Definition 1: a(x,y,z) is defined to be (x * (y * z)) \ ((x * y) * 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.
Proof. We have
K(y \ x,y) = (y * (y \ x)) \ ((y \ x) * y) by Definition
2.
Hence we are done by Axiom
4.
Proof. We have
K(y,x / y) = ((x / y) * y) \ (y * (x / y)) by Definition
2.
Hence we are done by Axiom
6.
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.
Proof. We have (x * y) / (y \ (x * y)) = y by Proposition
24.
Hence we are done by Definition
3.
Proof. We have (((x * y) * z) / (y * z)) \ ((x * y) * z) = y * z by Proposition
25.
Hence we are done by Definition
5.
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.
Proof. We have x * (x \ y) = y by Axiom
4.
Hence we are done by Proposition
46.
Proof. We have z *
T(y,z) = y * z by Proposition
46.
Hence we are done by Axiom
4.
Proof. We have (y / x) * x = y by Axiom
6.
Hence we are done by Proposition
46.
Proof. Assume
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.
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.
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.
Proof. We have (x / y) \ x = y by Proposition
25.
Hence we are done by Proposition
47.
Proof. We have (x * (x \ y)) /
T(x,x \ y) = x \ y by Theorem
5.
Hence we are done by Axiom
4.
Proof. We have (x *
T(
T(y,z),x)) / x =
T(y,z) by Proposition
48.
Hence we are done by Axiom
7.
Proof. We have (y * (y \ x)) *
K(y \ x,y) = (y \ x) * y by Proposition
82.
Hence we are done by Axiom
4.
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.
Proof. We have y * (y \ z) = z by Axiom
4.
Then
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).
Proof. Assume
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.
Proof. We have (x \ e) * x =
K(x \ e,x) by Proposition
76.
Hence we are done by Proposition
2.
Proof. We have x * (e / x) =
K(x,e / x) by Proposition
77.
Hence we are done by Proposition
2.
Proof. We have e * y = y by Axiom
1.
Then
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
by
(L2).
We have
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
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.
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
by Axiom
6.
We have ((x / y) * y) / (z \ ((x / y) * y)) = z by Proposition
24.
Hence we are done by
(L3).
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
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.
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.
Proof. We have ((y / x) \ y) * (((y / x) \ y) \ z) = z by Axiom
4.
Hence we are done by Theorem
26.
Proof. We have
L((y \ x) \ e,y \ x,y) = (y * (y \ x)) \ y by Proposition
78.
Hence we are done by Axiom
4.
Proof. We have
L(y \ e,y,x / y) = ((x / y) * y) \ (x / y) by Proposition
78.
Hence we are done by Axiom
6.
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.
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.
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.
Proof. We have x / ((x * y) \ x) = x * y by Proposition
24.
Hence we are done by Proposition
78.
Proof. We have
L((e / x) \ e,e / x,y) = (y * (e / x)) \ y by Proposition
78.
Hence we are done by Proposition
25.
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.
Proof. We have
L(x \ e,x,x \ e) = ((x \ e) * x) \ (x \ e) by Proposition
78.
Hence we are done by Proposition
76.
Proof. We have
R(e / x,x,x \ y) = (x \ y) / (x * (x \ y)) by Proposition
79.
Hence we are done by Axiom
4.
Proof. We have
R(e / (y / x),y / x,x) = x / ((y / x) * x) by Proposition
79.
Hence we are done by Axiom
6.
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.
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.
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.
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.
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.
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.
Proof. We have
R(e / (x \ e),x \ e,y) \ y = (x \ e) * y by Proposition
80.
Hence we are done by Proposition
24.
Proof. We have x \ ((y \ x) * y) =
K(y \ x,y) by Theorem
2.
Hence we are done by Proposition
46.
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.
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.
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.
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.
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.
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.
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.
Proof. Assume
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.
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.
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.
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.
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.
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.
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.
Proof. We have (y * x) / ((y * z) \ (y * x)) = y * z by Proposition
24.
Hence we are done by Proposition
53.
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.
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.
Proof. Assume
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.
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.
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.
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.
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.
Proof. We have y * ((y \ e) * x) =
L(x,y \ e,y) by Proposition
56.
Hence we are done by Proposition
1.
Proof. We have x * ((x \ e) * x) =
L(x,x \ e,x) by Proposition
56.
Hence we are done by Proposition
76.
Proof. We have (e / x) * (x * y) =
L(y,x,e / x) by Proposition
67.
Hence we are done by Proposition
2.
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.
Proof. We have (x * y) * (y \ e) =
R(x,y,y \ e) by Proposition
68.
Hence we are done by Proposition
1.
Proof. We have (x * (e / y)) * y =
R(x,e / y,y) by Proposition
69.
Hence we are done by Proposition
1.
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.
Proof. We have
L(x,e / x,x) = (x * (e / x)) \ x by Theorem
34.
Hence we are done by Proposition
77.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have y *
T(x / (y * x),y) = (x / (y * x)) * y by Proposition
46.
Hence we are done by Theorem
89.
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.
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.
Proof. We have z *
T((y * x) / y,z) = ((y * x) / y) * z by Proposition
46.
Hence we are done by Theorem
50.
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.
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.
Proof. We have e * y = y by Axiom
1.
Then
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.
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.
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.
Proof. We have
L(x \ e,x,y) * z = ((y * x) \ y) * z by Theorem
42.
Hence we are done by Proposition
2.
Proof. We have
L(x \ e,x,y) * (((y * x) \ y) \ z) = z by Theorem
43.
Hence we are done by Proposition
2.
Proof. We have
T(e / x,x) = x \ e by Proposition
47.
Hence we are done by Theorem
44.
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.
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.
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.
Proof. We have
T((x * (y \ z)) / z,z) = z \ (x * (y \ z)) by Proposition
47.
Hence we are done by Theorem
66.
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.
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.
Proof. We have
L(
T(x \ e,x),x,e / x) = x \ e by Theorem
101.
Hence we are done by Theorem
72.
Proof. We have (e / x) * ((x \ e) * x) = x \ e by Theorem
108.
Hence we are done by Proposition
76.
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.
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.
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.
Proof. We have
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.
Proof. We have
R(y,x,y) * (x * y) = (y * x) * y by Proposition
54.
Hence we are done by Theorem
112.
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.
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.
Proof. We have
by Theorem
114.
We have (e / x) * x = e by Axiom
6.
Then
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).
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.
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.
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.
Proof. We have
L(e / x,x,y) * x = x * ((y * x) \ y) by Theorem
120.
Hence we are done by Proposition
1.
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.
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.
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.
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.
Proof. We have (e / (e / x)) * (((e / x) \ e) * (e / x)) = (e / x) \ e by Theorem
108.
Then
by Proposition
25.
We have (e / x) \ e = x by Proposition
25.
Hence we are done by
(L2).
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.
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.
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.
Proof. We have (x * e) \ ((x * (e / y)) * y) =
a(x,e / y,y) by Theorem
1.
Hence we are done by Axiom
2.
Proof. We have x \ ((x * (e / y)) * y) =
a(x,e / y,y) by Theorem
130.
Hence we are done by Proposition
69.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have
L(e / x,x,y) * x = x * ((y * x) \ y) by Theorem
120.
Hence we are done by Theorem
160.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have (y * z) *
L(
T(
T(y,z),x),z,y) = y * (z *
T(
T(y,z),x)) by Proposition
52.
Then
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.
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.
Proof. We have y \ (y * z) = z by Axiom
3.
Then
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
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.
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
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.
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.
Proof. We have
T(x,y) \ (x * y) =
T(y,
T(x,y)) by Theorem
13.
Hence we are done by Theorem
174.
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.
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.
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.
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.
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.
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).
Proof. We have (x * y) *
L(
T(y \ e,x * y),y,x) =
L(y \ e,y,x) * (x * y) by Proposition
58.
Then
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
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.
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.
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.
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.
Proof. We have (y / (x * y)) * x = x *
R(x \ e,x,y) by Theorem
90.
Hence we are done by Theorem
187.
Proof. We have (e / y) \
R(e / y,y,x) = (x / (y * x)) * y by Theorem
188.
Then
by Proposition
79.
We have (x / (y * x)) / ((e / y) \ (x / (y * x))) = e / y by Proposition
24.
Hence we are done by
(L2).
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
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).
