`∀x,y,z,t:ℝ.  (y < t) `` ((x + y) < (z + t)) supposing x ≤ z`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rless: `x < y` radd: `a + b` real: `ℝ` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` top: `Top` uiff: `uiff(P;Q)`
Lemmas referenced :  less_than'_wf rsub_wf real_wf nat_plus_wf rless-iff-rpositive radd_wf rless_wf rleq_wf rpositive-radd2 rminus_wf rpositive_functionality req_transitivity real_term_polynomial itermSubtract_wf itermAdd_wf itermVar_wf itermMinus_wf int-to-real_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_add_lemma real_term_value_var_lemma real_term_value_minus_lemma req-iff-rsub-is-0 radd_functionality req_weakening
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality productElimination independent_pairEquality voidElimination extract_by_obid isectElimination applyEquality hypothesis setElimination rename minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry independent_functionElimination because_Cache independent_isectElimination computeAll int_eqEquality intEquality isect_memberEquality voidEquality

Latex:
\mforall{}x,y,z,t:\mBbbR{}.    (y  <  t)  {}\mRightarrow{}  ((x  +  y)  <  (z  +  t))  supposing  x  \mleq{}  z

Date html generated: 2017_10_03-AM-08_25_14
Last ObjectModification: 2017_07_28-AM-07_23_47

Theory : reals

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