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 Th41:
  ex A,B,C st {A,B,C} on P & not {A,B,C} is linear
proof
  consider A1,B1,C1,D1 being POINT of S such that
A1: A1 on P and
A2: not {A1,B1,C1,D1} is planar by Lm2;
  not {B1,D1,A1,C1} is planar by A2,ENUMSET1:69;
  then
A3: B1 <> D1 by Th16;
  not {C1,D1,A1,B1} is planar by A2,ENUMSET1:73;
  then
A4: C1 <> D1 by Th16;
  not {A1,B1,C1,D1} on P by A2;
  then not {B1,C1,D1,A1} on P by ENUMSET1:68;
  then not {B1,C1,D1} \/ {A1} on P by ENUMSET1:6;
  then not {B1,C1,D1} on P by A1,Th9;
  then not B1 on P or not C1 on P or not D1 on P by Th4;
  then consider X being POINT of S such that
A5: X = B1 or X = C1 or X = D1 and
A6: not X on P;
  not {B1,C1,A1,D1} is planar by A2,ENUMSET1:67;
  then B1 <> C1 by Th16;
  then consider Y,Z being POINT of S such that
A7: ( Y = B1 or Y = C1 or Y = D1)&( Z = B1 or Z = C1 or Z = D1) & Y <>
  X & Z <> X & Y <> Z by A5,A3,A4;
  set P1 = Plane(X,Y,A1), P2 = Plane(X,Z,A1);
A8: now
    assume {A1,X,Y,Z} is planar;
    then {A1,D1,B1,C1} is planar or {A1,D1,C1,B1} is planar by A2,A5,A7,
ENUMSET1:62;
    hence contradiction by A2,ENUMSET1:63,64;
  end;
  then not {A1,X,Y} is linear by Th17;
  then not {X,Y,A1} is linear by ENUMSET1:59;
  then
A9: {X,Y,A1} on P1 by Def20;
  then
A10: A1 on P1 by Th4;
  then consider B such that
A11: A1 <> B and
A12: B on P1 and
A13: B on P by A1,Def15;
  not {X,Z,A1,Y} is planar by A8,ENUMSET1:69;
  then not {X,Z,A1} is linear by Th17;
  then
A14: {X,Z,A1} on P2 by Def20;
  then
A15: A1 on P2 by Th4;
  then consider C such that
A16: A1 <> C and
A17: C on P and
A18: C on P2 by A1,Def15;
  take A1,B,C;
  thus {A1,B,C} on P by A1,A13,A17,Th4;
  given K such that
A19: {A1,B,C} on K;
A20: {A1,C} on P2 by A15,A18,Th3;
  {A1,C,B} on K by A19,ENUMSET1:57;
  then {A1,C} \/ {B} on K by ENUMSET1:3;
  then {A1,C} on K by Th10;
  then
A21: K on P2 by A16,A20,Def14;
  consider E such that
A22: B <> E and
A23: E on K by Lm1;
  {A1,B} \/ {C} on K by A19,ENUMSET1:3;
  then
A24: {A1,B} on K by Th10;
A25: now
    {A1,B} on P by A1,A13,Th3;
    then K on P by A11,A24,Def14;
    then E on P by A23,Def17;
    then
A26: {E,B} on P by A13,Th3;
    assume {X,B,E} is linear;
    then consider L such that
A27: {X,B,E} on L;
A28: X on L by A27,Th2;
    {E,B,X} on L by A27,ENUMSET1:60;
    then {E,B} \/ {X} on L by ENUMSET1:3;
    then {E,B} on L by Th8;
    then L on P by A22,A26,Def14;
    hence contradiction by A6,A28,Def17;
  end;
  B on K by A19,Th2;
  then
A29: B on P2 by A21,Def17;
A30: X on P2 by A14,Th4;
A31: X on P1 by A9,Th4;
  {A1,B} on P1 by A10,A12,Th3;
  then K on P1 by A11,A24,Def14;
  then E on P1 by A23,Def17;
  then
A32: {X,B,E} on P1 by A12,A31,Th4;
  E on P2 by A23,A21,Def17;
  then {X,B,E} on P2 by A29,A30,Th4;
  then P1 = P2 by A25,A32,Def13;
  then Z on P1 by A14,Th4;
  then {X,Y,A1} \/ {Z} on P1 by A9,Th9;
  then {X,Y,A1,Z} on P1 by ENUMSET1:6;
  then {X,Y,A1,Z} is planar;
  hence contradiction by A8,ENUMSET1:67;
end;
