let S be IncSpace; :: thesis: for A, B, C being POINT of S st not {A,B,C} is linear holds
Plane (A,B,C) = Plane (C,(Line (A,B)))
let A, B, C be POINT of S; :: thesis: ( not {A,B,C} is linear implies Plane (A,B,C) = Plane (C,(Line (A,B))) )
assume A1: not {A,B,C} is linear ; :: thesis: Plane (A,B,C) = Plane (C,(Line (A,B)))
then A <> B by Th15;
then A2: {A,B} on Line (A,B) by Def19;
then ( A on Line (A,B) & B on Line (A,B) ) by Th1;
then A3: ( C on Line (A,B) implies {A,B,C} on Line (A,B) ) by Th2;
then Line (A,B) on Plane (C,(Line (A,B))) by A1, Def21;
then A4: {A,B} on Plane (C,(Line (A,B))) by A2, Th14;
C on Plane (C,(Line (A,B))) by A1, A3, Def21;
then {A,B} \/ {C} on Plane (C,(Line (A,B))) by A4, Th9;
then {A,B,C} on Plane (C,(Line (A,B))) by ENUMSET1:3;
hence Plane (A,B,C) = Plane (C,(Line (A,B))) by A1, Def20; :: thesis: verum