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
  K <> L & (ex A st A on K & A on L) implies Plane(K,L) = Plane(L,K)
proof
  assume
A1: K <> L;
  set P2 = Plane(L,K);
  set P1 = Plane(K,L);
  given A such that
A2: A on K and
A3: A on L;
  consider C such that
A4: A <> C and
A5: C on L by Lm1;
  consider B such that
A6: A <> B and
A7: B on K by Lm1;
A8: K on P2 by A1,A2,A3,Def22;
  then
A9: B on P2 by A7,Def17;
A10: K on P1 by A1,A2,A3,Def22;
  then
A11: B on P1 by A7,Def17;
A12: now
    assume {A,B,C} is linear;
    then consider L1 such that
A13: {A,B,C} on L1;
A14: A on L1 by A13,Th2;
    C on L1 by A13,Th2;
    then
A15: {A,C} on L1 by A14,Th1;
A16: {A,B} on K by A2,A7,Th1;
    B on L1 by A13,Th2;
    then {A,B} on L1 by A14,Th1;
    then
A17: K = L1 by A6,A16,Def10;
    {A,C} on L by A3,A5,Th1;
    hence contradiction by A1,A4,A15,A17,Def10;
  end;
  L on P2 by A1,A2,A3,Def22;
  then
A18: C on P2 by A5,Def17;
  A on P2 by A2,A8,Def17;
  then
A19: {A,B,C} on P2 by A9,A18,Th4;
  L on P1 by A1,A2,A3,Def22;
  then
A20: C on P1 by A5,Def17;
  A on P1 by A2,A10,Def17;
  then {A,B,C} on P1 by A11,A20,Th4;
  hence thesis by A19,A12,Def13;
end;
