theorem
  not {A,B,C} is linear implies Plane(A,B,C) = Plane(C,Line(A,B))
proof
  assume
A1: not {A,B,C} is linear;
  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 thesis by A1,Def20;
end;