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.
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.
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.
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.
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.
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.
Proof. We have x *
T(x \ e,y) =
K(y \ (y / x),y) by Theorem
196.
Hence we are done by Proposition
2.
Proof. We have (x \ e) *
T((x \ e) \ e,y) = x \
T(x,y) by Theorem
145.
Hence we are done by Theorem
196.
Proof. We have (e / y) *
T((e / y) \ e,x) =
T(y,x) / y by Theorem
149.
Hence we are done by Theorem
196.
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.
Proof. We have (
T(x,y) / x) * x =
T(x,y) by Axiom
6.
Hence we are done by Theorem
199.
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.
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.
Proof. We have
T(x,x *
T(x \ e,x)) = x by Theorem
203.
Hence we are done by Proposition
46.
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.
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.
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.
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.
Proof. We have
T(x,x \ e) \ e =
T(x \ e,x) by Theorem
208.
Hence we are done by Proposition
49.
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.
Proof. We have
T(x,x \ e) = (x \ e) \ e by Proposition
49.
Hence we are done by Theorem
210.
Proof. We have
T(x,e / x) / x =
K(x,e / x) by Theorem
116.
Hence we are done by Theorem
210.
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.
Proof. We have
R(x,e / x,x) =
T(x,e / x) by Theorem
117.
Hence we are done by Theorem
211.
Proof. We have
T(x,e / x) / x =
K(x,e / x) by Theorem
116.
Hence we are done by Theorem
211.
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.
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.
Proof. We have x \
R(x,e / x,x) =
a(x,e / x,x) by Theorem
131.
Hence we are done by Theorem
214.
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.
Proof. We have (e / x) / ((x \ e) \ (e / x)) = x \ e by Proposition
24.
Then
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).
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.
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).
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.
Proof. We have
T(
K(y,x),y \ e) =
K(y,x) by Theorem
223.
Hence we are done by Proposition
21.
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).
Proof. We have
T((x * y) \ x,
K(y,z)) = (x * y) \ x by Theorem
225.
Hence we are done by Proposition
21.
Proof. We have (z * w) *
R(
T(z,z * w),x,y) =
R(z,x,y) * (z * w) by Proposition
59.
Then
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.
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.
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.
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.
Proof. We have
R(y,x,y) *
T(x,y) =
T(y * x,y) by Theorem
230.
Hence we are done by Proposition
2.
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).
Proof. We have (e / x) \
L((z * y) \ z,x,e / x) = x * ((z * y) \ z) by Theorem
71.
Then
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).
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.
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.
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.
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.
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.
Proof. We have (x * z) / z = x by Axiom
5.
Then
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
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.
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.
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.
Proof. We have ((e / x) *
T(x,x \ e)) / (e / x) =
T(e / (e / x),x \ e) by Theorem
51.
Then
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).
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.
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.
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).
Proof. We have (
T(x,x \ e) / x) * x =
T(x,x \ e) by Axiom
6.
Hence we are done by Theorem
244.
Proof. We have
T(x,x \ e) / x =
K(x,x \ e) by Theorem
244.
Hence we are done by Proposition
49.
Proof. We have
T(x,x \ e) / x =
K(x,e / x) by Theorem
215.
Hence we are done by Theorem
244.
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.
Proof. We have
K(x,e / x) = x * (e / x) by Proposition
77.
Hence we are done by Theorem
248.
Proof. We have x \
K(x,e / x) = e / x by Theorem
22.
Hence we are done by Theorem
248.
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.
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.
Proof. We have (x \ e) *
T(e / (e / x),x \ e) = (e / (e / x)) * (x \ e) by Proposition
46.
Then
by Theorem
242.
We have (e / (e / x)) * (x \ e) = (x \ e) / (e / x) by Theorem
153.
Then
by
(L2).
We have
K(x \ e,x) = (x \ e) * x by Proposition
76.
Hence we are done by
(L4).
Proof. We have ((x \ e) / (e / x)) * (e / x) = x \ e by Axiom
6.
Hence we are done by Theorem
254.
Proof. We have ((x \ e) / (e / x)) \ (x \ e) = e / x by Proposition
25.
Hence we are done by Theorem
254.
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
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.
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.
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.
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.
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.
Proof. We have x * (x \ (e / (e / x))) = e / (e / x) by Axiom
4.
Hence we are done by Theorem
257.
Proof. We have x *
K(x \ e,x) =
L(x,x \ e,x) by Theorem
70.
Hence we are done by Theorem
262.
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.
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.
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.
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).
Proof. We have
T(x,
K(x,e / x)) = x by Theorem
267.
Hence we are done by Theorem
248.
Proof. We have x \ ((x * (e / x)) * x) =
a(x,e / x,x) by Theorem
130.
Then
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).
Proof. We have x \
T(x,x \ e) =
a(x,e / x,x) by Theorem
218.
Hence we are done by Theorem
269.
Proof. We have
K(x,e / x) =
K(x,x \ e) by Theorem
248.
Hence we are done by Theorem
270.
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.
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.
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).
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.
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
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.
Proof. We have z \ (z * w) = w by Axiom
3.
Then
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.
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
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.
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.
Proof. We have x * (
L(x,y,z) \ e) =
L(x,y,z) \ x by Theorem
278.
Hence we are done by Proposition
2.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have ((y * (e / (e / x))) / y) * (e / x) = ((y * (e * x)) / y) / x by Theorem
232.
Then
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).
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.
Proof. We have (x * z) / z = x by Axiom
5.
Then
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
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.
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).
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.
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.
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.
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.
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.
Proof. We have
T(y,y *
R(y \ e,y,x)) = y by Theorem
298.
Hence we are done by Theorem
90.
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.
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.
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.
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.
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.
Proof. We have
T(y,y *
L(y \ e,y,x)) = y by Theorem
304.
Hence we are done by Proposition
78.
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.
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.
Proof. We have
T(e,z) = z \ (e * z) by Definition
3.
Then
by Axiom
1.
We have z \ z = e by Proposition
28.
Then
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).
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.
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.
Proof. We have (y \
T(x,y)) * y =
T((y \ x) * y,y) by Theorem
310.
Hence we are done by Proposition
1.
Proof. We have y * (y \
T(x,y)) =
T(x,y) by Axiom
4.
Hence we are done by Theorem
311.
Proof. We have (x / y) * y = x by Axiom
6.
Then
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).
Proof. We have (x / y) * y = x by Axiom
6.
Then
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).
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.
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.
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.
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.
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
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).
Proof. We have (e / (y / x)) * ((e / (y / x)) \ (x / y)) = x / y by Axiom
4.
Hence we are done by Theorem
319.
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.
Proof. We have (e / (y \ x)) * ((e / (y \ x)) \ (x \ y)) = x \ y by Axiom
4.
Hence we are done by Theorem
321.
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.
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.
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.
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.
Proof. We have x * (
K(x,x \ e) * (x \ y)) = (x * (e / x)) * y by Theorem
326.
Hence we are done by Theorem
250.
Proof. We have
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.
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.
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.
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.
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.
Proof. We have (y / x) \ (y * ((y * (y \ (y / x))) / y)) = x * ((y * (x \ e)) / y) by Theorem
330.
Then
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).
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.
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.
Proof. We have
T(y * (x / (y * x)),y) = y * (x / (y * x)) by Theorem
335.
Hence we are done by Proposition
21.
Proof. We have
T(x / y,(x / y) * (y / ((x / y) * y))) = x / y by Theorem
336.
Hence we are done by Axiom
6.
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.
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.
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
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.
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.
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.
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.
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.
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.
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).
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).
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).
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
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.
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
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).
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.
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
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
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).
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.
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.
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.
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.
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).
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
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).
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.
Proof. We have
T(e / (e / y),
R(e / y,y,x)) = y by Theorem
359.
Hence we are done by Proposition
79.
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.
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.
Proof. We have
T(y,(y \ x) / (y * (y \ x))) =
T(y,y \ e) by Theorem
362.
Hence we are done by Axiom
4.
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.
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
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
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
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.
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.
Proof. We have
T(
K(x,y),e / y) =
K(x,y) by Theorem
366.
Hence we are done by Proposition
21.
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.
Proof. We have
K(y,
K(x,y \ e)) = e by Theorem
368.
Hence we are done by Proposition
23.
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.
