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