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 Th61:
  X is being_plane & Y is being_plane & Z is being_plane & ( X
'||' Y & Y '||' Z or X '||' Y & Z '||' Y or Y '||' X & Y '||' Z or Y '||' X & Z
  '||' Y ) implies X '||' Z
proof
  assume that
A1: X is being_plane and
A2: Y is being_plane and
A3: Z is being_plane &( X '||' Y & Y '||' Z or X '||' Y & Z '||' Y or Y
  '||' X & Y '||' Z or Y '||' X & Z '||' Y);
  X '||' Y & Y '||' Z by A1,A2,A3,Th58;
  hence thesis by A2,Th59,Th60;
end;