Proof. We have
T(
T(z,x),
K(y,z \ e)) =
T(z,x) by Theorem
370.
Hence we are done by Proposition
21.
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.
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).
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
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
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.
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).
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.
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.
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
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
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.
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.
Proof. We have (x \ e) * (x * ((y * x) \ y)) = (y * x) \ y by Theorem
379.
Hence we are done by Proposition
2.
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
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
by Theorem
380.
We have
L(x \ e,x,y) / (x \ e) = x * ((y * x) \ y) by Theorem
162.
Then
by Proposition
78.
We have (((y * x) \ y) / (x \ e)) * (x \ e) = (y * x) \ y by Axiom
6.
Then
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.
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).
Proof. We have (y \ e) *
T(y,x \ (x / y)) = e by Theorem
382.
Hence we are done by Theorem
20.
Proof. We have (y \ e) *
T(y,x \ (x / y)) = e by Theorem
382.
Hence we are done by Proposition
2.
Proof. We have
T(y,(x * y) \ ((x * y) / y)) =
T(y,y \ e) by Theorem
383.
Hence we are done by Axiom
5.
Proof. We have
T(y,(x * y) \ ((x * y) / y)) = (y \ e) \ e by Theorem
384.
Hence we are done by Axiom
5.
Proof. We have
T(x \ y,y \ (y / (x \ y))) = ((x \ y) \ e) \ e by Theorem
384.
Hence we are done by Proposition
24.
Proof. We have
T(y,(x * y) \ x) =
T(y,y \ e) by Theorem
385.
Hence we are done by Proposition
78.
Proof. We have
T(y,(x * y) \ x) = (y \ e) \ e by Theorem
386.
Hence we are done by Proposition
78.
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
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).
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.
Proof. We have
T(y,(x * y) \ x) / y =
K(y,(x * y) \ x) by Theorem
364.
Then
by Theorem
386.
We have ((y \ e) \ e) / y =
K(y,y \ e) by Theorem
247.
Hence we are done by
(L2).
Proof. We have (((x * y) \ x) *
T(e / (e / y),(x * y) \ x)) / ((x * y) \ x) = e / (e / y) by Proposition
48.
Then
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).
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.
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).
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.
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.
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
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
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
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
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
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).
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
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).
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.
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.
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.
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.
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
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.
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).
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.
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.
Proof. We have (x * y) *
K(y,y \ e) = x * ((y \ e) \ e) by Theorem
407.
Hence we are done by Proposition
2.
Proof. We have (y *
K(x,x \ e)) /
K(x,x \ e) = y by Axiom
5.
Then
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
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
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
by
(L8).
We have (y / x) * x = y by Axiom
6.
Hence we are done by
(L13).
Proof. We have
R(y,
K(e / x,(e / x) \ e),e / x) = y by Theorem
409.
Hence we are done by Theorem
272.
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.
Proof. We have
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.
Proof. We have
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.
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.
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.
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.
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.
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.
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.
Proof. We have e / (x \ (x / ((y \ e) \ e))) = (x \ (x / (e / (y \ e)))) \ e by Theorem
419.
Then
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).
Proof. We have
T(x \ (x / y),y) = ((x \ (x / y)) \ e) \ e by Theorem
391.
Hence we are done by Theorem
420.
Proof. We have x * (x \ (x / ((y \ e) \ e))) = x / ((y \ e) \ e) by Axiom
4.
Hence we are done by Theorem
421.
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.
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.
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.
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.
Proof. We have
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.
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
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
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
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.
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.
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.
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.
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).
Proof. We have
L(y,e / x,
K(e / x,(e / x) \ e)) = y by Theorem
432.
Hence we are done by Theorem
272.
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.
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.
Proof. We have (x \ e) * (
K(x \ e,x) * ((x \ e) \ y)) =
K(x \ e,x) * y by Theorem
338.
Then
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.
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.
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.
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.
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).
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.
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.
Proof. We have
by Proposition
76.
We have (y \ e) *
T(
K(y \ e,y) / (y \ e),y \ e) =
K(y \ e,y) by Theorem
10.
Then
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
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
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
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
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).
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.
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.
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
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
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).
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.
Proof. We have
K(y / x,x) = (x * (y / x)) \ ((y / x) * x) by Definition
2.
Hence we are done by Axiom
6.
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.
Proof. We have x \ (x * x) = x by Axiom
3.
Hence we are done by Definition
3.
Proof. We have
T(
T(y,y),x) =
T(
T(y,x),y) by Axiom
7.
Hence we are done by Theorem
450.
Proof. We have (y * x) / ((x * y) \ (y * x)) = x * y by Proposition
24.
Hence we are done by Definition
2.
Proof. We have (y * (z * x)) / ((y * z) \ (y * (z * x))) = y * z by Proposition
24.
Hence we are done by Definition
4.
Proof. We have x * y = y *
T(x,y) by Proposition
46.
Hence we are done by Proposition
2.
Proof. We have
T(
T(y,x),y) =
T(y,x) by Theorem
451.
Hence we are done by Proposition
22.
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.
Proof. We have
K(
T((y * x) / y,y),(y * x) / y) = e by Theorem
455.
Hence we are done by Theorem
7.
Proof. We have
K(y,(x * y) / x) = e by Theorem
457.
Hence we are done by Proposition
23.
Proof. We have ((x / y) * y) *
K(y,x / y) = y * (x / y) by Proposition
82.
Hence we are done by Axiom
6.
Proof. Assume
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.
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.
Proof. We have (y * (y \ x)) /
K(y,y \ x) = (y \ x) * y by Theorem
452.
Hence we are done by Axiom
4.
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.
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.
Proof. We have (x * (e / x)) \ e =
K(e / x,x) by Theorem
448.
Hence we are done by Proposition
77.
Proof. We have z *
T(
T(x,z),y) =
T(x,y) * z by Proposition
50.
Hence we are done by Definition
3.
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.
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.
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.
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.
Proof. We have
T((y * z) / (x * z),x * z) = (x * z) \ (y * z) by Proposition
47.
Hence we are done by Proposition
55.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
Proof. We have
T((y * z) / (x * z),x * z) = (x * z) \ (y * z) by Proposition
47.
Hence we are done by Theorem
470.
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.
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.
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.
Proof. We have x *
T(y,x) = y * x by Proposition
46.
Then
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.
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.
Proof. We have
T((y / (x \ y)) *
T(x \ y,y),x \ y) = y by Theorem
493.
Hence we are done by Proposition
24.
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.
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.
Proof. We have
T(y / x,z) * x = x *
T(x \ y,z) by Proposition
61.
Hence we are done by Theorem
490.
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.
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.
Proof. We have
T(e / (e / x),y) * (e / x) =
T(x,y) / x by Theorem
147.
Hence we are done by Proposition
1.
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.
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.
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.
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.
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.
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.
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.
Proof. We have (x * y) *
T((y * x) / y,x * y) = ((y * x) / y) * (x * y) by Proposition
46.
Then
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
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
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.
Proof. We have
T(((y * x) * y) / (y * x),y * x) = y by Theorem
7.
Hence we are done by Theorem
508.
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.
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.
Proof. We have (
T(x,x / y) / y) * y =
T(x,x / y) by Axiom
6.
Hence we are done by Theorem
511.
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).
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.
Proof. We have ((y * (y / x)) / y) * x = x * ((y * (x \ y)) / y) by Theorem
125.
Hence we are done by Theorem
512.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
Proof. We have
T(x,y) * (x \ (
T(x,y) \ z)) = x \ z by Theorem
522.
Hence we are done by Proposition
2.
Proof. We have
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.
Proof. We have y * (
T(y,x) \ e) =
T(y,x) \ y by Theorem
175.
Then
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
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).
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.
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).
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.
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.
Proof. We have (e / z) *
R(z,x,y) =
R(z,x,y) / z by Theorem
192.
Hence we are done by Proposition
1.
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.
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.
Proof. We have (y / x) \ ((y / x) * z) = z by Axiom
3.
Then
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
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.
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.
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.
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.
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.
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.
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.
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
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).
Proof. We have
R(x,e / x,x) =
T(x,e / x) by Theorem
117.
Hence we are done by Theorem
210.
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).
Proof. We have ((y * x) / y) * (x \ e) = x \ ((y * x) / y) by Theorem
195.
Hence we are done by Theorem
483.
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.
