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 Th29:
  p is_orientedpath_of v1,v2,V & v1 <> v2 implies v1 in V
proof
  assume that
A1: p is_orientedpath_of v1,v2,V and
A2: v1 <> v2;
  p is_orientedpath_of v1,v2 by A1;
  then
A3: v1 in vertices p by Th27;
  not v1 in {v2} by A2,TARSKI:def 1;
  then
A4: v1 in vertices(p) \ {v2} by A3,XBOOLE_0:def 5;
  vertices(p) \ {v2} c= V by A1;
  hence thesis by A4;
end;
