theorem
  not C on L & {A,B} on L & A <> B implies Plane(C,L) = Plane(A,B,C)
proof
  assume that
A1: not C on L and
A2: {A,B} on L and
A3: A <> B;
  set P1 = Plane(C,L);
  L on P1 by A1,Def21;
  then
A4: {A,B} on P1 by A2,Th14;
  C on P1 by A1,Def21;
  then {A,B} \/ {C} on P1 by A4,Th9;
  then
A5: {A,B,C} on P1 by ENUMSET1:3;
  not {A,B,C} is linear by A1,A2,A3,Th18;
  hence thesis by A5,Def20;
end;
