`∀[p:{2...}]. ∀[n:ℕ]. ∀[a:p-adics(p)].  ((a/p^n) ∈ {x:basic-padic(p)| bpa-equiv(p;<n, a>;x)} )`

Proof

Definitions occuring in Statement :  mkpadic: `(a/p^n)` bpa-equiv: `bpa-equiv(p;x;y)` basic-padic: `basic-padic(p)` p-adics: `p-adics(p)` int_upper: `{i...}` nat: `ℕ` uall: `∀[x:A]. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` pair: `<a, b>` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` mkpadic: `(a/p^n)` all: `∀x:A. B[x]` basic-padic: `basic-padic(p)` nat_plus: `ℕ+` int_upper: `{i...}` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` prop: `ℙ` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` true: `True`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_pairEquality hypothesis dependent_set_memberEquality isectElimination setElimination rename productElimination natural_numberEquality unionElimination independent_pairFormation lambdaFormation voidElimination independent_functionElimination independent_isectElimination applyEquality because_Cache axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality

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