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
  X is being_plane implies a+X = a+(q+X)
proof
  assume
A1: X is being_plane;
  then
A2: a in a+X by Def6;
A3: a+X is being_plane by A1,Def6;
A4: q+X is being_plane by A1,Def6;
  then
A5: q+X '||' a+(q+X) by Def6;
A6: a in a+(q+X) by A4,Def6;
A7: a+(q+X) is being_plane by A4,Def6;
  X '||' q+X & X '||' a+X by A1,Def6;
  then a+X '||' q+X by A1,A4,A3,Th61;
  then a+X '||' a+(q+X) by A4,A5,A3,A7,Th61;
  hence thesis by A2,A6,A3,A7,Lm13;
end;
