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(ex P st L on P & L1 on P & L2 on P) & (ex A st A on L & A on L1 &
  A on L2) implies L <> L1
proof
  assume
A1: not(ex P st L on P & L1 on P & L2 on P);
  given A such that
A2: A on L and
A3: A on L1 and
A4: A on L2;
  consider C such that
A5: A <> C and
A6: C on L1 by Lm1;
  consider D such that
A7: A <> D and
A8: D on L2 by Lm1;
  consider B such that
A9: A <> B and
A10: B on L by Lm1;
  assume
A11: L = L1;
  then {A,C,B} on L1 by A3,A10,A6,Th2;
  then {A,C} \/ {B} on L1 by ENUMSET1:3;
  then
A12: {A,C} on L1 by Th10;
  {A,B,C} on L by A3,A11,A10,A6,Th2;
  then {A,B,C} is linear;
  then {A,B,C,D} is planar by Th17;
  then consider Q such that
A13: {A,B,C,D} on Q;
  A on Q & D on Q by A13,Th5;
  then
A14: {A,D} on Q by Th3;
  {A,D} on L2 by A4,A8,Th1;
  then
A15: L2 on Q by A7,A14,Def14;
  A on Q & C on Q by A13,Th5;
  then {A,C} on Q by Th3;
  then
A16: L1 on Q by A5,A12,Def14;
  {A,B} \/ {C,D} on Q by A13,ENUMSET1:5;
  then
A17: {A,B} on Q by Th11;
  {A,B} on L by A2,A10,Th1;
  then L on Q by A9,A17,Def14;
  hence contradiction by A1,A16,A15;
end;
