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 L,L1,L2 st A on L & A on L1 & A on L2 & not(ex P st L on P & L1 on
  P & L2 on P)
proof
  consider P such that
A1: {A,A,A} on P by Def12;
  A on P by A1,Th4;
  then consider L,L1,L2 such that
A2: L1 <> L2 and
A3: L1 on P and
A4: L2 on P and
A5: not L on P and
A6: A on L and
A7: A on L1 and
A8: A on L2 by Th50;
  consider B such that
A9: A <> B and
A10: B on L1 by Lm1;
  consider C such that
A11: A <> C and
A12: C on L2 by Lm1;
A13: C on P by A4,A12,Def17;
A14: {A,B} on L1 by A7,A10,Th1;
  then {A,B} on P by A3,Th14;
  then {A,B} \/ {C} on P by A13,Th9;
  then
A15: {A,B,C} on P by ENUMSET1:3;
  take L,L1,L2;
  thus A on L & A on L1 & A on L2 by A6,A7,A8;
  given Q such that
A16: L on Q and
A17: L1 on Q and
A18: L2 on Q;
A19: C on Q by A18,A12,Def17;
A20: {A,C} on L2 by A8,A12,Th1;
  now
    given K such that
A21: {A,B,C} on K;
    {A,C,B} on K by A21,ENUMSET1:57;
    then {A,C} \/ {B} on K by ENUMSET1:3;
    then
A22: {A,C} on K by Th8;
    {A,B} \/ {C} on K by A21,ENUMSET1:3;
    then {A,B} on K by Th8;
    then K = L1 by A9,A14,Def10;
    hence contradiction by A2,A11,A20,A22,Def10;
  end;
  then
A23: not {A,B,C} is linear;
  {A,B} on Q by A17,A14,Th14;
  then {A,B} \/ {C} on Q by A19,Th9;
  then {A,B,C} on Q by ENUMSET1:3;
  hence contradiction by A5,A16,A15,A23,Def13;
end;
