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 Th15:
  for x holds x in Dir_of_Planes(AS) iff ex X st x=PDir(X) & X is being_plane
proof
  let x;
A1: now
    assume
A2: x in Dir_of_Planes(AS);
    then reconsider x99= x as Subset of AfPlanes(AS);
    consider x9 being object such that
A3: x9 in AfPlanes(AS) and
A4: x99=Class(PlanesParallelity(AS),x9) by A2,EQREL_1:def 3;
    consider X such that
A5: x9=X and
A6: X is being_plane by A3;
    take X;
    thus x=PDir(X) by A4,A5;
    thus X is being_plane by A6;
  end;
  now
    given X such that
A7: x=PDir(X) and
A8: X is being_plane;
    X in AfPlanes(AS) by A8;
    hence x in Dir_of_Planes(AS) by A7,EQREL_1:def 3;
  end;
  hence thesis by A1;
end;
