Then
(L13) (((x * y) * (x \ e)) \ R(y,x,x \ e)) \ e = ((y * x) * (e / x)) \ ((x * y) * (e / x))
by Proposition 68. We have ((y * x) * (e / x)) \ ((x * y) * (e / x)) = ((y * x) * (x \ e)) \ ((x * y) * (x \ e)) by Theorem 1454. Then (((x * y) * (x \ e)) \ R(y,x,x \ e)) \ e = ((y * x) * (x \ e)) \ ((x * y) * (x \ e)) by (L13). Then (((x * y) * (x \ e)) \ R(y,x,x \ e)) \ e = R(y,x,x \ e) \ ((x * y) * (x \ e)) by Proposition 68. Then
(L17) (T(x,R(y,x,x \ e)) \ x) \ e = R(y,x,x \ e) \ ((x * y) * (x \ e))
by Theorem 1459. We have (T(x,R(y,x,x \ e)) \ x) \ e = x \ T(x,R(y,x,x \ e)) by Theorem 1197. Then R(y,x,x \ e) \ ((x * y) * (x \ e)) = x \ T(x,R(y,x,x \ e)) by (L17). Then R(y,x,x \ e) \ ((x * y) * (x \ e)) = K(x,y) by Theorem 1505. Then T(((x * y) * (x \ e)) \ T((x * y) * (x \ e),x),z) = T(K(x,y),z) \ e by (L38). Then
(L40) T(((x * y) * (x \ e)) \ R(y,x,x \ e),z) = T(K(x,y),z) \ e
by Theorem 1331. We have (x * K(x,y)) \ x = K(x,y) \ e by Theorem 1431. Then (x * K(x,y)) \ x = K(T(y,x),x) by Theorem 1037. Then
(L5) (x * K(x,y)) \ x = K(y,x)
by Theorem 1053. We have x * (x \ T(x,R(y,x,x \ e))) = T(x,R(y,x,x \ e)) by Axiom 4. Then x * K(x,y) = T(x,R(y,x,x \ e)) by Theorem 1505. Then
(L6) T(x,R(y,x,x \ e)) \ x = K(y,x)
by (L5). We have ((x * y) * (x \ e)) \ R(y,x,x \ e) = T(x,R(y,x,x \ e)) \ x by Theorem 1459. Then ((x * y) * (x \ e)) \ R(y,x,x \ e) = K(y,x) by (L6). Hence we are done by (L40).
Theorem 1507 (prov9_488ae1e27d): K(x,y * K(z,y \ e)) = T(K(x,y),z)
Proof. We have K(y * K(z,y \ e),x) \ e = K(x,y * K(z,y \ e)) by Theorem 1466. Then T(K(y,x),z) \ e = K(x,y * K(z,y \ e)) by Theorem 1499. Hence we are done by Theorem 1506.
Theorem 1508 (prov9_b9b95db83a): K(z,K(y,x)) = K(y,x) * T(K(x,y),z)
Proof. We have
(L1) K(y,x) * (T(K(y,x),z) \ e) = T(K(y,x),z) \ K(y,x)
by Theorem 175.
K(z,K(y,x))
=T(K(y,x),z) \ K(y,x) by Theorem 1482
=K(y,x) * T(K(x,y),z) by (L1), Theorem 1506

Hence we are done.
Theorem 1509 (prov9_d13604f895): K(z,K(y,x)) = K(K(x,y),z)
Proof. We have K(y,x) * T(K(x,y),z) = K(K(x,y),z) by Theorem 1493. Hence we are done by Theorem 1508.
Theorem 1510 (prov9_38fe0f9d72): K(x,x \ (x / L(y \ e,z,w))) = L(x \ T(x,y),z,w)
Proof. We have ((w * (z * (y \ e))) / L(y \ e,z,w)) \ ((w * (z * (y \ e))) * ((x * ((w * (z * (y \ e))) \ ((w * (z * (y \ e))) / L(y \ e,z,w)))) / x)) = L(y \ e,z,w) * ((x * (L(y \ e,z,w) \ e)) / x) by Theorem 330. Then (w * z) \ ((w * (z * (y \ e))) * ((x * ((w * (z * (y \ e))) \ ((w * (z * (y \ e))) / L(y \ e,z,w)))) / x)) = L(y \ e,z,w) * ((x * (L(y \ e,z,w) \ e)) / x) by Theorem 453. Then (w * z) \ ((w * (z * (y \ e))) * ((x * ((w * (z * (y \ e))) \ (w * z))) / x)) = L(y \ e,z,w) * ((x * (L(y \ e,z,w) \ e)) / x) by Theorem 453. Then (w * z) \ (w * ((z * (y \ e)) * ((x * ((z * (y \ e)) \ z)) / x))) = L(y \ e,z,w) * ((x * (L(y \ e,z,w) \ e)) / x) by Theorem 1102. Then
(L5) (w * z) \ (w * (z * ((y \ e) * ((x * ((y \ e) \ e)) / x)))) = L(y \ e,z,w) * ((x * (L(y \ e,z,w) \ e)) / x)
by Theorem 170. We have L((y \ e) * ((x * ((y \ e) \ e)) / x),z,w) = (w * z) \ (w * (z * ((y \ e) * ((x * ((y \ e) \ e)) / x)))) by Definition 4. Then
(L7) L((y \ e) * ((x * ((y \ e) \ e)) / x),z,w) = L(y \ e,z,w) * ((x * (L(y \ e,z,w) \ e)) / x)
by (L5). We have L(y \ e,z,w) * ((x * (L(y \ e,z,w) \ e)) / x) = x \ T(x,L(y \ e,z,w) \ e) by Theorem 980. Then
(L9) L((y \ e) * ((x * ((y \ e) \ e)) / x),z,w) = x \ T(x,L(y \ e,z,w) \ e)
by (L7).
K(x,L(y \ e,z,w) \ e)
=x \ T(x,L(y \ e,z,w) \ e) by Theorem 1308
=L(x \ T(x,(y \ e) \ e),z,w) by (L9), Theorem 980

Then
(L12) K(x,L(y \ e,z,w) \ e) = L(x \ T(x,(y \ e) \ e),z,w)
.
K(x,x \ (x / L(y \ e,z,w)))
=K(x,L(y \ e,z,w) \ e) by Theorem 1485
=L(x \ T(x,y),z,w) by (L12), Theorem 721

Hence we are done.
Theorem 1511 (prov9_ee0971eece): L(K(y,x),z,w) \ e = L(K(x,y),z,w)
Proof. We have K(x \ (x / L(y \ e,z,w)),x) \ e = K(x,x \ (x / L(y \ e,z,w))) by Theorem 1466. Then L(y \ T(y,x),z,w) \ e = K(x,x \ (x / L(y \ e,z,w))) by Theorem 1358. Then L(x \ T(x,y),z,w) = L(y \ T(y,x),z,w) \ e by Theorem 1510. Then L(K(x,y),z,w) = L(y \ T(y,x),z,w) \ e by Theorem 1308. Hence we are done by Theorem 1308.
Theorem 1512 (prov9_60734b6154): (x / (z * K(y,z))) * z = (x * z) / (z * K(y,z))
Proof. We have (x * z) / z = x by Axiom 5. Then
(L5) (((x * z) / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) / z = x
by Axiom 6. We have ((((x * z) / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) / z) * z = ((x * z) / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e))) by Axiom 6. Then ((((((x * z) / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) / z) / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z = ((x * z) / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e))) by Axiom 6. Then
(L8) ((x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z = ((x * z) / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))
by (L5). We have ((x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * ((L(((z * y) \ z) \ e,e / (((z * y) \ z) \ e),z) * z) / L(((z * y) \ z) \ e,e / (((z * y) \ z) \ e),z))) * z = ((x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z) * ((L(((z * y) \ z) \ e,e / (((z * y) \ z) \ e),z) * z) / L(((z * y) \ z) \ e,e / (((z * y) \ z) \ e),z)) by Theorem 860. Then ((x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z = ((x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z) * ((L(((z * y) \ z) \ e,e / (((z * y) \ z) \ e),z) * z) / L(((z * y) \ z) \ e,e / (((z * y) \ z) \ e),z)) by Theorem 883. Then ((x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e))) = ((x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z by Theorem 883. Then ((x * z) / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e))) = ((x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z) * (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e))) by (L8). Then (x * z) / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e))) = (x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z by Proposition 10. Then (x * z) / (z * ((((z * y) \ z) \ e) \ T(((z * y) \ z) \ e,z))) = (x / (z * (T(((z * y) \ z) \ e,z) / (((z * y) \ z) \ e)))) * z by Theorem 758. Then (x * z) / (z * ((((z * y) \ z) \ e) \ T(((z * y) \ z) \ e,z))) = (x / (z * ((((z * y) \ z) \ e) \ T(((z * y) \ z) \ e,z)))) * z by Theorem 758. Then (x * z) / (z * ((((z * y) \ z) \ e) \ (((z * y) \ z) \ K(y,z)))) = (x / (z * ((((z * y) \ z) \ e) \ T(((z * y) \ z) \ e,z)))) * z by Theorem 538. Then (x * z) / (z * K(y,z)) = (x / (z * ((((z * y) \ z) \ e) \ T(((z * y) \ z) \ e,z)))) * z by Theorem 735. Then (x / (z * (((z * y) \ z) * T(((z * y) \ z) \ e,z)))) * z = (x * z) / (z * K(y,z)) by Theorem 729. Hence we are done by Theorem 539.
Theorem 1513 (prov9_a3aea79ede): ((x * K(y,z)) / (z * K(y,z))) * z = R(x,K(y,z),z)
Proof. We have ((x * K(y,z)) * z) / (z * K(y,z)) = R(x,K(y,z),z) by Theorem 1420. Hence we are done by Theorem 1512.
Theorem 1514 (prov9_ce5c6a8f50): x * (z * K(y,z)) = (x * z) * K(y,z)
Proof. We have (((x * z) * K(y,z)) / (z * K(y,z))) * z = R(x * z,K(y,z),z) by Theorem 1513. Then
(L2) R(x,z,K(y,z)) * z = R(x * z,K(y,z),z)
by Definition 5. We have R(x * z,K(y,z),z) = x * z by Theorem 1504. Then R(x,z,K(y,z)) * z = x * z by (L2). Then
(L5) R(x,z,K(y,z)) = x
by Proposition 10. We have R(x,z,K(y,z)) * (z * K(y,z)) = (x * z) * K(y,z) by Proposition 54. Hence we are done by (L5).
Theorem 1515 (prov9_de069153c2): L(K(y,z),z,x) = K(y,z)
Proof. We have (x * z) * K(y,z) = x * (z * K(y,z)) by Theorem 1514. Hence we are done by Theorem 54.
Theorem 1516 (prov9_594a7279ad): a(x,z,K(y,z)) = e
Proof. We have L(K(y,z),z,x) = K(y,z) by Theorem 1515. Hence we are done by Proposition 20.
Theorem 1517 (prov9_1301c55252): K(x,z \ y) = K(x,(y \ z) \ e)
Proof. We have L(K(x,z \ y),z \ y,z) = K(x,(y \ z) \ e) by Theorem 1500. Hence we are done by Theorem 1515.
