reserve n,m,i,j,k for Nat,
  x,y,e,X,V,U for set,
  W,f,g for Function;
reserve p,q for FinSequence;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G;
reserve v,v1,v2,v3 for Element of G;
reserve p,q for oriented Chain of G;

theorem Th30:
  p is_orientedpath_of v1,v2,V & V c= U implies p is_orientedpath_of v1,v2,U
proof
  assume that
A1: p is_orientedpath_of v1,v2,V and
A2: V c= U;
  vertices(p) \ {v2} c= V by A1;
  then
A3: vertices(p) \ {v2} c= U by A2;
  p is_orientedpath_of v1,v2 by A1;
  hence thesis by A3;
end;
