reserve S for IncStruct;
reserve A,B,C,D for POINT of S;
reserve L for LINE of S;
reserve P for PLANE of S;
reserve F,G for Subset of the Points of S;
reserve a,b,c for Element of {0,1,2,3};
reserve S for IncSpace;
reserve A,B,C,D,E for POINT of S;
reserve K,L,L1,L2 for LINE of S;
reserve P,P1,P2,Q for PLANE of S;
reserve F for Subset of the Points of S;

theorem
  not {A,B,C} is linear implies Plane(A,B,C) = Plane(Line(A,B),Line(A,C) )
proof
  set P2 = Plane(Line(A,B),Line(A,C));
  set L1 = Line(A,B);
  set L2 = Line(A,C);
  assume
A1: not {A,B,C} is linear;
  then
A2: A <> B by Th15;
  then
A3: {A,B} on L1 by Def19;
  then
A4: A on L1 by Th1;
  not {A,C,B} is linear by A1,ENUMSET1:57;
  then
A5: A <> C by Th15;
  then
A6: {A,C} on L2 by Def19;
  then
A7: A on L2 by Th1;
  {A,C} on L2 by A5,Def19;
  then C on L2 by Th1;
  then
A8: L1 <> L2 by A1,A2,Th35;
  then L2 on P2 by A4,A7,Def22;
  then {A,C} on P2 by A6,Th14;
  then
A9: C on P2 by Th3;
  L1 on P2 by A4,A7,A8,Def22;
  then {A,B} on P2 by A3,Th14;
  then {A,B} \/ {C} on P2 by A9,Th9;
  then {A,B,C} on P2 by ENUMSET1:3;
  hence thesis by A1,Def20;
end;
