theorem Th7:
  p is_orientedpath_of v1,v2,V iff p is_orientedpath_of v1,v2,V \/ {v2}
proof
  set V9=V \/ {v2};
  thus p is_orientedpath_of v1,v2,V implies p is_orientedpath_of v1,v2,V9 by
GRAPH_5:32,XBOOLE_1:7;
  assume
A1: p is_orientedpath_of v1,v2,V9;
  then vertices(p) \ {v2} c= V9 by GRAPH_5:def 4;
  then vertices(p) \ {v2} \ {v2} c= V by XBOOLE_1:43;
  then
A2: vertices(p) \ ({v2} \/ {v2}) c= V by XBOOLE_1:41;
  p is_orientedpath_of v1,v2 by A1,GRAPH_5:def 4;
  hence thesis by A2,GRAPH_5:def 4;
end;