Theorem 1518 (prov9_6b2617f39e): L(K(y,z),y,x) = K(y,z)
Proof. We have L(K(z,y),y,x) \ e = L(K(y,z),y,x) by Theorem 1511. Then
(L2) K(z,y) \ e = L(K(y,z),y,x)
by Theorem 1515. We have K(z,y) \ e = K(y,z) by Theorem 1466. Hence we are done by (L2).
Theorem 1519 (prov9_6b6a692369): K(y,z) = x \ ((x / y) * T(y,z))
Proof. We have L(K(y,z),y,x / y) = x \ ((x / y) * T(y,z)) by Theorem 1488. Hence we are done by Theorem 1518.
Theorem 1520 (prov9_af595f30dc): K(x,z) = (y * T(z,x)) \ (y * z)
Proof. We have a(y,T(z,x),K(T(x,T(z,x)),T(z,x))) = e by Theorem 1516. Then a(y,T(z,x),K(x,z)) = e by Theorem 1313. Then
(L3) L(K(x,z),T(z,x),y) = K(x,z)
by Proposition 19. We have L(K(x,z),T(z,x),y) = (y * T(z,x)) \ (y * z) by Theorem 1483. Hence we are done by (L3).
Theorem 1521 (prov9_3fa109cc0e): (x * y) * K(y,z) = x * T(y,z)
Proof. We have (x * y) \ (x * T(y,z)) = L(K(y,z),y,x) by Theorem 1487. Then
(L2) (x * y) \ (x * T(y,z)) = K(y,z)
by Theorem 1518. We have (x * y) * ((x * y) \ (x * T(y,z))) = x * T(y,z) by Axiom 4. Hence we are done by (L2).
Theorem 1522 (prov9_76686f62f1): x * K(y,z) = (x / y) * T(y,z)
Proof. We have x * (x \ ((x / y) * T(y,z))) = (x / y) * T(y,z) by Axiom 4. Hence we are done by Theorem 1519.
Theorem 1523 (prov9_1e7f00e691): (y * T(z,x)) * K(x,z) = y * z
Proof. We have (y * T(z,x)) * ((y * T(z,x)) \ (y * z)) = y * z by Axiom 4. Hence we are done by Theorem 1520.
Theorem 1524 (prov9_c855606ca6): (x * K(y,z)) * y = x * T(y,z)
Proof. We have ((x / y) * T(y,z)) * y = x * T(y,z) by Theorem 1344. Hence we are done by Theorem 1522.
Theorem 1525 (prov9_4475f0ae2b): x * K(y,z) = (x / T(z,y)) * z
Proof. We have ((x / T(z,y)) * T(z,y)) * K(y,z) = (x / T(z,y)) * z by Theorem 1523. Hence we are done by Axiom 6.
Theorem 1526 (prov9_25acea06a9): (x * K(z,y)) \ x = K(y,z)
Proof. We have ((x / z) * T(z,y)) \ ((x / z) * z) = K(y,z) by Theorem 1520. Then ((x / z) * T(z,y)) \ x = K(y,z) by Axiom 6. Hence we are done by Theorem 1522.
Theorem 1527 (prov9_14fc043b2f): (z * K(y,x)) * K(x,y) = z
Proof. We have (z * K(y,x)) * ((z * K(y,x)) \ z) = z by Axiom 4. Hence we are done by Theorem 1526.
Theorem 1528 (prov9_d27013930c): x = L(x,K(x,y),z)
Proof. We have ((z / x) * T(x,y)) \ (z * T(x,y)) = x by Theorem 1346. Then
(L2) (z * K(x,y)) \ (z * T(x,y)) = x
by Theorem 1522. We have (z * K(x,y)) \ (z * T(x,y)) = L(x,K(x,y),z) by Theorem 1489. Hence we are done by (L2).
Theorem 1529 (prov9_8e65b32c51): (y \ x) * K(y,z) = y \ (x * K(y,z))
Proof. We have ((y \ x) * y) * K(y,z) = (y \ x) * T(y,z) by Theorem 1521. Then
(L2) ((y \ x) * y) * K(y,z) = ((y \ x) * K(y,z)) * y
by Theorem 1524. We have (y \ (x * K(y,z))) * y = ((y \ x) * y) * K(y,z) by Theorem 1501. Hence we are done by (L2) and Proposition 8.
Theorem 1530 (prov9_42d9e985f5): y * (x * K(y,z)) = (y * x) * K(y,z)
Proof. We have y * (y \ ((y * x) * K(y,z))) = (y * x) * K(y,z) by Axiom 4. Then y * ((y \ (y * x)) * K(y,z)) = (y * x) * K(y,z) by Theorem 1529. Hence we are done by Axiom 3.
Theorem 1531 (prov9_f21de7befc): T(x * y,K(y,z)) = y * T(x,T(y,z))
Proof. We have K(z \ (z / (y \ e)),z) \ (L(x,y,K(z \ (z / (y \ e)),z)) * (K(z \ (z / (y \ e)),z) * y)) = y * T(x,K(z \ (z / (y \ e)),z) * y) by Theorem 805. Then K(z \ (z / (y \ e)),z) \ (L(x,y,K(z \ (z / (y \ e)),z)) * T(y,z)) = y * T(x,K(z \ (z / (y \ e)),z) * y) by Theorem 1425. Then K(z \ (z / (y \ e)),z) \ (L(x,y,K(z \ (z / (y \ e)),z)) * T(y,z)) = y * T(x,T(y,z)) by Theorem 1425. Then (T(z,T(y,z)) \ z) \ (L(x,y,K(z \ (z / (y \ e)),z)) * T(y,z)) = y * T(x,T(y,z)) by Theorem 1456. Then
(L14) (T(z,T(y,z)) \ z) \ (L(x,y,T(z,T(y,z)) \ z) * T(y,z)) = y * T(x,T(y,z))
by Theorem 1456. We have L(x,y,y \ T(y,z)) = x by Theorem 1460. Then L(x,y,T(z,T(y,z)) \ z) = x by Theorem 986. Then
(L15) (T(z,T(y,z)) \ z) \ (x * T(y,z)) = y * T(x,T(y,z))
by (L14). We have (T(y,z) * (z \ T(z,T(y,z)))) * (T(z,T(y,z)) \ z) = T(y,z) by Theorem 1211. Then
(L5) y * (T(z,T(y,z)) \ z) = T(y,z)
by Theorem 978. We have T(y,z) = y * K(y,z) by Theorem 1309. Then y * (T(z,T(y,z)) \ z) = y * K(y,z) by (L5). Then T(z,T(y,z)) \ z = K(y,z) by Proposition 9. Then
(L16) K(y,z) \ (x * T(y,z)) = y * T(x,T(y,z))
by (L15). We have T(x * y,K(y,z)) = K(y,z) \ ((x * y) * K(y,z)) by Definition 3. Then T(x * y,K(y,z)) = K(y,z) \ (x * T(y,z)) by Theorem 1521. Hence we are done by (L16).
Theorem 1532 (prov9_9c8aa5b003): T(y,x) = T(y,x * K(y,z))
Proof. We have (T(y,z) * (T(T(y,z),y \ e) \ T(y,z))) \ (T(y,z) * T(y,z)) = T(T(y,z),y \ e) by Theorem 570. Then (T(y,z) * (y * T(y \ e,T(y,z)))) \ (T(y,z) * T(y,z)) = T(T(y,z),y \ e) by Theorem 1028. Then (y * (T(y,z) * T(y \ e,T(y,z)))) \ (T(y,z) * T(y,z)) = T(T(y,z),y \ e) by Theorem 172. Then (y * ((y \ e) * T(y,z))) \ (T(y,z) * T(y,z)) = T(T(y,z),y \ e) by Proposition 46. Then L(T(y,z),y \ e,y) \ (T(y,z) * T(y,z)) = T(T(y,z),y \ e) by Proposition 56. Then
(L13) T(e / (e / y),z) \ (T(y,z) * T(y,z)) = T(T(y,z),y \ e)
by Theorem 264. We have T(e / (e / y),z) \ (T(T(e / (e / y),z),e / y) * T(y,z)) = T(T(e / (e / y),z),e / y) * (T(e / (e / y),z) \ T(y,z)) by Theorem 173. Then T(e / (e / y),z) \ (T((e / y) \ e,z) * T(y,z)) = T(T(e / (e / y),z),e / y) * (T(e / (e / y),z) \ T(y,z)) by Proposition 51. Then T((e / y) \ e,z) * (T(e / (e / y),z) \ T(y,z)) = T(e / (e / y),z) \ (T((e / y) \ e,z) * T(y,z)) by Proposition 51. Then T(e / (e / y),z) \ (T((e / y) \ e,z) * T(y,z)) = T((e / y) \ e,z) * K(T(y,z),e / y) by Theorem 401. Then T(e / (e / y),z) \ (T(y,z) * T(y,z)) = T((e / y) \ e,z) * K(T(y,z),e / y) by Proposition 25. Then T((e / y) \ e,z) * K(T(y,z),e / y) = T(T(y,z),y \ e) by (L13). Then T(y,z) * K(T(y,z),e / y) = T(T(y,z),y \ e) by Proposition 25. Then T(y,z) * K(T(y,z),y \ e) = T(T(y,z),y \ e) by Theorem 780. Then
(L17) T(y,z) * K(y,y \ e) = T(T(y,z),y \ e)
by Theorem 1416. We have T(T(y,z),y \ e) = T((y \ e) \ e,z) by Theorem 18. Then
(L19) T(y,z) * K(y,y \ e) = T((y \ e) \ e,z)
by (L17). We have T(y,z) * K(y,y \ e) = y * (T(y,z) * (e / y)) by Theorem 1349. Then
(L21) T((y \ e) \ e,z) = y * (T(y,z) * (e / y))
by (L19). We have R((y * x) / T(y,y \ e),K(T(y,z),T(y,z) \ e),T(y,z)) = (y * x) / T(y,y \ e) by Theorem 409. Then
(L23) R((y * x) / T(y,y \ e),K(y,y \ e),T(y,z)) = (y * x) / T(y,y \ e)
by Theorem 744. We have R((y * x) / T(y,y \ e),K(y,y \ e),T(y,z)) * (K(y,y \ e) * T(y,z)) = (((y * x) / T(y,y \ e)) * K(y,y \ e)) * T(y,z) by Proposition 54. Then R((y * x) / T(y,y \ e),K(y,y \ e),T(y,z)) * (y * (T(y,z) * (e / y))) = (((y * x) / T(y,y \ e)) * K(y,y \ e)) * T(y,z) by Theorem 275. Then ((y * x) / T(y,y \ e)) * (y * (T(y,z) * (e / y))) = (((y * x) / T(y,y \ e)) * K(y,y \ e)) * T(y,z) by (L23). Then ((y * x) / y) * T(y,z) = ((y * x) / T(y,y \ e)) * (y * (T(y,z) * (e / y))) by Theorem 1361. Then ((y * x) / T(y,y \ e)) * T((y \ e) \ e,z) = ((y * x) / y) * T(y,z) by (L21). Then
(L29) ((y * x) / ((y \ e) \ e)) * T((y \ e) \ e,z) = ((y * x) / y) * T(y,z)
by Proposition 49. We have K(z \ (z / ((y * x) \ ((y * x) / ((y \ e) \ e)))),z) = (y * x) \ (((y * x) / ((y \ e) \ e)) * T((y \ e) \ e,z)) by Theorem 852. Then K(z \ (z / T((y * x) \ ((y * x) / y),y)),z) = (y * x) \ (((y * x) / ((y \ e) \ e)) * T((y \ e) \ e,z)) by Theorem 421. Then K(z \ (z / T((y * x) \ ((y * x) / y),y)),z) = (y * x) \ (((y * x) / y) * T(y,z)) by (L29). Then K(z \ (z / T(L(y \ e,T(y,x),x),y)),z) = (y * x) \ (((y * x) / y) * T(y,z)) by Theorem 1455. Then
(L34) K(z \ (z / L(T(y \ e,y),T(y,x),x)),z) = (y * x) \ (((y * x) / y) * T(y,z))
by Axiom 8. We have L(K(z \ (z / T(y \ e,y)),z),T(y,x),x) = K(z \ (z / L(T(y \ e,y),T(y,x),x)),z) by Theorem 1359. Then L(y \ T(y,z),T(y,x),x) = K(z \ (z / L(T(y \ e,y),T(y,x),x)),z) by Theorem 1426. Then
(L37) (y * x) \ (((y * x) / y) * T(y,z)) = L(y \ T(y,z),T(y,x),x)
by (L34). We have (y * x) * ((y * x) \ (((y * x) / y) * T(y,z))) = ((y * x) / y) * T(y,z) by Axiom 4. Then
(L39) (y * x) * L(y \ T(y,z),T(y,x),x) = ((y * x) / y) * T(y,z)
by (L37). We have (y * x) * L(y \ T(y,z),T(y,x),x) = x * (T(y,x) * (y \ T(y,z))) by Theorem 58. Then ((y * x) / y) * T(y,z) = x * (T(y,x) * (y \ T(y,z))) by (L39). Then x * (T(y,z) * (y \ T(y,x))) = ((y * x) / y) * T(y,z) by Theorem 833. Then x * (K(y,z) * T(y,x)) = ((y * x) / y) * T(y,z) by Theorem 1492. Then
(L44) (y * x) * K(y,z) = x * (K(y,z) * T(y,x))
by Theorem 1522. We have (y * x) * K(y,z) = y * (x * K(y,z)) by Theorem 1530. Then
(L46) x * (K(y,z) * T(y,x)) = y * (x * K(y,z))
by (L44). We have x \ (L(y,K(y,z),x) * (x * K(y,z))) = K(y,z) * T(y,x * K(y,z)) by Theorem 805. Then x \ (y * (x * K(y,z))) = K(y,z) * T(y,x * K(y,z)) by Theorem 1528. Then
(L47) x \ (x * (K(y,z) * T(y,x))) = K(y,z) * T(y,x * K(y,z))
by (L46). We have x \ (x * (K(y,z) * T(y,x))) = K(y,z) * T(y,x) by Axiom 3. Then K(y,z) * T(y,x * K(y,z)) = K(y,z) * T(y,x) by (L47). Hence we are done by Proposition 9.
