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 Th58:
  X is being_plane & Y is being_plane & X '||' Y implies Y '||' X
proof
  assume that
A1: X is being_plane & Y is being_plane and
A2: X '||' Y;
  consider A,P,M,N such that
A3: not A // P and
A4: A c= X & P c= X & M c= Y & N c= Y and
A5: A // M or M // A and
A6: P // N or N // P by A1,A2,Th55;
  not M // N
  proof
    assume M // N;
    then A // N by A5,AFF_1:44;
    hence contradiction by A3,A6,AFF_1:44;
  end;
  hence thesis by A1,A4,A5,A6,Th55;
end;