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.
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.
Proof. We have e * y = y by Axiom
1.
Then
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
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.
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.
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.
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.
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.
Proof. We have
K(x,x \ e) = x * (e / x) by Theorem
250.
Hence we are done by Theorem
245.
Proof. We have
K(x,x \ e) = x * (e / x) by Theorem
250.
Hence we are done by Theorem
265.
Proof. We have (x / (e / (e / x))) * (e / (e / x)) = x by Axiom
6.
Hence we are done by Theorem
556.
Proof. We have (x / (e / (e / x))) \ x = e / (e / x) by Proposition
25.
Hence we are done by Theorem
556.
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.
Proof. We have
K((x \ e) \ e,x \ e) = x \ ((x \ e) \ e) by Theorem
480.
Hence we are done by Theorem
559.
Proof. We have
K(x,e / x) =
K(x,x \ e) by Theorem
248.
Then
by Theorem
559.
We have
K((x \ e) \ e,x \ e) = x \ ((x \ e) \ e) by Theorem
480.
Hence we are done by
(L2).
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.
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.
Proof. We have
K(e / x,(e / x) \ e) =
K(x \ e,x) by Theorem
272.
Hence we are done by Proposition
25.
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.
Proof. We have
K((e / x) \ e,e / x) =
K(x,x \ e) by Theorem
253.
Hence we are done by Theorem
564.
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).
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.
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.
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.
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.
Proof. We have (y * y) /
T(y,x) = (y /
T(y,x)) * y by Theorem
239.
Hence we are done by Theorem
571.
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.
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.
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.
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.
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.
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.
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).
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).
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).
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.
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.
Proof. We have y * (y \
T(y * x,y)) =
T(y * x,y) by Axiom
4.
Hence we are done by Theorem
583.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
Proof. We have (x \ e) * ((y * ((x \ e) \ e)) / y) = x \ ((y * x) / y) by Theorem
546.
Hence we are done by Theorem
333.
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.
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.
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.
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.
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.
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.
Proof. We have
L(y \ e,y,((x / y) * y) \ (x / y)) = e / y by Theorem
342.
Hence we are done by Axiom
6.
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.
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.
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.
Proof. We have
T(
T(e / y,
K(x,y)),y) =
T(y \ e,
K(x,y)) by Proposition
51.
Then
by Theorem
367.
We have
T(e / y,y) = y \ e by Proposition
47.
Hence we are done by
(L2).
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.
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.
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.
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.
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.
Proof. We have (x \ y) \
K(x \ y,(x \ y) \ e) = e / (x \ y) by Theorem
251.
Hence we are done by Theorem
612.
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
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.
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.
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.
Proof. We have (e / (e / ((y * x) \ y))) \ e = e / ((y * x) \ y) by Proposition
25.
Hence we are done by Theorem
395.
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.
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.
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).
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.
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
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
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).
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.
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.
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.
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).
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.
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
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).
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.
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).
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
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).
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.
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.
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).
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
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.
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.
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
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.
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.
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).
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
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.
Proof. We have (y *
K(x,x \ e)) \ ((y * x) *
K(x,x \ e)) = x by Theorem
602.
Then
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).
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.
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.
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.
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.
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.
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.
Proof. We have
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.
Proof. We have
L(x,e / x,y) = (y * (e / x)) \ y by Theorem
34.
Hence we are done by Theorem
649.
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.
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.
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.
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.
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.
Proof. We have (x /
K(y \ e,y)) \ x =
K(y \ e,y) by Proposition
25.
Hence we are done by Theorem
656.
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.
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.
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.
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.
Proof. We have
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.
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.
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
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.
Proof. We have
K((x * y) \ x,x \ (x * y)) =
K(y \ e,y) by Theorem
664.
Hence we are done by Axiom
3.
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.
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.
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.
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.
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).
Proof. We have
R(x \ e,x,y) = ((y / (x * y)) \ e) \ e by Theorem
670.
Hence we are done by Proposition
79.
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).
Proof. We have
T((x *
K(x,y)) / x,x) =
K(x,y) by Theorem
7.
Hence we are done by Theorem
672.
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
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.
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
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
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).
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.
Proof. We have
T(x,x \ e) * (x \ y) =
K(x,x \ e) * y by Theorem
431.
Hence we are done by Proposition
49.
Proof. We have
L(y,e / x,
K(e / x,(e / x) \ e)) = y by Theorem
432.
Hence we are done by Proposition
25.
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.
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.
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.
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.
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.
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
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).
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.
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.
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.
Proof. We have ((y \ e) \ e) \ (((y \ e) \ e) * (
T(y,x) \ e)) =
T(y,x) \ e by Axiom
3.
Then
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
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.
Proof. We have
K(x,x \ e) * ((e / (e / x)) * y) = x * y by Theorem
685.
Hence we are done by Proposition
2.
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.
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.
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.
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.
Proof. We have
K(x,x \ e) * (
K(x \ e,x) * y) = y by Theorem
442.
Hence we are done by Proposition
2.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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).
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.
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.
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.
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).
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
by
(L5).
We have
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.
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.
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.
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.
Proof. We have
T(e / (e / (y / (x * y))),x * y) = (x * y) \ (
K(x,x \ e) * y) by Theorem
710.
Then
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).
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
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).
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
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.
Proof. We have
T(
K(x \ e,x),y) =
K(x \ e,x) by Theorem
716.
Hence we are done by Proposition
21.
Proof. We have
T(y,
K((e / x) \ e,e / x)) = y by Theorem
717.
Hence we are done by Theorem
253.
Proof. We have
T(z,
K((x \ y) \ e,x \ y)) = z by Theorem
717.
Hence we are done by Theorem
666.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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).
Proof. We have (x \ e) * (x *
T(x \ e,y)) =
T(x \ e,y) by Theorem
728.
Hence we are done by Proposition
2.
Proof. We have
K(y \ (y / x),y) = x *
T(x \ e,y) by Theorem
196.
Hence we are done by Theorem
729.
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.
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).
Proof. We have
K(e / (x \ e),y) =
K((x \ e) \ e,y) by Theorem
732.
Hence we are done by Proposition
24.
Proof. We have
T(
K(x,y),x) =
K((x \ e) \ e,y) by Theorem
674.
Hence we are done by Theorem
733.
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.
Proof. We have
T(
K(y,x),y) =
K(y,x) by Theorem
734.
Hence we are done by Proposition
21.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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).
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.
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.
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.
Proof. We have y * (y \ (y /
T(y,x))) = y /
T(y,x) by Axiom
4.
Then
by Theorem
752.
We have y * (
T(y,x) \ e) =
T(y,x) \ y by Theorem
175.
Hence we are done by
(L2).
Proof. We have y * (
T(y,x) \ y) = (y /
T(y,x)) * y by Theorem
572.
Then
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
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
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.
Proof. We have
T(
K(x,y),y) =
K(x,y) by Theorem
754.
Hence we are done by Proposition
21.
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.
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
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.
Proof. We have y * (
T(y,x) / y) =
T(y,x) by Theorem
757.
Hence we are done by Proposition
2.
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.
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.
Proof. We have (y \
T(y,x)) * y =
T(y,x) by Theorem
759.
Hence we are done by Theorem
20.
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
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).
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.
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.
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
by Theorem
755.
We have x * y = (y * x) *
K(x,y) by Proposition
82.
Hence we are done by
(L3) and Proposition
7.
Proof. We have (
K(x,y) \ y) *
K(x,y) = y by Theorem
765.
Hence we are done by Proposition
1.
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.
Proof. We have e / ((((x * y) / x) \ y) \ e) = ((x * y) / x) \ y by Proposition
24.
Hence we are done by Theorem
767.
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.
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.
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.
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.
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
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
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
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).
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
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).
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
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).
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.
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
by Definition
2.
We have y /
K(x,y) =
K(x,y) \ y by Theorem
766.
Hence we are done by
(L5).
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
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).
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
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
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
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).
Proof. We have
K(x,((e / y) \ e) \ e) =
K(x,e / y) by Theorem
779.
Hence we are done by Proposition
25.
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.
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.
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
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).
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.
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.
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.
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.
Proof. We have
T(
T(y,x),y) =
T(y,x) by Theorem
451.
Hence we are done by Proposition
21.
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.
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.
