### Nuprl Lemma : euclid-P3

`∀e:EuclideanPlane. ∀A,B,C1,C2:Point.  ∃E:Point. (A_E_B ∧ AE=C1C2) supposing (¬(C1 = C2 ∈ Point)) ∧ |C1C2| < |AB|`

Proof

Definitions occuring in Statement :  eu-lt: `p < q` eu-length: `|s|` eu-mk-seg: `ab` euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-congruent: `ab=cd` eu-point: `Point` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` prop: `ℙ` and: `P ∧ Q` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` not: `¬A` implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` exists: `∃x:A. B[x]` cand: `A c∧ B` stable: `Stable{P}` squash: `↓T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` eu-lt: `p < q`
Lemmas referenced :  not_wf equal_wf eu-point_wf eu-lt_wf eu-length_wf eu-mk-seg_wf euclidean-plane_wf eu-extend-exists eu-lt-null-segment eu-congruent_wf eu-between-eq_wf eu-congruence-identity-sym false_wf eu-between-eq-same-side2 eu-between-eq-symmetry stable__eu-between-eq eu-add-length-between eu-congruent-iff-length eu-O_wf eu-X_wf eu-add-length_wf squash_wf true_wf set_wf iff_weakening_equal eu-le-add1 eu-lt_transitivity
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation productEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis because_Cache dependent_functionElimination productElimination equalitySymmetry hyp_replacement Error :applyLambdaEquality,  sqequalRule independent_isectElimination voidElimination equalityTransitivity equalityEquality universeEquality dependent_set_memberEquality independent_functionElimination dependent_pairFormation independent_pairFormation setEquality applyEquality lambdaEquality imageElimination natural_numberEquality imageMemberEquality baseClosed

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}A,B,C1,C2:Point.
\mexists{}E:Point.  (A\_E\_B  \mwedge{}  AE=C1C2)  supposing  (\mneg{}(C1  =  C2))  \mwedge{}  |C1C2|  <  |AB|

Date html generated: 2016_10_26-AM-07_46_08
Last ObjectModification: 2016_08_29-PM-03_31_29

Theory : euclidean!geometry

Home Index