Theorem 1533 (prov9_1049e380a1): T(z,x) = T(z,x * K(y,z))
Proof. We have T(z,(x * K(y,z)) * K(z,y)) = T(z,x * K(y,z)) by Theorem 1532. Hence we are done by Theorem 1527.
Theorem 1534 (prov9_a864280f6d): T(y,K(x,y) * z) = T(y,z)
Proof. We have (x \ z) * T((x \ z) \ z,y) = T(x,y) * (x \ z) by Theorem 48. Then
(L2) z * K(y \ (y / (z \ (x \ z))),y) = T(x,y) * (x \ z)
by Theorem 284. We have T(y,z * K(y \ (y / (z \ (x \ z))),y)) = T(y,z) by Theorem 1533. Then T(y,T(x,y) * (x \ z)) = T(y,z) by (L2). Hence we are done by Theorem 1492.
Theorem 1535 (prov9_46b1e9d31b): T(T(z,w),x) = T(T(z,x),K(y,z) * w)
Proof. We have T(T(z,K(y,z) * w),x) = T(T(z,x),K(y,z) * w) by Axiom 7. Hence we are done by Theorem 1534.
Theorem 1536 (prov9_a15ba03d56): (y * z) / T(T(y,z),x) = K(x,y) * z
Proof. We have (y * z) / ((K(x,y) * z) \ (y * z)) = K(x,y) * z by Proposition 24. Then (y * z) / T((y * z) / (K(x,y) * z),K(x,y) * z) = K(x,y) * z by Proposition 47. Then (y * z) / T(T(y,x),K(x,y) * z) = K(x,y) * z by Theorem 1484. Hence we are done by Theorem 1535.
Theorem 1537 (prov9_c4ccb984cb): K(x,y) * z = z * K(x,T(y,z))
Proof. We have (z * T(y,z)) / T(T(y,z),x) = (z / T(T(y,z),x)) * T(y,z) by Theorem 239. Then (y * z) / T(T(y,z),x) = (z / T(T(y,z),x)) * T(y,z) by Proposition 46. Then z * K(x,T(y,z)) = (y * z) / T(T(y,z),x) by Theorem 1525. Hence we are done by Theorem 1536.
Theorem 1538 (prov9_64fb9a2ff1): T(K(x,y),z) = K(x,T(y,z))
Proof. We have z * K(x,T(y,z)) = K(x,y) * z by Theorem 1537. Hence we are done by Theorem 11.
Theorem 1539 (prov9_06b897a9c2): K(z,K(y,x)) = K(x,K(y,z))
Proof. We have K(y \ e,x / (y \ e)) = (y \ e) \ T(y \ e,R(x / (y \ e),y \ e,(y \ e) \ e)) by Theorem 1505. Then
(L36) K(y \ e,x / (y \ e)) = (y \ e) \ T(y \ e,x * ((y \ e) \ e))
by Proposition 71. We have K(y \ e,x / (y \ e)) = K(y \ e,(y \ e) \ x) by Theorem 1465. Then
(L38) (y \ e) \ T(y \ e,x * ((y \ e) \ e)) = K(y \ e,(y \ e) \ x)
by (L36). We have T(y \ e,x * (e / (y \ e))) = T(y \ e,x * ((y \ e) \ e)) by Theorem 1366. Then
(L40) T(y \ e,x * y) = T(y \ e,x * ((y \ e) \ e))
by Proposition 24. We have (y \ e) \ T(y \ e,x * ((y \ e) \ e)) = y * T(y \ e,x * ((y \ e) \ e)) by Theorem 729. Then (y \ e) \ T(y \ e,x * ((y \ e) \ e)) = y * T(y \ e,x * y) by (L40). Then
(L43) K(y \ e,(y \ e) \ x) = y * T(y \ e,x * y)
by (L38). We have K(y \ e,(y \ e) \ x) = K(y \ e,y * x) by Theorem 1478. Then y * T(y \ e,x * y) = K(y \ e,y * x) by (L43). Then
(L46) y \ K(y \ e,y * x) = T(y \ e,x * y)
by Proposition 2. We have (y \ e) \ (T(y \ e,(((y * x) * (y \ e)) \ (y \ e)) \ e) \ (y \ e)) = T(y \ e,(((y * x) * (y \ e)) \ (y \ e)) \ e) \ e by Theorem 1339. Then (y \ e) \ (((y \ e) * K(y \ e,y * x)) \ (y \ e)) = T(y \ e,(((y * x) * (y \ e)) \ (y \ e)) \ e) \ e by Theorem 1312. Then T(y \ e,(((y * x) * (y \ e)) \ (y \ e)) \ e) \ e = (y \ e) \ (K(y \ e,y * x) \ e) by Theorem 1431. Then
(L52) ((y \ e) * K(y \ e,y * x)) \ e = (y \ e) \ (K(y \ e,y * x) \ e)
by Theorem 1312. We have (y \ e) * K(y \ e,y * x) = y \ K(e / y,y * x) by Theorem 1386. Then (y \ e) * K(y \ e,y * x) = y \ K(y \ e,y * x) by Theorem 732. Then (y \ K(y \ e,y * x)) \ e = (y \ e) \ (K(y \ e,y * x) \ e) by (L52). Then (y \ e) \ K(y * x,y \ e) = (y \ K(y \ e,y * x)) \ e by Theorem 1466. Then
(L55) T(y \ e,x * y) \ e = (y \ e) \ K(y * x,y \ e)
by (L46). We have (e / (y \ e)) * K(y * x,y \ e) = K(y * x,y \ e) / (y \ e) by Theorem 1481. Then
(L57) y * K(y * x,y \ e) = K(y * x,y \ e) / (y \ e)
by Proposition 24. We have K(z,K(y * x,y \ e) / (y \ e)) = K(z,(y \ e) \ K(y * x,y \ e)) by Theorem 1453. Then K(z,y * K(y * x,y \ e)) = K(z,(y \ e) \ K(y * x,y \ e)) by (L57). Then
(L60) K(z,T(y \ e,x * y) \ e) = K(z,y * K(y * x,y \ e))
by (L55). We have K(z,T(y \ e,x * y) \ e) = T(K(z,y),x * y) by Theorem 1496. Then K(z,y * K(y * x,y \ e)) = T(K(z,y),x * y) by (L60). Then
(L63) T(K(z,y),y * x) = T(K(z,y),x * y)
by Theorem 1507. We have T(K(z,y),y * x) * K(y,z) = K(y,z) * T(K(z,y),y * x) by Theorem 1357. Then T(K(z,y),y * x) * K(y,z) = K(K(z,y),y * x) by Theorem 1493. Then
(L66) T(K(z,y),x * y) * K(y,z) = K(K(z,y),y * x)
by (L63). We have T(K(z,y),x * y) * K(y,z) = K(y,z) * T(K(z,y),x * y) by Theorem 1357. Then T(K(z,y),x * y) * K(y,z) = K(x * y,K(y,z)) by Theorem 1508. Then
(L69) K(K(z,y),y * x) = K(x * y,K(y,z))
by (L66). We have K(K(z,y),y * x) = K(y * x,K(y,z)) by Theorem 1509. Then
(L71) K(x * y,K(y,z)) = K(y * x,K(y,z))
by (L69). We have T((y * x) / x,z) * (((y * x) / x) \ (y * x)) = (T(z,(y * x) / x) \ z) * (y * x) by Theorem 1275. Then
(L14) T((y * x) / x,z) * (((y * x) / x) \ (y * x)) = z * (T(z,(y * x) / x) \ (y * x))
by Theorem 1463. We have K(z \ (z / ((y * x) \ (((y * x) / x) \ (y * x)))),z) = (y * x) \ ((((y * x) / x) \ (y * x)) * T((((y * x) / x) \ (y * x)) \ (y * x),z)) by Theorem 285. Then K(z \ (z / ((y * x) \ (((y * x) / x) \ (y * x)))),z) = (y * x) \ (T((y * x) / x,z) * (((y * x) / x) \ (y * x))) by Theorem 48. Then
(L17) K(z \ (z / ((y * x) \ (((y * x) / x) \ (y * x)))),z) = (y * x) \ (z * (T(z,(y * x) / x) \ (y * x)))
by (L14). We have T(T(z,(y * x) / x) \ z,y * x) = (y * x) \ ((T(z,(y * x) / x) \ z) * (y * x)) by Definition 3. Then T(T(z,(y * x) / x) \ z,y * x) = (y * x) \ (z * (T(z,(y * x) / x) \ (y * x))) by Theorem 1463. Then
(L20) K(z \ (z / ((y * x) \ (((y * x) / x) \ (y * x)))),z) = T(T(z,(y * x) / x) \ z,y * x)
by (L17). We have K(z \ (z / ((y * x) \ (((y * x) / x) \ (y * x)))),z) = T(z,((y * x) \ (((y * x) / x) \ (y * x))) \ e) \ z by Theorem 1029. Then T(T(z,(y * x) / x) \ z,y * x) = T(z,((y * x) \ (((y * x) / x) \ (y * x))) \ e) \ z by (L20). Then K(((y * x) \ (((y * x) / x) \ (y * x))) \ e,z) = T(T(z,(y * x) / x) \ z,y * x) by Theorem 1482. Then T(K((y * x) / x,z),y * x) = K(((y * x) \ (((y * x) / x) \ (y * x))) \ e,z) by Theorem 1482. Then
(L25) K(((y * x) \ x) \ e,z) = T(K((y * x) / x,z),y * x)
by Proposition 25. We have K(((y * x) \ x) \ e,z) \ e = K(z,((y * x) \ x) \ e) by Theorem 1466. Then T(K((y * x) / x,z),y * x) \ e = K(z,((y * x) \ x) \ e) by (L25). Then T(K(z,(y * x) / x),y * x) = K(z,((y * x) \ x) \ e) by Theorem 1506. Then
(L29) T(K(z,y),y * x) = K(z,((y * x) \ x) \ e)
by Axiom 5. We have K(y,z) * T(K(z,y),y * x) = K(y * x,K(y,z)) by Theorem 1508. Then K(y,z) * K(z,((y * x) \ x) \ e) = K(y * x,K(y,z)) by (L29). Then K(y,z) * K(z,x \ (y * x)) = K(y * x,K(y,z)) by Theorem 1517. Then
(L33) K(y,z) * K(z,T(y,x)) = K(y * x,K(y,z))
by Definition 3. We have K(y,z) * T(K(z,y),x) = K(x,K(y,z)) by Theorem 1508. Then K(y,z) * K(z,T(y,x)) = K(x,K(y,z)) by Theorem 1538. Then K(x,K(y,z)) = K(y * x,K(y,z)) by (L33). Then
(L72) K(x * y,K(y,z)) = K(x,K(y,z))
by (L71). We have (x * y) \ ((y * x) * T(K(x,y),z)) = K(y,x) * T(K(y,x) \ e,z) by Theorem 1127. Then (x * y) \ ((y * x) * T(K(x,y),z)) = K(y,x) * T(K(x,y),z) by Theorem 1466. Then (x * y) \ (y * (x * T(K(x,y),z))) = K(y,x) * T(K(x,y),z) by Theorem 1490. Then
(L6) (x * y) \ (y * (x * T(K(x,y),z))) = K(z,K(y,x))
by Theorem 1508. We have T(x,T(y,z)) = x * K(x,T(y,z)) by Theorem 1309. Then T(x,T(y,z)) = x * T(K(x,y),z) by Theorem 1538. Then (x * y) \ (y * T(x,T(y,z))) = K(z,K(y,x)) by (L6). Then
(L8) (x * y) \ T(x * y,K(y,z)) = K(z,K(y,x))
by Theorem 1531. We have K(x * y,K(y,z)) = (x * y) \ T(x * y,K(y,z)) by Theorem 1308. Then K(x * y,K(y,z)) = K(z,K(y,x)) by (L8). Hence we are done by (L72).
