
theorem
  for G1, G2, G3 being _Graph
  for f being PVertexMapping of G1, G2, g being PVertexMapping of G2, G3
  st f is directed & g is directed holds g*f is directed
proof
  let G1, G2, G3 be _Graph;
  let f be PVertexMapping of G1, G2, g be PVertexMapping of G2, G3;
  assume A1: f is directed & g is directed;
  now
    let v,w,e9 be object;
    assume A2: v in dom (g*f) & w in dom (g*f) & e9 DJoins v,w,G1;
    then v in dom f & w in dom f by FUNCT_1:11;
    then consider e8 being object such that
      A3: e8 DJoins f.v,f.w,G2 by A1, A2;
    f.v in dom g & f.w in dom g by A2, FUNCT_1:11;
    then consider e being object such that
      A4: e DJoins g.(f.v),g.(f.w),G3 by A1, A3;
    take e;
    e DJoins (g*f).v,g.(f.w),G3 by A2, A4, FUNCT_1:12;
    hence e DJoins (g*f).v,(g*f).w,G3 by A2, FUNCT_1:12;
  end;
  hence thesis;
end;
