
theorem Th80:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v2 = F_V.v1 & F is total
  holds F_V.:G1.reachableFrom(v1) c= G2.reachableFrom(v2)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v2 = F_V.v1 & F is total;
  then reconsider F0 = F as non empty PGraphMapping of G1, G2;
  now
    let y be object;
    assume y in F_V.:G1.reachableFrom(v1);
    then consider x being object such that
      A2: x in dom F_V & x in G1.reachableFrom(v1) & y=F_V.x by FUNCT_1:def 6;
    consider W1 being Walk of G1 such that
      A3: W1 is_Walk_from v1,x by A2, GLIB_002:def 5;
    reconsider W1 as F0-defined Walk of G1 by A1, GLIB_010:121;
    F0.:W1 is_Walk_from F0_V.v1,F0_V.x by A3, GLIB_010:132;
    hence y in G2.reachableFrom(v2) by A1, A2, GLIB_002:def 5;
  end;
  hence thesis by TARSKI:def 3;
end;