Theorem 1540 (Kassoc): K(K(x,y),z) = K(x,K(y,z))
Proof. We have K(K(x,y),z) = K(z,K(y,x)) by Theorem 1509. Hence we are done by Theorem 1539.
Theorem 1541 (KKcom_a): K(y,K(x,z)) = K(x,K(y,z))
Proof. We have K(K(z,y),x) * K(x,K(z,y)) = e by Theorem 1054. Then
(L2) K(x,K(y,z)) * K(x,K(z,y)) = e
by Theorem 1509.
K(y,K(x,z)) * K(K(x,z),y)
=e by Theorem 1054
=K(x,K(y,z)) * K(K(x,z),y) by (L2), Theorem 1540

Then K(y,K(x,z)) * K(K(x,z),y) = K(x,K(y,z)) * K(K(x,z),y). Hence we are done by Proposition 10.
Theorem 1542 (prov9_4b7616aa9b): K(y,K(z,x)) = K(x,K(y,z))
Proof. We have K(z,K(y,x)) = K(x,K(y,z)) by Theorem 1539. Hence we are done by Theorem 1541.
Theorem 1543 (KKcom_b): K(x,K(z,y)) = K(x,K(y,z))
Proof. We have K(z,K(y,x)) = K(x,K(y,z)) by Theorem 1539. Hence we are done by Theorem 1542.
Theorem 1544 (prov9_8d852d016c): R(x,T(y,z) \ y,y \ T(y,z)) = (x * (T(y,z) \ y)) * (y \ T(y,z))
Proof. We have (x * (T(y,z) \ y)) * ((T(y,z) \ y) \ e) = R(x,T(y,z) \ y,(T(y,z) \ y) \ e) by Proposition 68. Then (x * (T(y,z) \ y)) * (y \ T(y,z)) = R(x,T(y,z) \ y,(T(y,z) \ y) \ e) by Theorem 1197. Hence we are done by Theorem 1197.
Theorem 1545 (prov9_43ded6caa1): x \ T(y,x) = T(x,y) \ y
Proof. We have x * (T(x,y) \ y) = T(y,T(x,y)) by Theorem 177. Then x \ T(y,T(x,y)) = T(x,y) \ y by Proposition 2. Hence we are done by Theorem 1032.
Theorem 1546 (prov9_efb12f90d7): y / (x \ T(y,x)) = T(x,y)
Proof. We have y / (T(x,y) \ y) = T(x,y) by Proposition 24. Hence we are done by Theorem 1545.
Theorem 1547 (prov9_728a5f1dea): x / (K(z,y) * x) = K(y,z)
Proof. We have x / (K(y,z) \ x) = K(y,z) by Proposition 24. Hence we are done by Theorem 1467.
Theorem 1548 (prov9_736c9da693): ((x * y) / x) * K(y,x) = y
Proof. We have (y * (T(x,y) \ x)) * ((y * (T(x,y) \ x)) \ y) = y by Axiom 4. Then (y * (T(x,y) \ x)) * (x \ T(x,y)) = y by Theorem 771. Then
(L4) R(y,T(x,y) \ x,x \ T(x,y)) = y
by Theorem 1544. We have R((x * y) / x,T(x,y) \ x,x \ T(x,y)) = (x * R(y,T(x,y) \ x,x \ T(x,y))) / x by Theorem 123. Then
(L6) R((x * y) / x,T(x,y) \ x,x \ T(x,y)) = (x * y) / x
by (L4). We have (((x * y) / x) * (T(x,y) \ x)) * (x \ T(x,y)) = R((x * y) / x,T(x,y) \ x,x \ T(x,y)) by Theorem 1544. Then
(L8) (((x * y) / x) * (T(x,y) \ x)) * (x \ T(x,y)) = (x * y) / x
by (L6). We have y * (x \ T(x,y)) = (x * y) / x by Theorem 977. Then y = ((x * y) / x) * (T(x,y) \ x) by (L8) and Proposition 8. Hence we are done by Theorem 1482.
Theorem 1549 (prov9_d2c3d542fa): T(x,K(z,y)) = T(x,K(y,z))
Proof. We have
(L1) T(x,K(y,z)) = x * K(x,K(y,z))
by Theorem 1309.
T(x,K(z,y))
=x * K(x,K(z,y)) by Theorem 1309
=T(x,K(y,z)) by (L1), Theorem 1543

Hence we are done.
Theorem 1550 (prov9_094d0cb847): (z * K(x,y)) / z = T(K(x,y),z)
Proof. We have K(K(x,y),z) = K(x,K(y,z)) by Theorem 1540. Then
(L2) K(K(x,y),z) = K(z,K(x,y))
by Theorem 1542. We have T(K(x,y),z) * K(z,K(x,y)) = K(x,y) by Theorem 1307. Then
(L4) T(K(x,y),z) * K(K(x,y),z) = K(x,y)
by (L2). We have ((z * K(x,y)) / z) * K(K(x,y),z) = K(x,y) by Theorem 1548. Then ((z * K(x,y)) / z) * K(K(x,y),z) = T(K(x,y),z) * K(K(x,y),z) by (L4). Hence we are done by Proposition 10.
Theorem 1551 (prov9_06ec46d249): T(K(y,z),x) * x = x * K(y,z)
Proof. We have ((x * K(y,z)) / x) * x = x * K(y,z) by Axiom 6. Hence we are done by Theorem 1550.
Theorem 1552 (prov9_a1213f2bb0): K(y,T(z,x)) * x = x * K(y,z)
Proof. We have T(K(y,z),x) * x = x * K(y,z) by Theorem 1551. Hence we are done by Theorem 1538.
Theorem 1553 (prov9_ebbc24145b): K(y,T(T(z,x),x)) = K(y,z)
Proof. We have
(L1) K(y,T(T(z,x),x)) * x = x * K(y,T(z,x))
by Theorem 1552. We have K(y,z) * x = x * K(y,T(z,x)) by Theorem 1537. Hence we are done by (L1) and Proposition 8.
Theorem 1554 (prov9_72b31b1064): z / (z * K(y,x)) = T(K(x,y),z)
Proof. We have K(y,x) * T(z,K(y,x)) = z * K(y,x) by Proposition 46. Then
(L2) K(x,y) \ T(z,K(y,x)) = z * K(y,x)
by Theorem 1467. We have z / (K(x,y) \ T(z,K(x,y))) = T(K(x,y),z) by Theorem 1546. Then z / (K(x,y) \ T(z,K(y,x))) = T(K(x,y),z) by Theorem 1549. Hence we are done by (L2).
Theorem 1555 (prov9_351be98466): K(x,T(z,y)) = K(T(x,y),z)
Proof. We have y / (K(z,T(x,y)) * y) = K(T(x,y),z) by Theorem 1547. Then y / (y * K(z,x)) = K(T(x,y),z) by Theorem 1552. Then
(L3) T(K(x,z),y) = K(T(x,y),z)
by Theorem 1554. We have K(x,T(z,y)) = T(K(x,z),y) by Theorem 1538. Hence we are done by (L3).
Theorem 1556 (KTT): K(T(y,x),T(z,x)) = K(y,z)
Proof. We have K(y,T(T(z,x),x)) = K(y,z) by Theorem 1553. Hence we are done by Theorem 1555.
Theorem 1557 (prov9_a40d1a6f14): e / x = L(y,x,e / x) / (x * y)
Proof. We have e * L(y,x,e / x) = L(y,x,e / x) by Axiom 1. Then
(L3) L(y,x,e / x) * (L(y,x,e / x) \ (e * L(y,x,e / x))) = L(y,x,e / x)
by Axiom 4. We have L(y,x,e / x) * (L(y,x,e / x) \ (e * L(y,x,e / x))) = e * L(y,x,e / x) by Axiom 4. Then ((L(y,x,e / x) * (L(y,x,e / x) \ (e * L(y,x,e / x)))) / (x * y)) * (x * y) = e * L(y,x,e / x) by Axiom 6. Then (L(y,x,e / x) / (x * y)) * (x * y) = e * L(y,x,e / x) by (L3). Then
(L7) (L(y,x,e / x) / (x * y)) * (x * y) = ((e / x) * x) * L(y,x,e / x)
by Axiom 6. We have (e / x) * (x * y) = ((e / x) * x) * L(y,x,e / x) by Proposition 52. Hence we are done by (L7) and Proposition 8.
