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
  P <> Q implies not(ex A st A on P & A on Q) or ex L st for B holds B
  on P & B on Q iff B on L
proof
  assume
A1: P <> Q;
  given A such that
A2: A on P and
A3: A on Q;
  consider C such that
A4: A <> C and
A5: C on P and
A6: C on Q by A2,A3,Def15;
  take L = Line(A,C);
A7: {A,C} on L by A4,Def19;
  {A,C} on Q by A3,A6,Th3;
  then
A8: L on Q by A4,A7,Def14;
  let B;
  {A,C} on P by A2,A5,Th3;
  then
A9: L on P by A4,A7,Def14;
  thus B on P & B on Q implies B on L
  proof
    assume that
A10: B on P and
A11: B on Q and
A12: not B on L;
    consider P1 such that
A13: for P2 holds B on P2 & L on P2 iff P1 = P2 by A12,Th26;
    P = P1 by A9,A10,A13;
    hence contradiction by A1,A8,A11,A13;
  end;
  thus thesis by A9,A8,Def17;
end;
