reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem
  X is being_plane & Y is being_plane & Z is being_plane & X '||' Y & a
  in X /\ Z & b in X /\ Z & c in Y /\ Z & d in Y /\ Z & X<>Y & a<>b & c <>d
  implies X/\Z // Y/\Z
proof
  assume that
A1: X is being_plane and
A2: Y is being_plane and
A3: Z is being_plane and
A4: X '||' Y and
A5: a in X /\ Z and
A6: b in X /\ Z and
A7: c in Y /\ Z and
A8: d in Y /\ Z and
A9: X<>Y and
A10: a<>b and
A11: c <>d;
A12: d in Y & d in Z by A8,XBOOLE_0:def 4;
  set A = X /\ Z, C = Y /\ Z;
A13: c in Y & c in Z by A7,XBOOLE_0:def 4;
A14: a in X & a in Z by A5,XBOOLE_0:def 4;
  then Z<>Y by A1,A2,A4,A9,Lm13;
  then
A15: C is being_line by A2,A3,A11,A13,A12,Th24;
A16: b in X & b in Z by A6,XBOOLE_0:def 4;
  Z<>X by A1,A2,A4,A9,A13,Lm13;
  then
A17: A is being_line by A1,A3,A10,A14,A16,Th24;
  a,b // c,d by A1,A2,A3,A4,A5,A6,A7,A8,A9,Th63;
  hence thesis by A5,A6,A7,A8,A10,A11,A17,A15,AFF_1:38;
end;