Proof. We have
R(x \ e,x,y) = (((x \ e) * x) * y) / (x * y) by Definition
5.
Hence we are done by Proposition
76.
Proof. We have
T(x \ e,x) = x \ ((x \ e) * x) by Definition
3.
Hence we are done by Proposition
76.
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.
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
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).
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.
Proof. We have x /
L((e / y) \ e,e / y,x) = x * (e / y) by Theorem
33.
Hence we are done by Proposition
25.
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.
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.
Proof. We have (y * x) \ (y * ((y * x) / y)) =
K(y,(y * x) / y) by Theorem
3.
Hence we are done by Proposition
53.
Proof. We have ((x * z) / (y * z)) \ (x * z) = y * z by Proposition
25.
Hence we are done by Proposition
55.
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.
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.
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.
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.
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.
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.
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.
Proof. We have e * y = y by Axiom
1.
Then
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
by
(L2).
We have e \ (e * y) = y by Axiom
3.
Then
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.
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.
Proof. We have e * x = x by Axiom
1.
Then
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
Proof. We have
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have (
T(z,x) * y) / (z \ (
T(z,x) * y)) = z by Proposition
24.
Hence we are done by Theorem
173.
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.
Proof. We have y \ (
T(y,z) *
T(y,x)) =
T(y,z) * (y \
T(y,x)) by Theorem
173.
Then
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).
Proof. We have
R(x,y,z) * (x \ e) = x \
R(x,y,z) by Proposition
89.
Hence we are done by Proposition
1.
Proof. We have
R(z,x,y) * (z \ e) = z \
R(z,x,y) by Proposition
89.
Hence we are done by Proposition
2.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
Proof. We have y \ (y * z) = z by Axiom
3.
Then
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
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.
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.
Proof. We have y \ (y * z) = z by Axiom
3.
Then
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
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.
Proof. We have (e / y) * ((x * ((e / y) \ e)) / x) = ((x * y) / x) / y by Theorem
290.
Hence we are done by Proposition
25.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
Proof. We have e * z = z by Axiom
1.
Then
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
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.
Proof. We have x * (y * ((x * (y \ e)) / x)) =
T(x,x / (x * y)) by Theorem
845.
Hence we are done by Theorem
333.
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.
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).
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.
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.
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.
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.
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
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).
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
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).
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
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).
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.
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.
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.
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
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.
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.
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.
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
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).
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.
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.
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
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).
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.
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.
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.
Proof. We have y *
T((y \ e) * (y * x),y) = ((y \ e) * (y * x)) * y by Proposition
46.
Then
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).
Proof. We have ((x \ e) * (x * y)) * x =
L(y * x,x \ e,x) by Theorem
893.
Hence we are done by Proposition
1.
Proof. We have
K(y,((x / y) * y) \ (x / y)) =
K(y,y \ e) by Theorem
392.
Hence we are done by Axiom
6.
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.
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.
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.
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.
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
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.
Proof. We have e / ((
K(x,y) \ e) \ e) =
K(x,y) \ e by Proposition
24.
Hence we are done by Theorem
900.
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.
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.
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
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.
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
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.
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.
Proof. We have ((x * y) / ((y \ e) \ e)) * ((y \ e) \ e) = x * y by Axiom
6.
Hence we are done by Theorem
907.
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.
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.
Proof. We have ((x * (e / y)) *
K(y \ e,y)) * ((y \ e) \ e) = (x * (e / y)) * y by Theorem
908.
Then
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).
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.
Proof. We have (e / x) \
L(y,x,e / x) = x * y by Theorem
71.
Hence we are done by Theorem
683.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have (x /
T(x,y)) \ x =
T(x,y) by Proposition
25.
Hence we are done by Theorem
753.
Proof. We have (y /
T(y,x)) * y = y * (
T(y,x) \ y) by Theorem
572.
Hence we are done by Theorem
753.
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.
Proof. We have y *
T(
K(x,y),y) =
K(x,y) * y by Proposition
46.
Hence we are done by Theorem
754.
Proof. We have (y *
T(
K(x,y),y)) / y =
K(x,y) by Proposition
48.
Hence we are done by Theorem
754.
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.
Proof. We have
T(y,
K(y \ (y / (x \ e)),y)) = y by Theorem
755.
Hence we are done by Theorem
198.
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.
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.
Proof. We have
T(
T(z,x),
K(y,z)) =
T(z,x) by Theorem
935.
Hence we are done by Proposition
21.
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).
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.
Proof. We have x * (x \ ((y * x) / y)) = (y * x) / y by Axiom
4.
Hence we are done by Theorem
940.
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.
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.
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.
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).
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).
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.
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.
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).
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.
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
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
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
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.
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.
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).
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.
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
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
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.
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.
Proof. We have y * (
K(x,y) \ e) =
K(x,y) \ y by Theorem
777.
Hence we are done by Proposition
2.
Proof. We have y * (
K(x,y) \ e) =
K(x,y) \ y by Theorem
777.
Hence we are done by Theorem
763.
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
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).
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.
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.
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.
Proof. We have e /
R((x \ e) \ e,y,z) =
R(e / (x \ e),y,z) \ e by Theorem
963.
Then
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).
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.
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.
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
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.
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.
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.
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.
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).
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.
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.
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.
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).
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
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.
Proof. We have (y * (x \
T(x,y))) * x = x * y by Theorem
976.
Hence we are done by Proposition
1.
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.
Proof. We have y * (x \
T(x,y)) = (x * y) / x by Theorem
977.
Hence we are done by Proposition
2.
Proof. We have (x \ e) * (((y * (x \ e)) / y) / (x \ e)) = (y * (x \ e)) / y by Theorem
942.
Then
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.
Proof. We have
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.
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.
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
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.
Proof. We have (x * (y \
T(y,x))) \ x =
T(y,x) \ y by Theorem
952.
Hence we are done by Theorem
977.
Proof. We have
T(x,y) * (y \
T(y,
T(x,y))) = x by Theorem
978.
Hence we are done by Proposition
2.
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.
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.
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.
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.
Proof. We have ((y * (y \ x)) / y) \ (y \ x) =
T(y,y \ x) \ y by Theorem
984.
Hence we are done by Axiom
4.
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.
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.
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.
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.
Proof. We have (x \
T(x,y)) * (x * (
T(x,y) \ y)) =
L(y,x \
T(x,y),x) by Theorem
955.
Then
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
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.
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.
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.
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.
Proof. We have (x \
T(x,y)) * x =
T(x,y) by Theorem
759.
Hence we are done by Theorem
989.
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.
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.
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.
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.
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.
Proof. We have (x * ((x / y) / x)) * y =
T(x,x / (x * y)) by Theorem
881.
Hence we are done by Theorem
1003.
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.
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.
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.
Proof. We have (x / y) \
T(x / y,y) = y \ (x / (x / y)) by Theorem
987.
Hence we are done by Proposition
47.
Proof. We have
T(y,y \ x) \ y = (x / y) \ (y \ x) by Theorem
990.
Hence we are done by Theorem
1009.
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.
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.
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.
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).
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
by Theorem
759.
We have (z * y) /
T(z,y) = y by Theorem
5.
Then
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
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).
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.
Proof. We have
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
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.
Proof. We have
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.
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.
Proof. We have (y \
T(y,z)) * (y * x) =
T(y,z) * x by Theorem
1018.
Hence we are done by Proposition
1.
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.
Proof. We have x * ((x \
T(x,
T(y,x))) * (y * x)) =
T(x,
T(y,x)) * (y * x) by Theorem
1017.
Then
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.
Proof. We have (
T(y,x) \ y) * (y * x) = x * y by Theorem
1022.
Hence we are done by Proposition
1.
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
by Axiom
6.
We have (y / x) * ((y / x) \ (x * (x \ y))) = x * (x \ y) by Axiom
4.
Then
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
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).
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.
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
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).
Proof. We have (x *
R(e / y,y,y \ x)) / (y \ x) = ((x * (e / y)) / x) * y by Theorem
828.
Then
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
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
by
(L6).
We have (x \
T(x,y \ e)) * x =
T(x,y \ e) by Theorem
759.
Hence we are done by
(L9).
Proof. We have
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.
Proof. We have
K(y \ (y / x),y) = x *
T(x \ e,y) by Theorem
196.
Hence we are done by Theorem
1028.
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.
