`∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹].  ¬↑P[hd(remove_leading(x.P[x];L))] supposing ¬↑null(remove_leading(x.P[x];L))`

Proof

Definitions occuring in Statement :  remove_leading: `remove_leading(a.P[a];L)` null: `null(as)` hd: `hd(l)` list: `T List` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` not: `¬A` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` false: `False` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` top: `Top` prop: `ℙ` all: `∀x:A. B[x]` or: `P ∨ Q` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` cons: `[a / b]` bfalse: `ff` guard: `{T}` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` less_than': `less_than'(a;b)` listp: `A List+`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality applyEquality hypothesis cumulativity functionEquality independent_isectElimination isect_memberEquality voidElimination voidEquality because_Cache functionExtensionality dependent_functionElimination unionElimination independent_functionElimination natural_numberEquality promote_hyp hypothesis_subsumption productElimination setElimination rename addEquality independent_pairFormation intEquality minusEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].