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 Th19:
  for x holds (x in [:Dir_of_Planes(AS),{2}:] iff ex X st x=[PDir(
  X),2] & X is being_plane)
proof
  let x;
A1: now
    assume x in [:Dir_of_Planes(AS),{2}:];
    then consider x1,x2 being object such that
A2: x1 in Dir_of_Planes(AS) and
A3: x2 in {2} and
A4: x=[x1,x2] by ZFMISC_1:def 2;
    consider X such that
A5: x1=PDir(X) and
A6: X is being_plane by A2,Th15;
    take X;
    thus x=[PDir(X),2] by A3,A4,A5,TARSKI:def 1;
    thus X is being_plane by A6;
  end;
   (ex X st x=[PDir(X),2] & X is being_plane)
      implies x in [:Dir_of_Planes(AS),{2}:] by Th15,ZFMISC_1:106;
  hence thesis by A1;
end;
