let S be non empty satisfying_Tarski-model satisfying_Lower_Dimension_Axiom TarskiGeometryStruct ; :: thesis: for a being POINT of S
for A being Subset of S st A is_line & not a in A holds
a in half-plane (A,a)
let a be POINT of S; :: thesis: for A being Subset of S st A is_line & not a in A holds
a in half-plane (A,a)
let A be Subset of S; :: thesis: ( A is_line & not a in A implies a in half-plane (A,a) )
assume that
A1: A is_line and
A2: not a in A ; :: thesis: a in half-plane (A,a)
A out a,a by Th17, A1, A2;
hence a in half-plane (A,a) ; :: thesis: verum