Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Apply SepE with
SNoL x0,
λ x5 . SNoLt 0 x5,
x1,
SNoLt 1 (mul_SNo x0 x4) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply SNoL_E with
x0,
x1,
SNoLt 1 (mul_SNo x0 x4) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H9.
Apply SNo_mul_SNo with
x0,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H5.
Apply add_SNo_0L with
mul_SNo x0 x4,
λ x5 x6 . SNoLt 1 x5 leaving 2 subgoals.
Apply SNo_mul_SNo with
x0,
x4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H7.
Apply add_SNo_minus_Lt1 with
1,
mul_SNo x0 x4,
0 leaving 4 subgoals.
The subproof is completed by applying SNo_1.
Apply SNo_mul_SNo with
x0,
x4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H7.
The subproof is completed by applying SNo_0.
Apply H8 with
λ x5 x6 . SNoLt x6 0.
Apply mul_SNo_pos_neg with
add_SNo 1 (minus_SNo (mul_SNo x0 x3)),
mul_SNo (add_SNo x1 (minus_SNo x0)) x2 leaving 4 subgoals.
Apply SNo_add_SNo with
1,
minus_SNo (mul_SNo x0 x3) leaving 2 subgoals.
The subproof is completed by applying SNo_1.
Apply SNo_minus_SNo with
mul_SNo x0 x3.
The subproof is completed by applying L14.
Apply SNo_mul_SNo with
add_SNo x1 (minus_SNo x0),
x2 leaving 2 subgoals.
Apply SNo_add_SNo with
x1,
minus_SNo x0 leaving 2 subgoals.
The subproof is completed by applying H11.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply add_SNo_minus_Lt2b with
1,
mul_SNo x0 x3,
0 leaving 4 subgoals.
The subproof is completed by applying SNo_1.
The subproof is completed by applying L14.
The subproof is completed by applying SNo_0.
Apply add_SNo_0L with
mul_SNo x0 x3,
λ x5 x6 . SNoLt x6 1 leaving 2 subgoals.
The subproof is completed by applying L14.
The subproof is completed by applying H6.
Apply mul_SNo_neg_pos with
add_SNo x1 (minus_SNo x0),
x2 leaving 4 subgoals.
Apply SNo_add_SNo with
x1,
minus_SNo x0 leaving 2 subgoals.
The subproof is completed by applying H11.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply add_SNo_minus_Lt1b with
x1,
x0,
0 leaving 4 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply add_SNo_0L with
x0,
λ x5 x6 . SNoLt x1 x6 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H13.
Apply SNo_recip_pos_pos with
x1,
x2 leaving 4 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H3.
The subproof is completed by applying H10.
The subproof is completed by applying H4.