reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;

theorem Th10:
  X is being_plane implies for x holds x in PDir(X) iff ex Y st x=
  Y & Y is being_plane & X '||' Y
proof
  assume
A1: X is being_plane;
  let x;
A2: now
    assume x in PDir(X);
    then [x,X] in PlanesParallelity(AS) by EQREL_1:19;
    then consider K,M such that
A3: [x,X]=[K,M] and
A4: K is being_plane and
A5: M is being_plane and
A6: K '||' M;
    take Y=K;
    X=M by A3,XTUPLE_0:1;
    hence x=Y & Y is being_plane & X '||' Y by A3,A4,A5,A6,AFF_4:58,XTUPLE_0:1;
  end;
  now
    given Y such that
A7: x=Y and
A8: Y is being_plane and
A9: X '||' Y;
    Y '||' X by A1,A8,A9,AFF_4:58;
    then [Y,X] in { [K,M]: K is being_plane & M is being_plane & K '||' M} by
A1,A8;
    hence x in PDir(X) by A7,EQREL_1:19;
  end;
  hence thesis by A2;
end;