Theorem 1558 (prov9_db327f4903): y * ((x \ y) \ e) = R(x,x \ y,(x \ y) \ e)
Proof. We have (x * (x \ y)) * ((x \ y) \ e) = R(x,x \ y,(x \ y) \ e) by Proposition 68. Hence we are done by Axiom 4.
Theorem 1559 (prov9_395686f0bf): T(L(y,z,y),x) = T(T((z * y) / z,x),y * z)
Proof. We have T(T((z * y) / z,y * z),x) = T(T((z * y) / z,x),y * z) by Axiom 7. Hence we are done by Theorem 92.
Theorem 1560 (prov9_24c3e7cb0d): L(w \ (u * x),y,z) = L(L((u \ w) \ x,y,z),u \ w,u)
Proof. We have L(L((u \ w) \ x,u \ w,u),y,z) = L(L((u \ w) \ x,y,z),u \ w,u) by Axiom 11. Hence we are done by Theorem 60.
Theorem 1561 (prov9_1a0f721747): K(x \ e,x) * ((x \ e) \ e) = x
Proof. We have K(x \ e,x) * (K(x \ e,x) \ x) = x by Axiom 4. Then K(x \ e,x) * T(x,x \ e) = x by Theorem 1319. Hence we are done by Proposition 49.
Theorem 1562 (prov9_f6bdaef032): ((x * (y \ z)) / z) * y = y * R(y \ x,y,y \ z)
Proof. We have R(x / y,y,y \ z) * y = y * R(y \ x,y,y \ z) by Proposition 83. Hence we are done by Theorem 66.
Theorem 1563 (prov9_6823f7be0b): L(T(y,x),z,y) = T(T((z * y) / z,x),y * z)
Proof. We have L(T(y,x),z,y) = T(L(y,z,y),x) by Axiom 8. Hence we are done by Theorem 1559.
Theorem 1564 (prov9_49fc8a2f56): (e / y) \ (T(x,z) / y) = T((e / y) \ (x / y),z)
Proof. We have (e / y) * T((e / y) \ (x / y),z) = T((x / y) * y,z) / y by Theorem 498. Then (e / y) \ (T((x / y) * y,z) / y) = T((e / y) \ (x / y),z) by Proposition 2. Hence we are done by Axiom 6.
Theorem 1565 (prov9_321de95c64): L(z,x,y) \ (z \ L(z,x,y)) = z \ e
Proof. We have L(z,x,y) * (z \ e) = z \ L(z,x,y) by Theorem 837. Hence we are done by Proposition 2.
Theorem 1566 (prov9_1be66661fe): K(y \ (y / x),y) / x = T(e / x,y)
Proof. We have (x * T(x \ e,y)) / x = T(e / x,y) by Theorem 49. Hence we are done by Theorem 196.
Theorem 1567 (prov9_b95b45bf61): y * L(y \ (x * z),w,u) = x * ((x \ y) * L((x \ y) \ z,w,u))
Proof. We have y * L(L((x \ y) \ z,w,u),x \ y,x) = x * ((x \ y) * L((x \ y) \ z,w,u)) by Theorem 52. Hence we are done by Theorem 1560.
Theorem 1568 (prov9_cdb457b473): (e / T(y,z)) * T(y,x) = T(y,x) / T(y,z)
Proof. We have (e / T(y,z)) \ (T(y,x) / T(y,z)) = T((e / T(y,z)) \ (y / T(y,z)),x) by Theorem 1564. Then
(L2) (e / T(y,z)) \ (T(y,x) / T(y,z)) = T(y,x)
by Theorem 1345. We have (e / T(y,z)) * ((e / T(y,z)) \ (T(y,x) / T(y,z))) = T(y,x) / T(y,z) by Axiom 4. Hence we are done by (L2).
Theorem 1569 (prov9_97b58e2162): T(e / T(y,z),T(y,x)) = T(y,x) \ (T(y,x) / T(y,z))
Proof. We have T(e / T(y,z),T(y,x)) = T(y,x) \ ((e / T(y,z)) * T(y,x)) by Definition 3. Hence we are done by Theorem 1568.
Theorem 1570 (prov9_0b2bc0926a): L(e / (e / z),x,y) = L(L(z,x,y),z \ e,z)
Proof. We have L(L(z,z \ e,z),x,y) = L(L(z,x,y),z \ e,z) by Axiom 11. Hence we are done by Theorem 263.
Theorem 1571 (prov9_df2ff2c0f6): L(w,x,y) * (z * w) = w * (L(w,x,y) * T(z,w))
Proof. We have L(w,x,y) * (w * T(z,w)) = w * (L(w,x,y) * T(z,w)) by Theorem 276. Hence we are done by Proposition 46.
Theorem 1572 (prov9_6138ed9f6e): w * (L(w,y,z) \ ((w \ L(w,y,z)) * x)) = L(x,w \ L(w,y,z),w)
Proof. We have L(w,y,z) \ (w * ((w \ L(w,y,z)) * x)) = L(x,w \ L(w,y,z),w) by Theorem 449. Hence we are done by Theorem 277.
Theorem 1573 (prov9_32d201fcdc): x \ K(x,L(x,y,z) / x) = T(e / x,L(x,y,z))
Proof. We have x * T(e / x,L(x,y,z)) = K(x,L(x,y,z) / x) by Theorem 282. Hence we are done by Proposition 2.
Theorem 1574 (prov9_8d958c84f3): (x * (x * L(x \ e,y,z))) / x = x * L(x \ e,y,z)
Proof. We have (x * L(x \ e,y,z)) * x = x * (x * L(x \ e,y,z)) by Theorem 303. Hence we are done by Proposition 1.
Theorem 1575 (prov9_10b7fd6672): x \ (y * L(y \ (x * z),w,u)) = (x \ y) * L((x \ y) \ z,w,u)
Proof. We have x * ((x \ y) * L((x \ y) \ z,w,u)) = y * L(y \ (x * z),w,u) by Theorem 1567. Hence we are done by Proposition 2.
Theorem 1576 (prov9_9209bed939): x \ (y * L(y \ x,z,w)) = (x \ y) * L((x \ y) \ e,z,w)
Proof. We have x \ (y * L(y \ (x * e),z,w)) = (x \ y) * L((x \ y) \ e,z,w) by Theorem 1575. Hence we are done by Axiom 2.
Theorem 1577 (prov9_908fdfbdb6): (x \ L(x,y,z)) * ((x \ e) \ e) = L((x \ e) \ e,y,z)
Proof. We have (L(x,y,z) * (x \ e)) * ((x \ e) \ e) = R(L(x,y,z),x \ e,(x \ e) \ e) by Proposition 68. Then
(L2) (x \ L(x,y,z)) * ((x \ e) \ e) = R(L(x,y,z),x \ e,(x \ e) \ e)
by Theorem 837. We have R(L(x,y,z),x \ e,(x \ e) \ e) = L(R(x,x \ e,(x \ e) \ e),y,z) by Axiom 10. Then (x \ L(x,y,z)) * ((x \ e) \ e) = L(R(x,x \ e,(x \ e) \ e),y,z) by (L2). Then L(e * ((x \ e) \ e),y,z) = (x \ L(x,y,z)) * ((x \ e) \ e) by Theorem 1558. Hence we are done by Axiom 1.
Theorem 1578 (prov9_3a6beedbb6): L(x * L(x \ e,w,u),y,z) = L(x,y,z) * L(L(x,y,z) \ e,w,u)
Proof. We have ((z * y) \ (z * (y * x))) * L(((z * y) \ (z * (y * x))) \ e,w,u) = (z * y) \ ((z * (y * x)) * L((z * (y * x)) \ (z * y),w,u)) by Theorem 1576. Then L(x,y,z) * L(((z * y) \ (z * (y * x))) \ e,w,u) = (z * y) \ ((z * (y * x)) * L((z * (y * x)) \ (z * y),w,u)) by Definition 4. Then
(L5) (z * y) \ ((z * (y * x)) * L((z * (y * x)) \ (z * y),w,u)) = L(x,y,z) * L(L(x,y,z) \ e,w,u)
by Definition 4. We have (z * (y * x)) * L(L((y * x) \ y,y * x,z),w,u) = z * ((y * x) * L((y * x) \ y,w,u)) by Theorem 57. Then (z * (y * x)) * L((z * (y * x)) \ (z * y),w,u) = z * ((y * x) * L((y * x) \ y,w,u)) by Proposition 53. Then (z * y) \ (z * ((y * x) * L((y * x) \ y,w,u))) = L(x,y,z) * L(L(x,y,z) \ e,w,u) by (L5). Then
(L7) (z * y) \ (z * (y * (x * L(x \ e,w,u)))) = L(x,y,z) * L(L(x,y,z) \ e,w,u)
by Theorem 135. We have L(x * L(x \ e,w,u),y,z) = (z * y) \ (z * (y * (x * L(x \ e,w,u)))) by Definition 4. Hence we are done by (L7).
Theorem 1579 (prov9_7e1013058a): L(y,x,y) = L(y,x,e / (e / y))
Proof. We have T(y,K(y,y \ e) * ((e / (e / y)) * x)) = T(y,(e / (e / y)) * x) by Theorem 698. Then
(L2) T(y,y * x) = T(y,(e / (e / y)) * x)
by Theorem 685. We have (x * T(y,(e / (e / y)) * x)) / x = T((x * y) / x,(e / (e / y)) * x) by Theorem 50. Then
(L4) (x * T(y,y * x)) / x = T((x * y) / x,(e / (e / y)) * x)
by (L2). We have (x * T(y,y * x)) / x = L(y,x,y) by Theorem 80. Then
(L6) T((x * y) / x,(e / (e / y)) * x) = L(y,x,y)
by (L4). We have (x * T(e / (e / y),y \ e)) / x = T((x * (e / (e / y))) / x,y \ e) by Theorem 50. Then
(L8) (x * y) / x = T((x * (e / (e / y))) / x,y \ e)
by Theorem 242. We have T(T((x * (e / (e / y))) / x,y \ e),(e / (e / y)) * x) = L(T(e / (e / y),y \ e),x,e / (e / y)) by Theorem 1563. Then T(T((x * (e / (e / y))) / x,y \ e),(e / (e / y)) * x) = L(y,x,e / (e / y)) by Theorem 242. Then T((x * y) / x,(e / (e / y)) * x) = L(y,x,e / (e / y)) by (L8). Hence we are done by (L6).
Theorem 1580 (prov9_8d95cd3da6): (L(x,y,x) \ e) \ e = T((((y * x) / y) \ e) \ e,x * y)
Proof. We have (T((y * x) / y,x * y) \ e) \ e = T((((y * x) / y) \ e) \ e,x * y) by Theorem 919. Hence we are done by Theorem 92.
Theorem 1581 (prov9_498addac08): L((y \ e) \ e,x,y) = (L(y,x,y) \ e) \ e
Proof. We have T(T((x * y) / x,((x * y) / x) \ e),y * x) = L(T(y,((x * y) / x) \ e),x,y) by Theorem 1563. Then T((((x * y) / x) \ e) \ e,y * x) = L(T(y,((x * y) / x) \ e),x,y) by Proposition 49. Then (L(y,x,y) \ e) \ e = L(T(y,((x * y) / x) \ e),x,y) by Theorem 1580. Hence we are done by Theorem 1392.
