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 Th39:
  p is_orientedpath_of v1,v2,V implies AcyclicPaths(p) c= AcyclicPaths(v1,v2,V)
proof
  assume
A1: p is_orientedpath_of v1,v2,V;
  let x be object;
  assume x in AcyclicPaths(p);
  then consider q being Simple oriented Chain of G such that
A2: x=q and
A3: q <> {} & (the Source of G).(q.1) = (the Source of G).(p.1) & (the
  Target of G).(q.(len q)) = (the Target of G).(p.(len p)) and
A4: rng q c= rng p;
  vertices(p) \ {v2} c= V by A1;
  then
A5: vertices(q) \ {v2} c= V by A4,Th20;
  p is_orientedpath_of v1,v2 by A1;
  then (the Source of G).(p.1) = v1 & (the Target of G).(p.(len p)) = v2;
  then q is_orientedpath_of v1,v2 by A3;
  then q is_orientedpath_of v1,v2,V by A5;
  then q is_acyclicpath_of v1,v2,V;
  hence thesis by A2;
end;
