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 Th13:
  X is being_plane & Y is being_plane implies (PDir(X)=PDir(Y) iff X '||' Y)
proof
  assume that
A1: X is being_plane and
A2: Y is being_plane;
A3: PDir(Y)= Class(PlanesParallelity(AS),Y);
A4: Y in AfPlanes(AS) by A2;
A5: now
    assume PDir(X)=PDir(Y);
    then X in Class(PlanesParallelity(AS),Y) by A4,EQREL_1:23;
    then ex Y9 st X=Y9 & Y9 is being_plane & Y '||' Y9 by A2,A3,Th10;
    hence X '||' Y by A2,AFF_4:58;
  end;
A6: PDir(X)=Class(PlanesParallelity(AS),X);
A7: X in AfPlanes(AS) by A1;
  now
    assume X '||' Y;
    then Y in Class(PlanesParallelity(AS),X) by A1,A2,A6,Th10;
    hence PDir(X)=PDir(Y) by A7,EQREL_1:23;
  end;
  hence thesis by A5;
end;
