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
  L1 on P & L2 on P & not L on P & L1 <> L2 implies not(ex Q st L on Q &
  L1 on Q & L2 on Q)
proof
  assume that
A1: L1 on P and
A2: L2 on P and
A3: not L on P and
A4: L1 <> L2;
  consider A,B such that
A5: A <> B and
A6: {A,B} on L1 by Def8;
A7: {A,B} on P by A1,A6,Th14;
  consider C,C1 being POINT of S such that
A8: C <> C1 and
A9: {C,C1} on L2 by Def8;
A10: now
    assume C on L1 & C1 on L1;
    then {C,C1} on L1 by Th1;
    hence contradiction by A4,A8,A9,Def10;
  end;
A11: {C,C1} on P by A2,A9,Th14;
  then C on P by Th3;
  then {A,B} \/ {C} on P by A7,Th9;
  then
A12: {A,B,C} on P by ENUMSET1:3;
  C1 on P by A11,Th3;
  then {A,B} \/ {C1} on P by A7,Th9;
  then
A13: {A,B,C1} on P by ENUMSET1:3;
  consider D,E such that
A14: D <> E and
A15: {D,E} on L by Def8;
  given Q such that
A16: L on Q and
A17: L1 on Q and
A18: L2 on Q;
A19: {A,B} on Q by A17,A6,Th14;
A20: {C,C1} on Q by A18,A9,Th14;
  then
A21: C on Q by Th3;
A22: {D,E} on Q by A16,A15,Th14;
  then
A23: D on Q by Th3;
  then {C,D} on Q by A21,Th3;
  then {A,B} \/ {C,D} on Q by A19,Th11;
  then {A,B,C,D} on Q by ENUMSET1:5;
  then
A24: {A,B,C,D} is planar;
A25: E on Q by A22,Th3;
  then {C,E} on Q by A21,Th3;
  then {A,B} \/ {C,E} on Q by A19,Th11;
  then {A,B,C,E} on Q by ENUMSET1:5;
  then
A26: {A,B,C,E} is planar;
A27: C1 on Q by A20,Th3;
  then {C1,D} on Q by A23,Th3;
  then {A,B} \/ {C1,D} on Q by A19,Th11;
  then {A,B,C1,D} on Q by ENUMSET1:5;
  then
A28: {A,B,C1,D} is planar;
  {C1,E} on Q by A27,A25,Th3;
  then {A,B} \/ {C1,E} on Q by A19,Th11;
  then {A,B,C1,E} on Q by ENUMSET1:5;
  then
A29: {A,B,C1,E} is planar;
  not {D,E} on P by A3,A14,A15,Def14;
  then not D on P or not E on P by Th3;
  hence contradiction by A5,A6,A24,A26,A28,A29,A10,A12,A13,Th18,Th19;
end;
