
theorem
  for G1, G2, G3 being _Graph
  for f being PVertexMapping of G1, G2, g being PVertexMapping of G2, G3
  st f is continuous & g is continuous holds g*f is continuous
proof
  let G1, G2, G3 be _Graph;
  let f be PVertexMapping of G1, G2, g be PVertexMapping of G2, G3;
  assume A1: f is continuous & g is continuous;
  now
    let v,w,e9 be object;
    assume A2: v in dom (g*f) & w in dom (g*f) & e9 Joins (g*f).v,(g*f).w,G3;
    then e9 Joins g.(f.v),(g*f).w,G3 by FUNCT_1:12;
    then A3: e9 Joins g.(f.v),g.(f.w),G3 by A2, FUNCT_1:12;
    f.v in dom g & f.w in dom g by A2, FUNCT_1:11;
    then consider e8 being object such that
      A4: e8 Joins f.v,f.w,G2 by A1, A3, Th2;
    v in dom f & w in dom f by A2, FUNCT_1:11;
    then consider e being object such that
      A5: e Joins v,w,G1 by A1, A4, Th2;
    take e;
    thus e Joins v,w,G1 by A5;
  end;
  hence thesis by Th2;
end;
