Let x0 of type ι → ο be given.
Claim L1:
∀ x1 x2 x3 . x0 x1 ⟶ x0 x2 ⟶ MagmaHom x1 x2 x3 ⟶ x3 ∈ setexp (ap x2 0) (ap x1 0)Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H1: x0 x1.
Assume H2: x0 x2.
Apply H0 with
x1,
λ x4 . MagmaHom x4 x2 x3 ⟶ x3 ∈ setexp (ap x2 0) (ap x4 0) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Let x5 of type ι → ι → ι be given.
Assume H3: ∀ x6 . x6 ∈ x4 ⟶ ∀ x7 . x7 ∈ x4 ⟶ x5 x6 x7 ∈ x4.
Apply H0 with
x2,
λ x6 . MagmaHom (pack_b x4 x5) x6 x3 ⟶ x3 ∈ setexp (ap x6 0) (ap (pack_b x4 x5) 0) leaving 2 subgoals.
The subproof is completed by applying H2.
Let x6 of type ι be given.
Let x7 of type ι → ι → ι be given.
Assume H4: ∀ x8 . x8 ∈ x6 ⟶ ∀ x9 . x9 ∈ x6 ⟶ x7 x8 x9 ∈ x6.
Apply unknownprop_7ee20a9b005b9d1cb4acab7f037a1615344131a99780aaa35f8fa78a1fc7660f with
x4,
x6,
x5,
x7,
x3,
λ x8 x9 : ο . x9 ⟶ x3 ∈ setexp (ap (pack_b x6 x7) 0) (ap (pack_b x4 x5) 0).
Assume H5:
and (x3 ∈ setexp x6 x4) (∀ x8 . x8 ∈ x4 ⟶ ∀ x9 . x9 ∈ x4 ⟶ ap x3 (x5 x8 x9) = x7 (ap x3 x8) (ap x3 x9)).
Apply pack_b_0_eq2 with
x6,
x7,
λ x8 x9 . x3 ∈ setexp x8 (ap (pack_b x4 x5) 0).
Apply pack_b_0_eq2 with
x4,
x5,
λ x8 x9 . x3 ∈ setexp x6 x8.
Apply H5 with
x3 ∈ setexp x6 x4.
Assume H7:
∀ x8 . x8 ∈ x4 ⟶ ∀ x9 . x9 ∈ x4 ⟶ ap x3 (x5 x8 x9) = x7 (ap x3 x8) (ap x3 x9).
The subproof is completed by applying H6.
Apply unknownprop_cb7abf829499aec888363ff9292dd7680786c42dc92f10fdd88dc16ada048723 with
x0,
λ x1 . ap x1 0,
MagmaHom.
The subproof is completed by applying L1.