Nuprl Lemma : eu-colinear-between2
∀e:EuclideanPlane
  ∀[A,B,C,D:Point].  (Colinear(B;C;D)) supposing ((¬(B = C ∈ Point)) and (¬(A = B ∈ Point)) and A_C_B and A_D_B)
Proof
Definitions occuring in Statement : 
euclidean-plane: EuclideanPlane
, 
eu-between-eq: a_b_c
, 
eu-colinear: Colinear(a;b;c)
, 
eu-point: Point
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
false: False
, 
euclidean-plane: EuclideanPlane
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
prop: ℙ
, 
and: P ∧ Q
, 
exists: ∃x:A. B[x]
Lemmas referenced : 
eu-point_wf, 
sq_stable__colinear, 
not_wf, 
equal_wf, 
eu-between-eq_wf, 
euclidean-plane_wf, 
eu-proper-extend-exists, 
eu-O_wf, 
eu-not-colinear-OXY, 
eu-X_wf, 
eu-colinear-same-side, 
eu-between-eq-symmetry, 
eu-between-implies-between-eq, 
eu-between-eq-exchange3, 
eu-between-eq-exchange4, 
eu-between_wf, 
not-eu-between-same
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
isect_memberFormation, 
cut, 
introduction, 
sqequalRule, 
sqequalHypSubstitution, 
lambdaEquality, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
voidElimination, 
equalityEquality, 
extract_by_obid, 
isectElimination, 
setElimination, 
rename, 
hypothesis, 
because_Cache, 
independent_functionElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
dependent_set_memberEquality, 
productElimination, 
independent_isectElimination, 
equalitySymmetry, 
hyp_replacement, 
Error :applyLambdaEquality
Latex:
\mforall{}e:EuclideanPlane
    \mforall{}[A,B,C,D:Point].    (Colinear(B;C;D))  supposing  ((\mneg{}(B  =  C))  and  (\mneg{}(A  =  B))  and  A\_C\_B  and  A\_D\_B)
Date html generated:
2016_10_26-AM-07_43_22
Last ObjectModification:
2016_07_12-AM-08_09_26
Theory : euclidean!geometry
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