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
  ex P,Q st P <> Q & L on P & L on Q
proof
  consider A,B such that
A1: A <> B and
A2: {A,B} on L by Def8;
  consider C,D such that
A3: not {A,B,C,D} is planar by A1,Th47;
  take P = Plane (A,B,C), Q = Plane(A,B,D);
  not {A,B,C} is linear by A3,Th17;
  then
A4: {A,B,C} on P by Def20;
  not {A,B,D,C} is planar by A3,ENUMSET1:61;
  then not {A,B,D} is linear by Th17;
  then
A5: {A,B,D} on Q by Def20;
  then {A,B} \/ {D} on Q by ENUMSET1:3;
  then
A6: {A,B} on Q by Th11;
  D on Q by A5,Th4;
  then P = Q implies {A,B,C} \/ {D} on P by A4,Th9;
  then P = Q implies {A,B,C,D} on P by ENUMSET1:6;
  hence P <> Q by A3;
  {A,B} \/ {C} on P by A4,ENUMSET1:3;
  then {A,B} on P by Th11;
  hence thesis by A1,A2,A6,Def14;
end;
