### Nuprl Lemma : fun-connected-step-back

`∀[T:Type]. ∀f:T ⟶ T. ∀x,y:T.  (x is f*(y) `` x is f*(f y) supposing ¬(x = y ∈ T))`

Proof

Definitions occuring in Statement :  fun-connected: `y is f*(x)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` uimplies: `b supposing a` member: `t ∈ T` not: `¬A` false: `False` fun-connected: `y is f*(x)` exists: `∃x:A. B[x]` or: `P ∨ Q` fun-path: `y=f*(x) via L` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` subtract: `n - m` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` and: `P ∧ Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` cons: `[a / b]` bfalse: `ff` prop: `ℙ` append: `as @ bs` so_lambda: so_lambda3 so_apply: `x[s1;s2;s3]` ge: `i ≥ j ` true: `True` guard: `{T}` nat: `ℕ` le: `A ≤ B` cand: `A c∧ B` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` listp: `A List+` subtype_rel: `A ⊆r B` satisfiable_int_formula: `satisfiable_int_formula(fmla)` int_seg: `{i..j-}` lelt: `i ≤ j < k` last: `last(L)` sq_type: `SQType(T)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt lambdaFormation_alt cut introduction sqequalRule sqequalHypSubstitution lambdaEquality_alt dependent_functionElimination thin hypothesisEquality voidElimination functionIsTypeImplies inhabitedIsType rename productElimination extract_by_obid isectElimination hypothesis unionElimination baseClosed independent_isectElimination Error :memTop,  imageElimination promote_hyp hypothesis_subsumption functionIsType equalityIstype universeIsType because_Cache instantiate universeEquality dependent_pairFormation_alt applyEquality independent_functionElimination equalityTransitivity equalitySymmetry natural_numberEquality setElimination independent_pairFormation addEquality minusEquality dependent_set_memberEquality_alt imageMemberEquality independent_pairEquality axiomEquality hyp_replacement applyLambdaEquality voidEquality approximateComputation int_eqEquality pointwiseFunctionality baseApply closedConclusion productIsType cumulativity intEquality

Latex:
\mforall{}[T:Type].  \mforall{}f:T  {}\mrightarrow{}  T.  \mforall{}x,y:T.    (x  is  f*(y)  {}\mRightarrow{}  x  is  f*(f  y)  supposing  \mneg{}(x  =  y))

Date html generated: 2020_05_20-AM-08_10_38
Last ObjectModification: 2019_12_31-PM-06_30_08

Theory : general

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