induction' P using Finset.induction with p S hS hSi
R : Type u_2
ι : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq ι
P : Finset (MvPolynomial ι R)
⊢ (P.prod id).support ⊆ P.sum support
case empty
R : Type u_2
ι : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq ι
⊢ (∅.prod id).support ⊆ ∅.sum support

case insert
R : Type u_2
ι : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq ι
p : MvPolynomial ι R
S : Finset (MvPolynomial ι R)
hS : p ∉ S
hSi : (S.prod id).support ⊆ S.sum support
⊢ ((insert p S).prod id).support ⊆ (insert p S).sum support