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.
Apply SNoLe_tra with
add_SNo x0 x1,
add_SNo x2 x1,
add_SNo x2 x3 leaving 5 subgoals.
Apply SNo_add_SNo with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNo_add_SNo with
x2,
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply SNo_add_SNo with
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply add_SNo_Le1 with
x0,
x1,
x2 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Apply add_SNo_Le2 with
x2,
x1,
x3 leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H5.