let S be non empty satisfying_Tarski-model satisfying_Lower_Dimension_Axiom TarskiGeometryStruct ; :: thesis: for a, b being POINT of S
for A being Subset of S st A is_plane & not a in A & not b in A & b in half-space3 (A,a) holds
half-space3 (A,b) c= half-space3 (A,a)
let a, b be POINT of S; :: thesis: for A being Subset of S st A is_plane & not a in A & not b in A & b in half-space3 (A,a) holds
half-space3 (A,b) c= half-space3 (A,a)
let A be Subset of S; :: thesis: ( A is_plane & not a in A & not b in A & b in half-space3 (A,a) implies half-space3 (A,b) c= half-space3 (A,a) )
assume that
A1: A is_plane and
A2: not a in A and
A3: not b in A and
A4: b in half-space3 (A,a) ; :: thesis: half-space3 (A,b) c= half-space3 (A,a)
b in { x where x is POINT of S : A out2 x,a } by A1, A2, A4, Def18;
then consider t being POINT of S such that
A5: b = t and
A6: A out2 t,a ;
half-space3 (A,b) c= half-space3 (A,a)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in half-space3 (A,b) or x in half-space3 (A,a) )
assume x in half-space3 (A,b) ; :: thesis: x in half-space3 (A,a)
then x in { x where x is POINT of S : A out2 x,b } by A1, A3, Def18;
then consider y being POINT of S such that
A7: x = y and
A8: A out2 y,b ;
A out2 y,a by A5, A6, A8, Th79;
then x in { x where x is POINT of S : A out2 x,a } by A7;
hence x in half-space3 (A,a) by A1, A2, Def18; :: thesis: verum

end;

hence half-space3 (A,b) c= half-space3 (A,a) ; :: thesis: verum