Theorem 1582 (prov9_13e3abdc26): K(y \ (y / T(y,x)),y) = K(y \ e,y)
Proof. We have K(y \ (y / T(y,x)),y) = T(K(y \ e,y),x) by Theorem 577. Hence we are done by Theorem 716.
Theorem 1583 (prov9_92b060f170): L(y,T(y,x) \ e,T(y,x)) = e / (e / y)
Proof. We have y * (T(y,x) * T(T(y,x) \ e,y)) = T(y,x) * ((T(y,x) \ e) * y) by Theorem 520. Then
(L4) y * K(y \ (y / T(y,x)),y) = T(y,x) * ((T(y,x) \ e) * y)
by Theorem 196. We have T(y,x) * ((T(y,x) \ e) * y) = L(y,T(y,x) \ e,T(y,x)) by Proposition 56. Then
(L6) y * K(y \ (y / T(y,x)),y) = L(y,T(y,x) \ e,T(y,x))
by (L4). We have y * K(y \ e,y) = e / (e / y) by Theorem 262. Then y * K(y \ (y / T(y,x)),y) = e / (e / y) by Theorem 1582. Hence we are done by (L6).
Theorem 1584 (prov9_b976f92a73): K(y \ x,(x / y) \ e) = K(x / y,(x / y) \ e)
Proof. We have K(T(x / y,y),(x / y) \ e) = K(x / y,(x / y) \ e) by Theorem 1416. Hence we are done by Proposition 47.
Theorem 1585 (prov9_bb68516067): T(e / T(x,y),x) = T(x,y) \ e
Proof. We have T(e / T(x,y),x) = x \ (x / T(x,y)) by Theorem 551. Hence we are done by Theorem 752.
Theorem 1586 (prov9_38d0e1e562): K(x \ e,x) / T(x,y) = T(x,y) \ e
Proof. We have K(x \ (x / T(x,y)),x) / T(x,y) = T(e / T(x,y),x) by Theorem 1566. Then K(x \ e,x) / T(x,y) = T(e / T(x,y),x) by Theorem 1582. Hence we are done by Theorem 1585.
Theorem 1587 (prov9_e5d79dc755): T(e / T(y,z),T(y,x)) = T(y,z) \ e
Proof. We have T(y * T(y \ e,T(y,x)),z) = K(T(y,x) \ (T(y,x) / T(y,z)),T(y,x)) by Theorem 296. Then T(y * T(y \ e,y),z) = K(T(y,x) \ (T(y,x) / T(y,z)),T(y,x)) by Theorem 1201. Then K(T(e / T(y,z),T(y,x)),T(y,x)) = T(y * T(y \ e,y),z) by Theorem 1569. Then K(T(e / T(y,z),T(y,x)),T(y,x)) = T((y \ e) * y,z) by Proposition 46. Then
(L5) K(T(e / T(y,z),T(y,x)),T(y,x)) = T(K(y \ e,y),z)
by Proposition 76. We have T(K(y \ e,y),z) = K(y \ e,y) by Theorem 716. Then
(L7) K(T(e / T(y,z),T(y,x)),T(y,x)) = K(y \ e,y)
by (L5). We have K(T(y,x) \ (T(y,x) / T(y,z)),T(y,x)) / T(y,z) = T(e / T(y,z),T(y,x)) by Theorem 1566. Then K(T(e / T(y,z),T(y,x)),T(y,x)) / T(y,z) = T(e / T(y,z),T(y,x)) by Theorem 1569. Then
(L10) K(y \ e,y) / T(y,z) = T(e / T(y,z),T(y,x))
by (L7). We have K(y \ e,y) / T(y,z) = T(y,z) \ e by Theorem 1586. Hence we are done by (L10).
Theorem 1588 (prov9_aec70ec74c): T(e / T(x / y,z),y \ x) = T(x / y,z) \ e
Proof. We have T(e / T(x / y,z),T(x / y,y)) = T(x / y,z) \ e by Theorem 1587. Hence we are done by Proposition 47.
Theorem 1589 (prov9_cd112355ef): K(y,((x * y) / x) \ e) = K(y,y \ e)
Proof. We have K((x * y) \ x,x \ (x * y)) = K(y \ e,y) by Theorem 664. Then
(L5) K(((x * y) / x) \ e,(x * y) / x) = K(y \ e,y)
by Theorem 1421. We have K(((x * y) / x) \ e,(x * y) / x) * ((((x * y) / x) \ e) \ e) = (x * y) / x by Theorem 1561. Then
(L7) K(y \ e,y) * ((((x * y) / x) \ e) \ e) = (x * y) / x
by (L5). We have K(y \ e,y) * (K(y,y \ e) * ((x * y) / x)) = (x * y) / x by Theorem 693. Then K(y \ e,y) * (K(y,y \ e) * ((x * y) / x)) = K(y \ e,y) * ((((x * y) / x) \ e) \ e) by (L7). Then
(L10) K(y,y \ e) * ((x * y) / x) = (((x * y) / x) \ e) \ e
by Proposition 9. We have K((x * y) / x,((x * y) / x) \ e) * ((x * y) / x) = (((x * y) / x) \ e) \ e by Theorem 552. Then K(x \ (x * y),((x * y) / x) \ e) * ((x * y) / x) = (((x * y) / x) \ e) \ e by Theorem 1584. Then K(y,((x * y) / x) \ e) * ((x * y) / x) = (((x * y) / x) \ e) \ e by Axiom 3. Hence we are done by (L10) and Proposition 8.
Theorem 1590 (prov9_358a1d6e28): R(e / y,y,K(x,y)) = e / y
Proof. We have (e / y) * (y * R(y \ (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y)))) = L(R(y \ (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))),y,e / y) by Proposition 67. Then (e / y) * (y * R(y \ (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y)))) = R((e / y) * (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))) by Theorem 81. Then (e / y) * ((((((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))) * (y \ (e / ((y * x) \ y)))) / (e / ((y * x) \ y))) * y) = R((e / y) * (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))) by Theorem 1562. Then (e / y) * ((T(e / ((y * x) \ y),y) / (e / ((y * x) \ y))) * y) = R((e / y) * (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))) by Theorem 179. Then R((e / y) * (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))) = (e / y) * (((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y)) * y) by Theorem 758. Then
(L8) R((e / y) * (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))) = (e / y) * (y * ((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y)))
by Theorem 937. We have (e / y) * (y * ((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y))) = L((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y,e / y) by Proposition 67. Then
(L10) R((e / y) * (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))) = L((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y,e / y)
by (L8). We have (e / y) * (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))) = (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))) / y by Theorem 858. Then (e / y) * (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))) = y \ (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))) by Theorem 940. Then
(L11) R(y \ (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))) = L((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y,e / y)
by (L10). We have T(((((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))) * (y \ (e / ((y * x) \ y)))) / (y * (y \ (e / ((y * x) \ y)))),y) = R(y \ (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))) by Proposition 64. Then T(T(e / ((y * x) \ y),y) / (y * (y \ (e / ((y * x) \ y)))),y) = R(y \ (((e / ((y * x) \ y)) * y) / (e / ((y * x) \ y))),y,y \ (e / ((y * x) \ y))) by Theorem 179. Then L((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y,e / y) = T(T(e / ((y * x) \ y),y) / (y * (y \ (e / ((y * x) \ y)))),y) by (L11). Then T(T(e / ((y * x) \ y),y) / (e / ((y * x) \ y)),y) = L((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y,e / y) by Axiom 4. Then
(L16) L((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y,e / y) = T((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y)
by Theorem 758. We have T(y,(e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y)) = y by Theorem 936. Then T((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y) = (e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y) by Proposition 21. Then
(L19) L((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y,e / y) = (e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y)
by (L16). We have L((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y),y,e / y) / (y * ((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y))) = e / y by Theorem 1557. Then
(L21) ((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y)) / (y * ((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y))) = e / y
by (L19). We have R(e / y,y,(e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y)) = ((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y)) / (y * ((e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y))) by Proposition 79. Then R(e / y,y,(e / ((y * x) \ y)) \ T(e / ((y * x) \ y),y)) = e / y by (L21). Then R(e / y,y,((y * x) \ y) * T(((y * x) \ y) \ e,y)) = e / y by Theorem 102. Hence we are done by Theorem 539.
