### Nuprl Lemma : rsum-difference

`∀[k,n,m:ℤ]. ∀[x:{k..m + 1-} ⟶ ℝ].`
`  ((Σ{x[i] | k≤i≤m} - Σ{x[i] | k≤i≤n}) = Σ{x[i] | n + 1≤i≤m}) supposing ((n ≤ m) and (k ≤ n))`

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` rsub: `x - y` req: `x = y` real: `ℝ` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` rsub: `x - y`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination hypothesis sqequalRule lambdaEquality applyEquality because_Cache addEquality natural_numberEquality setElimination rename dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination equalityTransitivity equalitySymmetry functionEquality

Latex:
\mforall{}[k,n,m:\mBbbZ{}].  \mforall{}[x:\{k..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].
((\mSigma{}\{x[i]  |  k\mleq{}i\mleq{}m\}  -  \mSigma{}\{x[i]  |  k\mleq{}i\mleq{}n\})  =  \mSigma{}\{x[i]  |  n  +  1\mleq{}i\mleq{}m\})  supposing  ((n  \mleq{}  m)  and  (k  \mleq{}  n))

Date html generated: 2016_05_18-AM-07_46_32
Last ObjectModification: 2016_01_17-AM-02_11_11

Theory : reals

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