Let x0 of type ι be given.
Apply Field_Hom_I with
x0,
x0,
lam_id (ap x0 0) leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H0.
The subproof is completed by applying lam_id_exp_In with
ap x0 0.
Apply beta with
ap x0 0,
λ x1 . x1,
field3 x0.
Apply Field_zero_In with
x0.
The subproof is completed by applying H0.
Apply beta with
ap x0 0,
λ x1 . x1,
field4 x0.
Apply Field_one_In with
x0.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply beta with
ap x0 0,
λ x3 . x3,
x1,
λ x3 x4 . ap (lam (ap x0 0) (λ x5 . x5)) (field1b x0 x1 x2) = field1b x0 x4 (ap (lam (ap x0 0) (λ x5 . x5)) x2) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply beta with
ap x0 0,
λ x3 . x3,
x2,
λ x3 x4 . ap (lam (ap x0 0) (λ x5 . x5)) (field1b x0 x1 x2) = field1b x0 x1 x4 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply beta with
ap x0 0,
λ x3 . x3,
field1b x0 x1 x2.
Apply Field_plus_clos with
x0,
x1,
x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply beta with
ap x0 0,
λ x3 . x3,
x1,
λ x3 x4 . ap (lam (ap x0 0) (λ x5 . x5)) (field2b x0 x1 x2) = field2b x0 x4 (ap (lam (ap x0 0) (λ x5 . x5)) x2) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply beta with
ap x0 0,
λ x3 . x3,
x2,
λ x3 x4 . ap (lam (ap x0 0) (λ x5 . x5)) (field2b x0 x1 x2) = field2b x0 x1 x4 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply beta with
ap x0 0,
λ x3 . x3,
field2b x0 x1 x2.
Apply Field_mult_clos with
x0,
x1,
x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.