Proof. We have
T(y,x) * (
T(y,x) \ y) = y by Axiom
4.
Hence we are done by Theorem
1030.
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.
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.
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
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.
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.
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.
Proof. We have
K(
T(y,x),x) =
K(y,(y * x) / y) by Theorem
1008.
Hence we are done by Theorem
1035.
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.
Proof. We have
R(z,
K(
T(x,z),z) \ e,y) = z by Theorem
1019.
Hence we are done by Theorem
1036.
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.
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
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).
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
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.
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.
Proof. We have
L(z,
T(y,y),
K(x,y)) = z by Theorem
1043.
Hence we are done by Theorem
450.
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
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).
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.
Proof. We have (z *
K(x,z)) * y = z * (
K(x,z) * y) by Theorem
1046.
Hence we are done by Theorem
64.
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
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.
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
by Theorem
1045.
We have
(L4) T(x *
K(x,y),y) = y \ ((x *
K(x,y)) * y)
by Definition
3.
Hence we are done.
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).
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
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
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.
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).
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.
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.
Proof. We have
K(x,x \ y) = ((x \ y) * x) \ (x * (x \ y)) by Definition
2.
Hence we are done by Axiom
4.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
Proof. We have
R(x,x \ e,x) = x / ((x \ e) * x) by Proposition
66.
Hence we are done by Proposition
76.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have
T(y,z) * (y * x) = y * (
T(y,z) * x) by Theorem
172.
Hence we are done by Proposition
1.
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.
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.
Proof. We have ((y * x) / y) * (x \ y) =
T(y,x) by Theorem
179.
Hence we are done by Proposition
1.
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.
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).
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.
Proof. We have
T(x,x \ y) \ (x * z) = x * (((x \ y) \ y) \ z) by Theorem
1090.
Hence we are done by Proposition
49.
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.
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.
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.
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.
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.
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.
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.
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).
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).
Proof. We have (x *
T(z,y)) * z = (x * z) *
T(z,y) by Theorem
238.
Hence we are done by Proposition
2.
Proof. We have (x * (y \ z)) / (y \ z) = x by Axiom
5.
Then
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
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.
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.
Proof. We have
K(x \ e,x) * x = x *
K(x \ e,x) by Theorem
933.
Hence we are done by Theorem
262.
Proof. We have
K(x \ e,x) * x = e / (e / x) by Theorem
1108.
Hence we are done by Proposition
1.
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.
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.
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.
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.
Proof. We have
T(x,x / (x / y)) * (x \ y) = (x * y) / x by Theorem
861.
Hence we are done by Proposition
1.
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.
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).
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.
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).
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).
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
Proof. We have
T(x,x *
T(e / x,y)) = x by Theorem
1131.
Hence we are done by Proposition
21.
Proof. We have (y \ e) * (y * ((x * y) \ x)) = (x * y) \ x by Theorem
379.
Then
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).
Proof. We have
R(e / y,y,((x / y) * y) \ (x / y)) = y \ e by Theorem
1133.
Hence we are done by Axiom
6.
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.
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
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).
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.
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).
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.
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.
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.
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.
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.
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.
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.
Proof. We have x *
T(x \ (x / y),y) = x / ((y \ e) \ e) by Theorem
422.
Hence we are done by Proposition
49.
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.
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.
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.
Proof. We have ((x \ y) \ e) \
K((x \ y) \ e,x \ y) = x \ y by Theorem
21.
Hence we are done by Theorem
666.
Proof. We have (x * (e / y)) * y =
R(x,e / y,y) by Proposition
69.
Hence we are done by Theorem
911.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have (e / (e / x)) * (
K(x,x \ e) * y) = x * y by Theorem
684.
Hence we are done by Proposition
2.
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.
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).
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.
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.
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.
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
by
(L2).
We have (x /
K(x \ e,x)) *
K(x \ e,x) = x by Axiom
6.
Then
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.
Proof. We have x /
K(x \ e,x) =
R(x,x \ e,x) by Theorem
1069.
Then
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).
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.
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.
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.
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.
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.
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.
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.
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
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).
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.
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).
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.
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.
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.
Proof. We have ((e / y) * x) *
K(y \ e,y) = (e / y) * (x *
K(y \ e,y)) by Theorem
1138.
Then
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.
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.
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.
Proof. We have
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.
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.
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.
Proof. We have (y \ e) * ((e / y) \ x) =
K(y \ e,y) * x by Theorem
1161.
Hence we are done by Theorem
923.
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.
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.
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.
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.
Proof. We have
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
.
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).
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.
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
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.
Proof. We have x * (x \
T(x,y)) =
T(x,y) by Axiom
4.
Hence we are done by Theorem
1197.
Proof. We have (e / x) *
T((e / x) \ e,y) =
T(x,y) / x by Theorem
149.
Hence we are done by Theorem
758.
Proof. We have
T(e / (y \ e),
T(y \ e,x)) = (y \ e) \ e by Theorem
762.
Hence we are done by Proposition
24.
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.
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.
Proof. We have
T(x,x \ e) = (x \ e) \ e by Proposition
49.
Hence we are done by Theorem
1200.
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.
Proof. We have x *
T(x \ e,
T(x,y)) =
K(x \ e,
T(x,y)) by Theorem
1202.
Then
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).
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.
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.
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.
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).
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.
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.
Proof. We have (e / (e / (((e / x) * y) \ y))) \ e = e / (((e / x) * y) \ y) by Proposition
25.
Then
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
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.
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.
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
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.
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.
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.
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).
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
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).
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.
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).
Proof. We have
L(z,x,
K((e / y) \ e,e / y)) = z by Theorem
967.
Hence we are done by Theorem
253.
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.
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.
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.
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.
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.
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.
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.
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
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).
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.
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.
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.
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.
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.
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.
Proof. We have x * (
T(x,y) * (x \ y)) =
T(x,y) * y by Theorem
830.
Hence we are done by Theorem
996.
Proof. We have
T(y,y / (y / (x \ y))) =
T(y,x \ y) by Theorem
997.
Hence we are done by Proposition
24.
Proof. We have (y / x) \
T(y,y / (y / x)) = (y * x) / y by Theorem
1087.
Hence we are done by Theorem
997.
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.
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.
Proof. We have
T(y,y / (x \ y)) =
T(y,(x \ y) \ y) by Theorem
1237.
Hence we are done by Proposition
24.
Proof. We have (x \ e) * ((y * ((x \ e) \ y)) / y) =
T(y,y / (x \ e)) by Theorem
515.
Then
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.
Proof. We have
T(x,(y \ x) \ x) =
T(x,y) by Theorem
1241.
Hence we are done by Proposition
49.
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.
Proof. We have
T(y,y \ x) * (y \ x) = y * ((y * (y \ x)) / y) by Theorem
1236.
Hence we are done by Axiom
4.
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.
Proof. We have (x * y) * (((x * y) * ((x * y) \ y)) / (x * y)) = (x * (((x * y) * (x \ e)) / (x * y))) * y by Theorem
343.
Then
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.
Proof. We have (x * y) * ((y * ((x * y) \ y)) / y) =
T(y,y / (x * y)) by Theorem
515.
Then
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.
Proof. We have y * ((x \ y) / y) = x \
T(y,x \ e) by Theorem
1248.
Hence we are done by Proposition
2.
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.
Proof. We have
K(
T(y,y \ x),y \ x) =
K(y,x / y) by Theorem
1250.
Hence we are done by Proposition
49.
Proof. We have
T(y,y / (y /
T(x,y))) * (x * y) = y * (y * x) by Theorem
848.
Then
by Proposition
1.
We have
(L5) T(y,y / (y /
T(x,y))) = (y /
T(x,y)) * x
by Theorem
825.
Then
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).
Hence we are done.
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.
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.
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.
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
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
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).
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.
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.
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).
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.
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.
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
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
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.
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
by Theorem
1262.
We have (y / (y *
K(x,y))) \ y = y *
K(x,y) by Proposition
25.
Hence we are done by
(L3).
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
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
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
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.
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
by Theorem
425.
We have (x \ ((y \ e) \ e)) *
K(y \ e,y) = ((x \ y) / y) * y by Theorem
1264.
Then
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).
Proof. We have x * (x \
T(e / (e / y),(e / (e / y)) \ e)) = ((e / (e / y)) \ e) \ e by Theorem
795.
Then
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
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.
