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.
Assume H4: x0 ⊆ x2.
Assume H5: x1 ⊆ x3.
Apply Subq_tra with
add_nat x0 x1,
add_nat x2 x1,
add_nat x2 x3 leaving 2 subgoals.
Apply unknownprop_db0df2488a5bd91eb2b38c3c1e292417630b2f14c788682eee7dea2a657a90b2 with
x0,
x2,
x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
Apply add_nat_com with
x2,
x1,
λ x4 x5 . x5 ⊆ add_nat x2 x3 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply add_nat_com with
x2,
x3,
λ x4 x5 . add_nat x1 x2 ⊆ x5 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply unknownprop_db0df2488a5bd91eb2b38c3c1e292417630b2f14c788682eee7dea2a657a90b2 with
x1,
x3,
x2 leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
The subproof is completed by applying H5.