### Nuprl Lemma : eu-lt_transitivity

`∀e:EuclideanPlane. ∀[p,q,r:{p:Point| O_X_p} ].  (p < r) supposing (q < r and p ≤ q)`

Proof

Definitions occuring in Statement :  eu-lt: `p < q` eu-le: `p ≤ q` euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-X: `X` eu-O: `O` eu-point: `Point` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` set: `{x:A| B[x]} `
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` eu-lt: `p < q` eu-le: `p ≤ q` member: `t ∈ T` prop: `ℙ` euclidean-plane: `EuclideanPlane` so_lambda: `λ2x.t[x]` so_apply: `x[s]` and: `P ∧ Q` cand: `A c∧ B` not: `¬A` implies: `P `` Q` sq_stable: `SqStable(P)` false: `False` squash: `↓T`
Lemmas referenced :  eu-lt_wf eu-between-eq_wf eu-O_wf eu-X_wf eu-le_wf set_wf eu-point_wf euclidean-plane_wf not_wf equal_wf squash_wf eu-between-eq-symmetry eu-between-eq-inner-trans eu-between-eq-exchange3 eu-between-eq-exchange4 sq_stable__and sq_stable__eu-between-eq sq_stable__not eu-between-eq-same
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation sqequalHypSubstitution cut introduction extract_by_obid isectElimination thin hypothesisEquality setElimination rename dependent_set_memberEquality hypothesis dependent_functionElimination because_Cache sqequalRule lambdaEquality isect_memberEquality productElimination equalityEquality independent_isectElimination independent_pairFormation equalitySymmetry hyp_replacement Error :applyLambdaEquality,  independent_functionElimination voidElimination imageMemberEquality baseClosed imageElimination equalityTransitivity

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}[p,q,r:\{p:Point|  O\_X\_p\}  ].    (p  <  r)  supposing  (q  <  r  and  p  \mleq{}  q)

Date html generated: 2016_10_26-AM-07_41_41
Last ObjectModification: 2016_07_12-AM-08_07_55

Theory : euclidean!geometry

Home Index