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 Th45:
  not {A,B,C} is linear implies ex D st not {A,B,C,D} is planar
proof
  assume
A1: not {A,B,C} is linear;
  consider P such that
A2: {A,B,C} on P by Def12;
  consider A1,B1,C1,D1 being POINT of S such that
A3: not {A1,B1,C1,D1} is planar by Def16;
  not {A1,B1,C1,D1} on P by A3;
  then not A1 on P or not B1 on P or not C1 on P or not D1 on P by Th5;
  then not {A,B,C,A1} is planar or not {A,B,C,B1} is planar or not {A,B,C,C1}
  is planar or not {A,B,C,D1} is planar by A1,A2,Th19;
  hence thesis;
end;
