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 Th63:
  X is being_plane & Y is being_plane & Z is being_plane & X '||'
Y & X<>Y & a in X /\ Z & b in X /\ Z & c in Y /\ Z & d in Y /\ Z implies a,b //
  c,d
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: X<>Y and
A6: a in X /\ Z and
A7: b in X /\ Z and
A8: c in Y /\ Z and
A9: d in Y /\ Z;
A10: c in Z by A8,XBOOLE_0:def 4;
A11: a in X & a in Z by A6,XBOOLE_0:def 4;
  then
A12: Z<>Y by A1,A2,A4,A5,Lm13;
A13: c in Y by A8,XBOOLE_0:def 4;
  then
A14: Z<>X by A1,A2,A4,A5,A10,Lm13;
  set A = X /\ Z, C = Y /\ Z;
A15: b in X & b in Z by A7,XBOOLE_0:def 4;
A16: d in Y & d in Z by A9,XBOOLE_0:def 4;
  now
A17: C c= Y & C c= Z by XBOOLE_1:17;
    set K=c*A;
    assume that
A18: a<>b and
A19: c <>d;
A20: A is being_line by A1,A3,A11,A15,A14,A18,Th24;
    then
A21: A // K by Def3;
    A c= X by XBOOLE_1:17;
    then
A22: K c= Y by A4,A13,A20;
A23: K c= Z by A3,A10,A20,Th28,XBOOLE_1:17;
    C is being_line & K is being_line by A1,A2,A3,A11,A15,A13,A10,A16,A12,A14
,A18,A19,Th24,Th27;
    then K=C by A2,A3,A12,A17,A23,A22,Th26;
    hence thesis by A6,A7,A8,A9,A21,AFF_1:39;
  end;
  hence thesis by AFF_1:3;
end;