Theorem 1591 (prov9_093c13fe26): L((y \ e) \ e,x,y) = y * (L(y,x,y) * (e / y))
Proof. We have (T((x * y) / x,y * x) * (T(T((x * y) / x,y * x),((x * y) / x) \ e) \ T((x * y) / x,y * x))) \ (T((x * y) / x,y * x) * T((x * y) / x,y * x)) = T(T((x * y) / x,y * x),((x * y) / x) \ e) by Theorem 570. Then (T((x * y) / x,y * x) * (((x * y) / x) * T(((x * y) / x) \ e,T((x * y) / x,y * x)))) \ (T((x * y) / x,y * x) * T((x * y) / x,y * x)) = T(T((x * y) / x,y * x),((x * y) / x) \ e) by Theorem 1028. Then (((x * y) / x) * (T((x * y) / x,y * x) * T(((x * y) / x) \ e,T((x * y) / x,y * x)))) \ (T((x * y) / x,y * x) * T((x * y) / x,y * x)) = T(T((x * y) / x,y * x),((x * y) / x) \ e) by Theorem 172. Then (((x * y) / x) * ((((x * y) / x) \ e) * T((x * y) / x,y * x))) \ (T((x * y) / x,y * x) * T((x * y) / x,y * x)) = T(T((x * y) / x,y * x),((x * y) / x) \ e) by Proposition 46. Then L(T((x * y) / x,y * x),((x * y) / x) \ e,(x * y) / x) \ (T((x * y) / x,y * x) * T((x * y) / x,y * x)) = T(T((x * y) / x,y * x),((x * y) / x) \ e) by Proposition 56. Then
(L11) T(e / (e / ((x * y) / x)),y * x) \ (T((x * y) / x,y * x) * T((x * y) / x,y * x)) = T(T((x * y) / x,y * x),((x * y) / x) \ e)
by Theorem 264. We have T(e / (e / ((x * y) / x)),y * x) \ (T(T(e / (e / ((x * y) / x)),y * x),e / ((x * y) / x)) * T((x * y) / x,y * x)) = T(T(e / (e / ((x * y) / x)),y * x),e / ((x * y) / x)) * (T(e / (e / ((x * y) / x)),y * x) \ T((x * y) / x,y * x)) by Theorem 173. Then T(e / (e / ((x * y) / x)),y * x) \ (T((e / ((x * y) / x)) \ e,y * x) * T((x * y) / x,y * x)) = T(T(e / (e / ((x * y) / x)),y * x),e / ((x * y) / x)) * (T(e / (e / ((x * y) / x)),y * x) \ T((x * y) / x,y * x)) by Proposition 51. Then T((e / ((x * y) / x)) \ e,y * x) * (T(e / (e / ((x * y) / x)),y * x) \ T((x * y) / x,y * x)) = T(e / (e / ((x * y) / x)),y * x) \ (T((e / ((x * y) / x)) \ e,y * x) * T((x * y) / x,y * x)) by Proposition 51. Then T(e / (e / ((x * y) / x)),y * x) \ (T((e / ((x * y) / x)) \ e,y * x) * T((x * y) / x,y * x)) = T((e / ((x * y) / x)) \ e,y * x) * K(T((x * y) / x,y * x),e / ((x * y) / x)) by Theorem 401. Then T(e / (e / ((x * y) / x)),y * x) \ (T((x * y) / x,y * x) * T((x * y) / x,y * x)) = T((e / ((x * y) / x)) \ e,y * x) * K(T((x * y) / x,y * x),e / ((x * y) / x)) by Proposition 25. Then T((e / ((x * y) / x)) \ e,y * x) * K(T((x * y) / x,y * x),e / ((x * y) / x)) = T(T((x * y) / x,y * x),((x * y) / x) \ e) by (L11). Then T((x * y) / x,y * x) * K(T((x * y) / x,y * x),e / ((x * y) / x)) = T(T((x * y) / x,y * x),((x * y) / x) \ e) by Proposition 25. Then T((x * y) / x,y * x) * K(T((x * y) / x,y * x),((x * y) / x) \ e) = T(T((x * y) / x,y * x),((x * y) / x) \ e) by Theorem 780. Then
(L15) T((x * y) / x,y * x) * K((x * y) / x,((x * y) / x) \ e) = T(T((x * y) / x,y * x),((x * y) / x) \ e)
by Theorem 1416. We have T(T((x * y) / x,y * x),((x * y) / x) \ e) = T((((x * y) / x) \ e) \ e,y * x) by Theorem 18. Then
(L17) T((x * y) / x,y * x) * K((x * y) / x,((x * y) / x) \ e) = T((((x * y) / x) \ e) \ e,y * x)
by (L15). We have K((x * y) / x,((x * y) / x) \ e) * T((x * y) / x,y * x) = T((x * y) / x,y * x) * K((x * y) / x,((x * y) / x) \ e) by Theorem 1311. Then K((x * y) / x,((x * y) / x) \ e) * T((x * y) / x,y * x) = T((((x * y) / x) \ e) \ e,y * x) by (L17). Then K((x * y) / x,((x * y) / x) \ e) * L(y,x,y) = T((((x * y) / x) \ e) \ e,y * x) by Theorem 92. Then K(x \ (x * y),((x * y) / x) \ e) * L(y,x,y) = T((((x * y) / x) \ e) \ e,y * x) by Theorem 1584. Then K(y,((x * y) / x) \ e) * L(y,x,y) = T((((x * y) / x) \ e) \ e,y * x) by Axiom 3. Then
(L23) K(y,y \ e) * L(y,x,y) = T((((x * y) / x) \ e) \ e,y * x)
by Theorem 1589. We have T((((x * y) / x) \ e) \ e,y * x) = (L(y,x,y) \ e) \ e by Theorem 1580. Then
(L25) K(y,y \ e) * L(y,x,y) = (L(y,x,y) \ e) \ e
by (L23). We have K(y,y \ e) * L(y,x,y) = y * (L(y,x,y) * (e / y)) by Theorem 281. Then
(L27) (L(y,x,y) \ e) \ e = y * (L(y,x,y) * (e / y))
by (L25). We have L((y \ e) \ e,x,y) = (L(y,x,y) \ e) \ e by Theorem 1581. Hence we are done by (L27).
Theorem 1592 (prov9_a9223895ff): x \ K(L(x,y,z),x \ e) = (x \ L(x,y,z)) / L(x,y,z)
Proof. We have T((K(x,L(x,y,z) / x) * K(L(x,y,z) / x,x)) / (x * K(L(x,y,z) / x,x)),x) = R(x \ K(x,L(x,y,z) / x),x,K(L(x,y,z) / x,x)) by Proposition 64. Then T(e / (x * K(L(x,y,z) / x,x)),x) = R(x \ K(x,L(x,y,z) / x),x,K(L(x,y,z) / x,x)) by Theorem 1054. Then
(L5) R(T(e / x,L(x,y,z)),x,K(L(x,y,z) / x,x)) = T(e / (x * K(L(x,y,z) / x,x)),x)
by Theorem 1573. We have R(T(e / x,L(x,y,z)),x,K(L(x,y,z) / x,x)) = T(R(e / x,x,K(L(x,y,z) / x,x)),L(x,y,z)) by Axiom 9. Then R(T(e / x,L(x,y,z)),x,K(L(x,y,z) / x,x)) = T(e / x,L(x,y,z)) by Theorem 1590. Then T(e / (x * K(L(x,y,z) / x,x)),x) = T(e / x,L(x,y,z)) by (L5). Then
(L9) T(e / (x * K(x \ L(x,y,z),x)),x) = T(e / x,L(x,y,z))
by Theorem 1464. We have x * (L(x,y,z) \ ((x \ L(x,y,z)) * x)) = L(x,x \ L(x,y,z),x) by Theorem 1572. Then x * K(x \ L(x,y,z),x) = L(x,x \ L(x,y,z),x) by Theorem 2. Then
(L10) T(e / L(x,x \ L(x,y,z),x),x) = T(e / x,L(x,y,z))
by (L9). We have T(e / T(((x \ L(x,y,z)) * x) / (x \ L(x,y,z)),x * (x \ L(x,y,z))),(x \ L(x,y,z)) \ ((x \ L(x,y,z)) * x)) = T(((x \ L(x,y,z)) * x) / (x \ L(x,y,z)),x * (x \ L(x,y,z))) \ e by Theorem 1588. Then T(e / L(x,x \ L(x,y,z),x),(x \ L(x,y,z)) \ ((x \ L(x,y,z)) * x)) = T(((x \ L(x,y,z)) * x) / (x \ L(x,y,z)),x * (x \ L(x,y,z))) \ e by Theorem 92. Then T(e / L(x,x \ L(x,y,z),x),x) = T(((x \ L(x,y,z)) * x) / (x \ L(x,y,z)),x * (x \ L(x,y,z))) \ e by Axiom 3. Then T(e / L(x,x \ L(x,y,z),x),x) = L(x,x \ L(x,y,z),x) \ e by Theorem 92. Then
(L15) T(e / x,L(x,y,z)) = L(x,x \ L(x,y,z),x) \ e
by (L10). We have L((x \ e) \ e,x \ L(x,y,z),x) = (L(x,x \ L(x,y,z),x) \ e) \ e by Theorem 1581. Then
(L17) L((x \ e) \ e,x \ L(x,y,z),x) = T(e / x,L(x,y,z)) \ e
by (L15). We have
(L18) x * (L(x,x \ L(x,y,z),x) * (e / x)) = L((x \ e) \ e,x \ L(x,y,z),x)
by Theorem 1591. Then x \ L((x \ e) \ e,x \ L(x,y,z),x) = L(x,x \ L(x,y,z),x) * (e / x) by Proposition 2. Then
(L20) x \ (T(e / x,L(x,y,z)) \ e) = L(x,x \ L(x,y,z),x) * (e / x)
by (L17). We have x * (L(x,y,z) \ ((x \ L(x,y,z)) * ((x \ e) \ e))) = L((x \ e) \ e,x \ L(x,y,z),x) by Theorem 1572. Then L(x,x \ L(x,y,z),x) * (e / x) = L(x,y,z) \ ((x \ L(x,y,z)) * ((x \ e) \ e)) by (L18) and Proposition 7. Then L(x,y,z) \ L((x \ e) \ e,y,z) = L(x,x \ L(x,y,z),x) * (e / x) by Theorem 1577. Then
(L24) L(x,y,z) \ L((x \ e) \ e,y,z) = x \ (T(e / x,L(x,y,z)) \ e)
by (L20). We have ((e / x) \ T(e / x,L(x,y,z))) \ e = T(e / x,L(x,y,z)) \ (e / x) by Theorem 749. Then (x * T(x \ e,L(x,y,z))) \ e = T(e / x,L(x,y,z)) \ (e / x) by Theorem 102. Then K(x \ e,L(x,y,z)) \ e = T(e / x,L(x,y,z)) \ (e / x) by Theorem 1486. Then
(L28) T(e / x,L(x,y,z)) \ (e / x) = K(L(x,y,z),x \ e)
by Theorem 1466. We have ((e / x) \ e) \ (T(e / x,L(x,y,z)) \ e) = T(e / x,L(x,y,z)) \ (e / x) by Theorem 1418. Then x \ (T(e / x,L(x,y,z)) \ e) = T(e / x,L(x,y,z)) \ (e / x) by Proposition 25. Then K(L(x,y,z),x \ e) = x \ (T(e / x,L(x,y,z)) \ e) by (L28). Then
(L32) L(x,y,z) \ L((x \ e) \ e,y,z) = K(L(x,y,z),x \ e)
by (L24). We have L(x,y,z) \ T(L(x,y,z),R(x \ e,(x \ e) \ e,L(x,y,z))) = (x \ e) \ ((L(x,y,z) * (x \ e)) / L(x,y,z)) by Theorem 760. Then L(x,y,z) \ T(L(x,y,z),R(x \ e,(x \ e) \ e,L(x,y,z))) = (x \ e) \ ((x \ L(x,y,z)) / L(x,y,z)) by Theorem 837. Then L(x,y,z) \ T(L(x,y,z),R(e / x,x,L(x,y,z))) = (x \ e) \ ((x \ L(x,y,z)) / L(x,y,z)) by Theorem 727. Then L(x,y,z) \ L(T(x,R(e / x,x,L(x,y,z))),y,z) = (x \ e) \ ((x \ L(x,y,z)) / L(x,y,z)) by Axiom 8. Then
(L37) L(x,y,z) \ L(T(x,x \ e),y,z) = (x \ e) \ ((x \ L(x,y,z)) / L(x,y,z))
by Theorem 361. We have (x \ e) \ ((x \ L(x,y,z)) / L(x,y,z)) = x * ((x \ L(x,y,z)) / L(x,y,z)) by Theorem 709. Then L(x,y,z) \ L(T(x,x \ e),y,z) = x * ((x \ L(x,y,z)) / L(x,y,z)) by (L37). Then L(x,y,z) \ L((x \ e) \ e,y,z) = x * ((x \ L(x,y,z)) / L(x,y,z)) by Proposition 49. Then K(L(x,y,z),x \ e) = x * ((x \ L(x,y,z)) / L(x,y,z)) by (L32). Hence we are done by Proposition 2.
Theorem 1593 (prov9_d2fd2c22a9): L(K(x,y \ e),z,w) = K(x,L(y,z,w) \ e)
Proof. We have ((w * (z * y)) / L(y,z,w)) \ ((w * (z * y)) * ((x * ((w * (z * y)) \ ((w * (z * y)) / L(y,z,w)))) / x)) = L(y,z,w) * ((x * (L(y,z,w) \ e)) / x) by Theorem 330. Then (w * z) \ ((w * (z * y)) * ((x * ((w * (z * y)) \ ((w * (z * y)) / L(y,z,w)))) / x)) = L(y,z,w) * ((x * (L(y,z,w) \ e)) / x) by Theorem 453. Then (w * z) \ ((w * (z * y)) * ((x * ((w * (z * y)) \ (w * z))) / x)) = L(y,z,w) * ((x * (L(y,z,w) \ e)) / x) by Theorem 453. Then (w * z) \ (w * ((z * y) * ((x * ((z * y) \ z)) / x))) = L(y,z,w) * ((x * (L(y,z,w) \ e)) / x) by Theorem 1102. Then
(L5) (w * z) \ (w * (z * (y * ((x * (y \ e)) / x)))) = L(y,z,w) * ((x * (L(y,z,w) \ e)) / x)
by Theorem 170. We have L(y * ((x * (y \ e)) / x),z,w) = (w * z) \ (w * (z * (y * ((x * (y \ e)) / x)))) by Definition 4. Then
(L7) L(y * ((x * (y \ e)) / x),z,w) = L(y,z,w) * ((x * (L(y,z,w) \ e)) / x)
by (L5). We have L(y,z,w) * ((x * (L(y,z,w) \ e)) / x) = x \ T(x,L(y,z,w) \ e) by Theorem 980. Then
(L9) L(y * ((x * (y \ e)) / x),z,w) = x \ T(x,L(y,z,w) \ e)
by (L7).
