Let x0 of type ι be given.
Apply SNoS_E2 with
ordsucc omega,
x0,
∃ x1 . and (x1 ∈ omega) (SNoLt (minus_SNo x1) x0) leaving 3 subgoals.
The subproof is completed by applying ordsucc_omega_ordinal.
The subproof is completed by applying H0.
Apply SNoS_ordsucc_omega_bdd_above with
minus_SNo x0,
∃ x1 . and (x1 ∈ omega) (SNoLt (minus_SNo x1) x0) leaving 3 subgoals.
Apply minus_SNo_SNoS_ with
ordsucc omega,
x0 leaving 2 subgoals.
The subproof is completed by applying ordsucc_omega_ordinal.
The subproof is completed by applying H0.
Apply minus_SNo_Lt_contra1 with
omega,
x0 leaving 3 subgoals.
The subproof is completed by applying SNo_omega.
The subproof is completed by applying H4.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Apply H6 with
∃ x2 . and (x2 ∈ omega) (SNoLt (minus_SNo x2) x0).
Let x2 of type ο be given.
Apply H9 with
x1.
Apply andI with
x1 ∈ omega,
SNoLt (minus_SNo x1) x0 leaving 2 subgoals.
The subproof is completed by applying H7.
Apply minus_SNo_Lt_contra1 with
x0,
x1 leaving 3 subgoals.
The subproof is completed by applying H4.
Apply omega_SNo with
x1.
The subproof is completed by applying H7.
The subproof is completed by applying H8.