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 Th27:
  K <>L & (ex A st A on K & A on L) implies ex P st for Q holds K
  on Q & L on Q iff P = Q
proof
  assume that
A1: K <> L and
A2: ex A st A on K & A on L;
  consider A such that
A3: A on K and
A4: A on L by A2;
  consider C such that
A5: A <> C and
A6: C on L by Lm1;
  consider B such that
A7: A <> B and
A8: B on K by Lm1;
  consider P such that
A9: {A,B,C} on P by Def12;
A10: A on P by A9,Th4;
  take P;
  let Q;
  thus K on Q & L on Q implies P = Q
  proof
    {A,C} on L by A4,A6,Th1;
    then not {A,C} on K by A1,A5,Def10;
    then
A11: not C on K by A3,Th1;
    assume that
A12: K on Q and
A13: L on Q;
A14: C on Q by A6,A13,Def17;
    A on Q & B on Q by A3,A8,A12,Def17;
    then
A15: {A,B,C} on Q by A14,Th4;
    {A,B} on K by A3,A8,Th1;
    then not {A,B,C} is linear by A7,A11,Th18;
    hence thesis by A9,A15,Def13;
  end;
  B on P by A9,Th4;
  then
A16: {A,B} on P by A10,Th3;
  C on P by A9,Th4;
  then
A17: {A,C} on P by A10,Th3;
A18: {A,C} on L by A4,A6,Th1;
  {A,B} on K by A3,A8,Th1;
  hence thesis by A7,A5,A18,A16,A17,Def14;
end;