K(x,L(y,z,w) \ e)
=x \ T(x,L(y,z,w) \ e) by Theorem 1308
=L(x \ T(x,y \ e),z,w) by (L9), Theorem 980

Then K(x,L(y,z,w) \ e) = L(x \ T(x,y \ e),z,w). Hence we are done by Theorem 1308.
Theorem 1594 (prov9_b9c6a37e08): (L(x,y,z) / x) \ (e / x) = e / L(x,y,z)
Proof. We have (e / x) * ((x \ e) * L((x \ e) \ ((e / x) \ (x \ e)),y,z)) = ((e / x) * L((e / x) \ e,y,z)) * (x \ e) by Theorem 325. Then (e / x) * ((x \ e) * L((x \ e) \ K(x \ e,x),y,z)) = ((e / x) * L((e / x) \ e,y,z)) * (x \ e) by Proposition 90. Then
(L5) (e / x) * ((x \ e) * L(x,y,z)) = ((e / x) * L((e / x) \ e,y,z)) * (x \ e)
by Theorem 21. We have (e / x) * ((x \ e) * L(x,y,z)) = (x \ e) * ((e / x) * L(x,y,z)) by Theorem 535. Then ((e / x) * L((e / x) \ e,y,z)) * (x \ e) = (x \ e) * ((e / x) * L(x,y,z)) by (L5). Then
(L8) T((e / x) * L((e / x) \ e,y,z),x \ e) = (e / x) * L(x,y,z)
by Theorem 11. We have (e / x) * L(x,y,z) = L(x,y,z) / x by Theorem 166. Then T((e / x) * L((e / x) \ e,y,z),x \ e) = L(x,y,z) / x by (L8). Then T((e / x) * L(x,y,z),x \ e) = L(x,y,z) / x by Proposition 25. Then
(L12) T(L(x,y,z) / x,x \ e) = L(x,y,z) / x
by Theorem 166. We have (x \ e) * (T(((L(x,y,z) / x) * (e / (x \ e))) * (x \ e),x \ e) / (x \ e)) = T((x \ e) * ((L(x,y,z) / x) * (e / (x \ e))),x \ e) by Theorem 584. Then (x \ e) * ((e / (x \ e)) * T((e / (x \ e)) \ ((L(x,y,z) / x) * (e / (x \ e))),x \ e)) = T((x \ e) * ((L(x,y,z) / x) * (e / (x \ e))),x \ e) by Theorem 498. Then (x \ e) * (T(L(x,y,z) / x,x \ e) * (e / (x \ e))) = T((x \ e) * ((L(x,y,z) / x) * (e / (x \ e))),x \ e) by Theorem 466. Then T((x \ e) * ((L(x,y,z) / x) * (e / (x \ e))),x \ e) = (x \ e) * ((L(x,y,z) / x) * (e / (x \ e))) by (L12). Then T((x \ e) * ((L(x,y,z) / x) * (e / (x \ e))),x \ e) = (x \ e) * ((L(x,y,z) / x) * x) by Proposition 24. Then T((x \ e) * ((L(x,y,z) / x) * (e / (x \ e))),x \ e) = (x \ e) * L(x,y,z) by Axiom 6. Then T((x \ e) * ((L(x,y,z) / x) * x),x \ e) = (x \ e) * L(x,y,z) by Proposition 24. Then T((x \ e) * L(x,y,z),x \ e) = (x \ e) * L(x,y,z) by Axiom 6. Then
(L21) T(x \ e,(x \ e) * L(x,y,z)) = x \ e
by Proposition 21. We have L(x,y,z) * T(x \ e,(x \ e) * L(x,y,z)) = L(x \ e,L(x,y,z),x \ e) * L(x,y,z) by Theorem 79. Then
(L23) L(x,y,z) * (x \ e) = L(x \ e,L(x,y,z),x \ e) * L(x,y,z)
by (L21). We have L(x,y,z) * (x \ e) = x \ L(x,y,z) by Theorem 837. Then L(x \ e,L(x,y,z),x \ e) * L(x,y,z) = x \ L(x,y,z) by (L23). Then (x \ L(x,y,z)) / L(x,y,z) = L(x \ e,L(x,y,z),x \ e) by Proposition 1. Then
(L27) L(x \ e,L(x,y,z),x \ e) = x \ K(L(x,y,z),x \ e)
by Theorem 1592. We have L(L(x,y,z) \ e,L(x,y,z),e / x) = ((e / x) * L(x,y,z)) \ ((e / x) * e) by Proposition 53. Then
(L29) L(L(x,y,z) \ e,L(x,y,z),e / x) = (L(x,y,z) / x) \ ((e / x) * e)
by Theorem 166. We have L(x,y,z) * L(L(x,y,z) \ e,L(x,y,z),e / x) = L(x * L(x \ e,L(x,y,z),e / x),y,z) by Theorem 1578. Then L(x,y,z) * ((L(x,y,z) / x) \ ((e / x) * e)) = L(x * L(x \ e,L(x,y,z),e / x),y,z) by (L29). Then L(x,y,z) * ((L(x,y,z) / x) \ (e / x)) = L(x * L(x \ e,L(x,y,z),e / x),y,z) by Axiom 2. Then L(x * L(x \ e,L(x,y,z),x \ e),y,z) = L(x,y,z) * ((L(x,y,z) / x) \ (e / x)) by Theorem 1224. Then
(L34) L(x * (x \ K(L(x,y,z),x \ e)),y,z) = L(x,y,z) * ((L(x,y,z) / x) \ (e / x))
by (L27). We have L(K(L(x,y,z),x \ e),y,z) = K(L(x,y,z),L(x,y,z) \ e) by Theorem 1593. Then L(x * (x \ K(L(x,y,z),x \ e)),y,z) = K(L(x,y,z),L(x,y,z) \ e) by Axiom 4. Then
(L35) K(L(x,y,z),L(x,y,z) \ e) = L(x,y,z) * ((L(x,y,z) / x) \ (e / x))
by (L34). We have L(x,y,z) * (e / L(x,y,z)) = K(L(x,y,z),L(x,y,z) \ e) by Theorem 250. Hence we are done by (L35) and Proposition 7.
Theorem 1595 (prov9_d8239410ef): L(e / x,y,z) \ e = e / L(x \ e,y,z)
Proof. We have L(x \ T(x,x * L(x \ e,y,z)),y,z) = K((x * L(x \ e,y,z)) \ ((x * L(x \ e,y,z)) / L(x \ e,y,z)),x * L(x \ e,y,z)) by Theorem 1358. Then L(x \ T(x,x * L(x \ e,y,z)),y,z) = K((x * L(x \ e,y,z)) \ x,x * L(x \ e,y,z)) by Axiom 5. Then L(x \ x,y,z) = K((x * L(x \ e,y,z)) \ x,x * L(x \ e,y,z)) by Theorem 304. Then
(L4) K((x * L(x \ e,y,z)) \ x,x * L(x \ e,y,z)) = L(e,y,z)
by Proposition 28. We have L(e,y,z) = (z * y) \ (z * (y * e)) by Definition 4. Then
(L6) L(e,y,z) = (z * y) \ (z * y)
by Axiom 2. We have (z * y) \ (z * y) = e by Proposition 28. Then L(e,y,z) = e by (L6). Then
(L9) K((x * L(x \ e,y,z)) \ x,x * L(x \ e,y,z)) = e
by (L4). We have T(x,x * L(x \ e,y,z)) * K((x * L(x \ e,y,z)) \ x,(x * (x * L(x \ e,y,z))) / x) = ((x * L(x \ e,y,z)) \ x) * ((x * (x * L(x \ e,y,z))) / x) by Theorem 1092. Then x * K((x * L(x \ e,y,z)) \ x,(x * (x * L(x \ e,y,z))) / x) = ((x * L(x \ e,y,z)) \ x) * ((x * (x * L(x \ e,y,z))) / x) by Theorem 304. Then x * K((x * L(x \ e,y,z)) \ x,x * L(x \ e,y,z)) = ((x * L(x \ e,y,z)) \ x) * ((x * (x * L(x \ e,y,z))) / x) by Theorem 1574. Then
(L13) ((x * L(x \ e,y,z)) \ x) * ((x * (x * L(x \ e,y,z))) / x) = x * e
by (L9). We have (((x * L(x \ e,y,z)) \ x) * ((x * (x * L(x \ e,y,z))) / x)) * e = ((x * L(x \ e,y,z)) \ x) * ((x * (x * L(x \ e,y,z))) / x) by Axiom 2. Then x * e = (((x * L(x \ e,y,z)) \ x) * ((x * (x * L(x \ e,y,z))) / x)) * e by (L13). Then x = ((x * L(x \ e,y,z)) \ x) * ((x * (x * L(x \ e,y,z))) / x) by Proposition 10. Then
(L17) ((x * L(x \ e,y,z)) \ x) * (x * L(x \ e,y,z)) = x
by Theorem 1574. We have ((e / x) \ e) / ((e / x) \ L(e / x,y,z)) = L(e / x,y,z) \ e by Theorem 906. Then x / ((e / x) \ L(e / x,y,z)) = L(e / x,y,z) \ e by Proposition 25. Then
(L20) x / (x * L(x \ e,y,z)) = L(e / x,y,z) \ e
by Theorem 159. We have (x / (x * L(x \ e,y,z))) * (x * L(x \ e,y,z)) = x by Axiom 6. Then (L(e / x,y,z) \ e) * (x * L(x \ e,y,z)) = x by (L20). Then (L(e / x,y,z) \ e) * (x * L(x \ e,y,z)) = ((x * L(x \ e,y,z)) \ x) * (x * L(x \ e,y,z)) by (L17). Then
(L24) L(e / x,y,z) \ e = (x * L(x \ e,y,z)) \ x
by Proposition 10. We have (L(x \ e,y,z) / (x \ e)) \ (e / (x \ e)) = e / L(x \ e,y,z) by Theorem 1594. Then (L(x \ e,y,z) / (x \ e)) \ x = e / L(x \ e,y,z) by Proposition 24. Then (x * L(x \ e,y,z)) \ x = e / L(x \ e,y,z) by Theorem 161. Hence we are done by (L24).
Theorem 1596 (prov9_b25e958cd8): e / L(x,y,z) = L(e / (e / x),y,z) \ e
Proof. We have e / L((e / x) \ e,y,z) = L(e / (e / x),y,z) \ e by Theorem 1595. Hence we are done by Proposition 25.
Theorem 1597 (L_prop): e / (e / L(x,y,z)) = L(e / (e / x),y,z)
Proof. We have e / (L(e / (e / x),y,z) \ e) = L(e / (e / x),y,z) by Proposition 24. Hence we are done by Theorem 1596.