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 Th65:
  for a,X st X is being_plane ex Y st a in Y & X '||' Y & Y is being_plane
proof
  let a,X;
  assume
A1: X is being_plane;
  then consider p,q,r such that
A2: p in X and
A3: q in X and
A4: r in X and
A5: not LIN p,q,r by Th34;
  set M=Line(p,q),N=Line(p,r);
A6: p<>r by A5,AFF_1:7;
  then
A7: N c= X by A1,A2,A4,Th19;
  set M9=a*M,N9=a*N;
A8: p<>q by A5,AFF_1:7;
  then
A9: M is being_line by AFF_1:def 3;
  then
A10: M9 is being_line by Th27;
A11: p in N & r in N by AFF_1:15;
A12: p in M by AFF_1:15;
A13: q in M by AFF_1:15;
A14: not M // N
  proof
    assume M // N;
    then r in M by A12,A11,AFF_1:45;
    hence contradiction by A5,A12,A13,A9,AFF_1:21;
  end;
A15: N is being_line by A6,AFF_1:def 3;
  then
A16: N // N9 by Def3;
A17: M // M9 by A9,Def3;
A18: a in M9 by A9,Def3;
  N9 is being_line & a in N9 by A15,Def3,Th27;
  then consider Y such that
A19: M9 c= Y and
A20: N9 c= Y and
A21: Y is being_plane by A10,A18,Th38;
  M c= X by A1,A2,A3,A8,Th19;
  then X '||' Y by A1,A17,A16,A19,A20,A21,A7,A14,Th55;
  hence thesis by A18,A19,A21;
end;
