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 Th23:
  (ex A st A on K & A on L) implies ex P st K on P & L on P
proof
  given A such that
A1: A on K and
A2: A on L;
  consider C such that
A3: A <> C and
A4: C on L by Lm1;
A5: {A,C} on L by A2,A4,Th1;
  consider B such that
A6: A <> B and
A7: B on K by Lm1;
  consider P such that
A8: {A,B,C} on P by Def12;
  take P;
A9: A on P by A8,Th4;
  C on P by A8,Th4;
  then
A10: {A,C} on P by A9,Th3;
  B on P by A8,Th4;
  then
A11: {A,B} on P by A9,Th3;
  {A,B} on K by A1,A7,Th1;
  hence thesis by A6,A3,A5,A11,A10,Def14;
end;
