let S be non empty satisfying_Tarski-model satisfying_Lower_Dimension_Axiom TarskiGeometryStruct ; :: thesis: for r being POINT of S
for A being Subset of S st A is_line & not r in A holds
( A c= Plane (A,r) & r in Plane (A,r) )
let r be POINT of S; :: thesis: for A being Subset of S st A is_line & not r in A holds
( A c= Plane (A,r) & r in Plane (A,r) )
let A be Subset of S; :: thesis: ( A is_line & not r in A implies ( A c= Plane (A,r) & r in Plane (A,r) ) )
assume that
A1: A is_line and
A2: not r in A ; :: thesis: ( A c= Plane (A,r) & r in Plane (A,r) )
ex r9 being POINT of S st
( between r,A,r9 & Plane (A,r) = ((half-plane (A,r)) \/ A) \/ (half-plane (A,r9)) ) by A1, A2, Def10;
then A3: ( A c= (half-plane (A,r)) \/ A & (half-plane (A,r)) \/ A c= Plane (A,r) ) by XBOOLE_1:7;
( r in half-plane (A,r) & half-plane (A,r) c= Plane (A,r) ) by Th20, Th30, A1, A2;
hence ( A c= Plane (A,r) & r in Plane (A,r) ) by A3; :: thesis: verum