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
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
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.
Proof. We have ((y \ (e / (x \ e))) * y) * (x \ e) = (x \ e) * ((y \ ((x \ e) \ e)) * y) by Theorem
1267.
Then
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).
Proof. We have
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.
Proof. We have
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.
Proof. We have
L(z,y \
T(y,x),y) = z by Theorem
1269.
Hence we are done by Theorem
986.
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.
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.
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.
Proof. We have (y \
T(y,x)) * z =
T(y,x) * (y \ z) by Theorem
1270.
Hence we are done by Theorem
1030.
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.
Proof. We have y * (x \
T(x,y)) = (x * y) / x by Theorem
977.
Then
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).
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.
Proof. We have y * (x \
T(x,y)) = (x * y) / x by Theorem
977.
Then
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).
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
by Theorem
1273.
We have (y * x) *
T(
K(x,y),y * x) =
K(x,y) * (y * x) by Proposition
46.
Then
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).
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
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
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
by
(L9).
We have x * (x \ ((
T(y,x) \ y) * z)) = (
T(y,x) \ y) * z by Axiom
4.
Then
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.
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.
Proof. We have
K(
T(y,x),x) \ x = x *
K(x,y) by Theorem
1038.
Hence we are done by Theorem
1053.
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
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
by
(L14).
We have
K(y,x) * (
K(y,x) \ e) = e by Axiom
4.
Then
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
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.
Proof. We have (x * y) *
R(
T(x,x * y),y,x) =
R(x,y,x) * (x * y) by Proposition
59.
Then
by Theorem
1252.
We have
R(x,y,x) * (x * y) = x * (
R(x,y,x) * y) by Theorem
227.
Then
by
(L7).
We have
T(x * y,
T(x,y)) =
T(x * y,x) by Theorem
1244.
Then
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
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
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).
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.
Proof. We have ((((y \ x) * y) * y) / ((y \ x) * y)) * (y \ x) = (y \ x) *
R(y,y \ x,y) by Theorem
482.
Then
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
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.
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.
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
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
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
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.
Proof. We have
T(x,(y \ x) * y) = x by Theorem
1289.
Hence we are done by Proposition
21.
Proof. We have (x / y) * y = x by Axiom
6.
Then
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).
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
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
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
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
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.
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.
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.
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
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).
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
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).
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.
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.
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.
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
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
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
by
(L15).
We have (e / y) \
T((e / y) * x,(e / y) \ e) = y *
T(y \ x,(e / y) \ e) by Theorem
1080.
Then
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
by Proposition
25.
We have y *
T(y \ x,y) = (y \ x) * y by Proposition
46.
Hence we are done by
(L23).
Proof. We have ((y \ e) * x) * (x / ((y \ e) * x)) = (y \ x) * y by Theorem
1300.
Hence we are done by Proposition
66.
Proof. We have x * ((e / x) * ((x \ y) * x)) =
L(x \ y,x,e / x) * x by Theorem
1074.
Then
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
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
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
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
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
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).
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
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
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).
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
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
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
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
by
(L21).
We have ((y \ x) * y) * (x \ e) = (e / x) * ((y \ x) * y) by Theorem
1268.
Then
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
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
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
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
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
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
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.
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.
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.
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
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.
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.
Proof. We have x \
T(x,y) =
K(x,y) by Theorem
1308.
Hence we are done by Proposition
5.
Proof. We have (x *
T(
K(x,y),x)) / x =
K(x,y) by Proposition
48.
Hence we are done by Theorem
734.
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.
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.
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.
Proof. We have x * ((x \
T(x,y)) * z) =
T(x,y) * z by Theorem
1017.
Hence we are done by Theorem
1030.
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
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).
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.
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.
Proof. We have x *
R(
T(y,x),x,
K(x,y)) =
R(y,x,
K(x,y)) * x by Proposition
59.
Then
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.
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.
Proof. We have y / (
K(x,y) \ y) =
K(x,y) by Proposition
24.
Hence we are done by Theorem
1319.
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.
Proof. We have y *
T(
T(y,x),y) =
T(y,x) * y by Proposition
46.
Hence we are done by Theorem
451.
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.
Proof. We have x / (((x / y) * z) \ ((x / y) * y)) = (x / y) * z by Theorem
24.
Hence we are done by Proposition
53.
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.
Proof. We have
T(x * y,y \ e) = (y \ e) \ ((x * y) * (y \ e)) by Definition
3.
Hence we are done by Proposition
68.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have x * (
T(x,y) \ e) =
T(x,y) \ x by Theorem
175.
Hence we are done by Proposition
2.
Proof. We have ((y * x) / y) * (x \ y) =
T(y,x) by Theorem
179.
Hence we are done by Proposition
2.
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).
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).
Proof. We have (x *
T(z,y)) /
T(z,y) = x by Axiom
5.
Then
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
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.
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.
Proof. We have (e /
T(y,x)) * y = y /
T(y,x) by Theorem
550.
Hence we are done by Proposition
2.
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.
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.
Proof. We have (x * (z / y)) / (z / y) = x by Axiom
5.
Then
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
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.
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.
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
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
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
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have (x * (e / y)) *
K(y \ e,y) = x * (y \ e) by Theorem
412.
Hence we are done by Proposition
1.
Proof. We have (x * (y \ e)) /
K(y \ e,y) = x * (e / y) by Theorem
1363.
Hence we are done by Theorem
656.
Proof. We have
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.
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.
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.
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.
Proof. We have (x * y) * (y \ e) =
R(x,y,y \ e) by Proposition
68.
Hence we are done by Theorem
1367.
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.
Proof. We have
K(((x / y) * y) \ (x / y),y) =
K(y \ e,y) by Theorem
665.
Hence we are done by Axiom
6.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have (e / (((x \ e) \ y) / y)) \ e = ((x \ e) \ y) / y by Proposition
25.
Hence we are done by Theorem
1379.
Proof. We have
R(x \ e,x,y) = y / (((x \ e) \ e) * y) by Theorem
703.
Hence we are done by Proposition
49.
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.
Proof. We have y / (
R(x,e / x,y) \ y) =
R(x,e / x,y) by Proposition
24.
Then
by Theorem
1382.
We have
R(x,x \ e,y) = y / ((x \ e) * y) by Proposition
66.
Hence we are done by
(L2).
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).
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).
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
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
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
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
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).
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.
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).
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.
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.
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.
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.
Proof. We have
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
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.
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.
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.
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.
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.
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.
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
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.
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.
Proof. We have
T(x,((y \ e) \ e) \ e) =
T(x,y \ e) by Theorem
721.
Hence we are done by Theorem
209.
Proof. We have
T(x,((y / z) \ e) \ e) =
T(x,y / z) by Theorem
721.
Hence we are done by Theorem
1391.
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.
Proof. We have (((e / y) * x) * y) / e = ((e / y) * x) * y by Proposition
27.
Then
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.
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.
Proof. We have (x *
K(y \ e,y)) * ((y \ e) \ e) = x * y by Theorem
908.
Hence we are done by Theorem
923.
Proof. We have (x * y) *
K(y \ e,y) = x * (e / (e / y)) by Theorem
1365.
Hence we are done by Theorem
923.
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.
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.
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
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.
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.
Proof. We have y *
K(y,y \ e) =
T(y,y \ e) by Theorem
562.
Hence we are done by Theorem
1405.
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.
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
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
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
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.
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.
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.
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.
Proof. We have (y \ e) \ (
T(y,x) \ e) = (y \
T(y,x)) \ e by Theorem
638.
Hence we are done by Theorem
749.
Proof. We have
T(y,
K(y \ (y / x),y)) = y by Theorem
755.
Hence we are done by Theorem
196.
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.
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.
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).
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.
Proof. We have
T(
T(x,y) / x,x) = x \
T(x,y) by Proposition
47.
Hence we are done by Theorem
758.
Proof. We have (x \
T(x,y)) * x =
T(x,y) by Theorem
759.
Hence we are done by Theorem
198.
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.
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.
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.
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.
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.
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.
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
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
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).
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.
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.
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.
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.
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.
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.
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.
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).
Proof. We have y /
K(x,y) =
K(x,y) \ y by Theorem
766.
Hence we are done by Theorem
776.
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.
Proof. We have (e / (
K(x,y) \ y)) \ e =
K(x,y) \ y by Proposition
25.
