Let x0 of type ι be given.
Let x1 of type ι be given.
Apply ordinal_ordsucc_SNo_eq with
x0,
λ x2 x3 . add_SNo (minus_SNo x3) (ordsucc x1) = add_SNo (minus_SNo x0) x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply ordinal_ordsucc_SNo_eq with
x1,
λ x2 x3 . add_SNo (minus_SNo (add_SNo 1 x0)) x3 = add_SNo (minus_SNo x0) x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply minus_add_SNo_distr with
1,
x0,
λ x2 x3 . add_SNo x3 (add_SNo 1 x1) = add_SNo (minus_SNo x0) x1 leaving 3 subgoals.
The subproof is completed by applying SNo_1.
Apply ordinal_SNo with
x0.
The subproof is completed by applying H0.
Apply add_SNo_com_4_inner_mid with
minus_SNo 1,
minus_SNo x0,
1,
x1,
λ x2 x3 . x3 = add_SNo (minus_SNo x0) x1 leaving 5 subgoals.
Apply SNo_minus_SNo with
1.
The subproof is completed by applying SNo_1.
Apply SNo_minus_SNo with
x0.
Apply ordinal_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_1.
Apply ordinal_SNo with
x1.
The subproof is completed by applying H1.
Apply add_SNo_minus_SNo_linv with
1,
λ x2 x3 . add_SNo x3 (add_SNo (minus_SNo x0) x1) = add_SNo (minus_SNo x0) x1 leaving 2 subgoals.
The subproof is completed by applying SNo_1.
Apply add_SNo_0L with
add_SNo (minus_SNo x0) x1.
Apply SNo_add_SNo with
minus_SNo x0,
x1 leaving 2 subgoals.
Apply SNo_minus_SNo with
x0.
Apply ordinal_SNo with
x0.
The subproof is completed by applying H0.
Apply ordinal_SNo with
x1.
The subproof is completed by applying H1.