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 Th46:
  ex B,C st {B,C} on P & not {A,B,C} is linear
proof
A1: now
    assume
A2: not A on P;
    consider B,C,D such that
A3: {B,C,D} on P and
A4: not {B,C,D} is linear by Th41;
A5: B <> C by A4,Th15;
    take B, C;
    {B,C} \/ {D} on P by A3,ENUMSET1:3;
    hence
A6: {B,C} on P by Th9;
    assume {A,B,C} is linear;
    then consider K such that
A7: {A,B,C} on K;
    {B,C,A} on K by A7,ENUMSET1:59;
    then
A8: {B,C} \/ {A} on K by ENUMSET1:3;
    then
A9: A on K by Th8;
    {B,C} on K by A8,Th10;
    then K on P by A6,A5,Def14;
    hence contradiction by A2,A9,Def17;
  end;
  now
    assume A on P;
    then consider B such that
A10: A <> B and
A11: B on P and
    B on P by Def15;
    consider C such that
A12: C on P and
A13: not {A,B,C} is linear by A10,Th44;
    {B,C} on P by A11,A12,Th3;
    hence thesis by A13;
  end;
  hence thesis by A1;
end;