Hence we are done by Theorem
1442.
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.
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.
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.
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.
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.
Proof. We have e / ((
K(x,y) \ y) \ e) =
K(x,y) \ y by Proposition
24.
Hence we are done by Theorem
1448.
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).
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.
Proof. We have
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
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.
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
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
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).
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.
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.
Proof. We have
K(y \ (y / (x \ e)),y) = x \
T(x,y) by Theorem
198.
Hence we are done by Theorem
986.
Proof. We have x *
K(x,(x / y) / x) =
T(x,x / (x * y)) by Theorem
870.
Hence we are done by Theorem
1003.
Proof. We have x \
T(y,x \ e) = y * ((x \ y) / y) by Theorem
1248.
Hence we are done by Theorem
37.
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.
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.
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.
Proof. We have
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.
Proof. We have (x \
T(x,y)) \ z = x * (
T(x,y) \ z) by Theorem
1272.
Hence we are done by Theorem
1281.
Proof. We have
K(
T(x / y,y),y) =
K(x / y,y) by Theorem
1053.
Hence we are done by Proposition
47.
Proof. We have
K(
T(y,y \ x),y \ x) =
K(y,x / y) by Theorem
1250.
Hence we are done by Theorem
1053.
Proof. We have
K(
T(y,x),x) \ e =
K(x,y) by Theorem
1036.
Hence we are done by Theorem
1053.
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.
Proof. We have (y * (x / y)) \ x =
K(x / y,y) by Theorem
448.
Hence we are done by Theorem
1464.
Proof. We have (x * (y / x)) * ((x * (y / x)) \ y) = y by Axiom
4.
Hence we are done by Theorem
1468.
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.
Proof. We have (e / z) *
T((e / z) \ (x / z),y) =
T((x / z) * z,y) / z by Theorem
498.
Then
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
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).
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.
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).
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
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
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
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).
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.
Proof. We have
K((e / y) * x,y) =
K((y \ e) * x,y) by Theorem
1447.
Hence we are done by Theorem
1475.
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).
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.
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.
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
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
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
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
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
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.
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.
Proof. We have
T(y,x) *
K(x,y) = y by Theorem
1307.
Hence we are done by Proposition
2.
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.
Proof. We have (y * x) / (
K(z,y) * x) =
K(z,y) \ y by Theorem
1050.
Hence we are done by Theorem
1319.
Proof. We have x *
K(x,(x / y) / x) =
T(x,y \ e) by Theorem
1457.
Then
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.
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.
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.
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.
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.
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.
Proof. We have
K(y \ (y / x),y) = (x \ e) \
T(x \ e,y) by Theorem
730.
Hence we are done by Theorem
1308.
Proof. We have (y \
T(y,x)) * z =
T(y,x) * (y \ z) by Theorem
1270.
Hence we are done by Theorem
1308.
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.
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.
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.
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.
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).
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.
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
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
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.
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
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.
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.
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
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.
Proof. We have
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.
Proof. We have
R(z,
K((e / x) \ e,y),y) = z by Theorem
1503.
Hence we are done by Proposition
25.
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
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
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
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
by
(L34).
We have
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
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
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
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.
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
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
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).
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.
Proof. We have
(L1) K(y,x) * (
T(
K(y,x),z) \ e) =
T(
K(y,x),z) \
K(y,x)
by Theorem
175.
Hence we are done.
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.
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
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.
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.
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.
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.
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
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).
Proof. We have (x * z) *
K(y,z) = x * (z *
K(y,z)) by Theorem
1514.
Hence we are done by Theorem
54.
Proof. We have
L(
K(y,z),z,x) =
K(y,z) by Theorem
1515.
Hence we are done by Proposition
20.
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.
Proof. We have
L(
K(z,y),y,x) \ e =
L(
K(y,z),y,x) by Theorem
1511.
Then
by Theorem
1515.
We have
K(z,y) \ e =
K(y,z) by Theorem
1466.
Hence we are done by
(L2).
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.
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
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).
Proof. We have (x * y) \ (x *
T(y,z)) =
L(
K(y,z),y,x) by Theorem
1487.
Then
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).
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.
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.
Proof. We have ((x / y) *
T(y,z)) * y = x *
T(y,z) by Theorem
1344.
Hence we are done by Theorem
1522.
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.
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.
Proof. We have (z *
K(y,x)) * ((z *
K(y,x)) \ z) = z by Axiom
4.
Hence we are done by Theorem
1526.
Proof. We have ((z / x) *
T(x,y)) \ (z *
T(x,y)) = x by Theorem
1346.
Then
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).
Proof. We have ((y \ x) * y) *
K(y,z) = (y \ x) *
T(y,z) by Theorem
1521.
Then
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.
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.
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
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
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).
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
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
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
by
(L34).
We have (y * x) * ((y * x) \ (((y * x) / y) *
T(y,z))) = ((y * x) / y) *
T(y,z) by Axiom
4.
Then
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
by Theorem
1522.
We have (y * x) *
K(y,z) = y * (x *
K(y,z)) by Theorem
1530.
Then
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
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.
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.
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.
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.
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.
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.
Proof. We have z *
K(x,
T(y,z)) =
K(x,y) * z by Theorem
1537.
Hence we are done by Theorem
11.
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
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
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
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
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
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
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
by
(L66).
We have
K(
K(z,y),y * x) =
K(y * x,
K(y,z)) by Theorem
1509.
Then
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
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
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
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
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).
Proof. We have
K(
K(x,y),z) =
K(z,
K(y,x)) by Theorem
1509.
Hence we are done by Theorem
1539.
Proof. We have
K(
K(z,y),x) *
K(x,
K(z,y)) = e by Theorem
1054.
Then
by Theorem
1509.
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.
Proof. We have
K(z,
K(y,x)) =
K(x,
K(y,z)) by Theorem
1539.
Hence we are done by Theorem
1541.
Proof. We have
K(z,
K(y,x)) =
K(x,
K(y,z)) by Theorem
1539.
Hence we are done by Theorem
1542.
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.
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.
Proof. We have y / (
T(x,y) \ y) =
T(x,y) by Proposition
24.
Hence we are done by Theorem
1545.
Proof. We have x / (
K(y,z) \ x) =
K(y,z) by Proposition
24.
Hence we are done by Theorem
1467.
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
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.
Proof. We have
by Theorem
1309.
Hence we are done.
Proof. We have
K(
K(x,y),z) =
K(x,
K(y,z)) by Theorem
1540.
Then
by Theorem
1542.
We have
T(
K(x,y),z) *
K(z,
K(x,y)) =
K(x,y) by Theorem
1307.
Then
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.
Proof. We have ((x *
K(y,z)) / x) * x = x *
K(y,z) by Axiom
6.
Hence we are done by Theorem
1550.
Proof. We have
T(
K(y,z),x) * x = x *
K(y,z) by Theorem
1551.
Hence we are done by Theorem
1538.
Proof. We have
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.
Proof. We have
K(y,x) *
T(z,
K(y,x)) = z *
K(y,x) by Proposition
46.
Then
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).
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
by Theorem
1554.
We have
K(x,
T(z,y)) =
T(
K(x,z),y) by Theorem
1538.
Hence we are done by
(L3).
Proof. We have
K(y,
T(
T(z,x),x)) =
K(y,z) by Theorem
1553.
Hence we are done by Theorem
1555.
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.
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.
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.
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.
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.
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.
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.
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.
Proof. We have
L(z,x,y) * (z \ e) = z \
L(z,x,y) by Theorem
837.
Hence we are done by Proposition
2.
Proof. We have (x *
T(x \ e,y)) / x =
T(e / x,y) by Theorem
49.
Hence we are done by Theorem
196.
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.
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
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).
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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).
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
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).
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.
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.
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.
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
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).
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.
Proof. We have
T(e /
T(x,y),x) = x \ (x /
T(x,y)) by Theorem
551.
Hence we are done by Theorem
752.
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.
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
by Proposition
76.
We have
T(
K(y \ e,y),z) =
K(y \ e,y) by Theorem
716.
Then
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
by
(L7).
We have
K(y \ e,y) /
T(y,z) =
T(y,z) \ e by Theorem
1586.
Hence we are done by
(L10).
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.
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.
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
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.
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).
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
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
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.
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
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.
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
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.
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
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
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).
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.
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.