
theorem Th5:
  for G being _Graph holds id the_Vertices_of G is
    directed continuous Dcontinuous PVertexMapping of G, G
proof
  let G be _Graph;
  set f = id the_Vertices_of G;
  A1: now
    let v,w,e be object;
    assume A2: v in dom f & w in dom f & e Joins v,w,G;
    take e;
    e Joins v,f.w,G by A2, FUNCT_1:18;
    hence e Joins f.v,f.w,G by A2, FUNCT_1:18;
  end;
  A3: now
    let v,w,e be object;
    assume A4: v in dom f & w in dom f & e DJoins v,w,G;
    take e;
    e DJoins v,f.w,G by A4, FUNCT_1:18;
    hence e DJoins f.v,f.w,G by A4, FUNCT_1:18;
  end;
  A5: now
    let v,w,e be object;
    assume A6: v in dom f & w in dom f & e Joins f.v,f.w,G;
    take e;
    e Joins v,f.w,G by A6, FUNCT_1:18;
    hence e Joins v,w,G by A6, FUNCT_1:18;
  end;
  now
    let v,w,e be object;
    assume A7: v in dom f & w in dom f & e DJoins f.v,f.w,G;
    take e;
    e DJoins v,f.w,G by A7, FUNCT_1:18;
    hence e DJoins v,w,G by A7, FUNCT_1:18;
  end;
  hence thesis by A1, A3, A5, Th1, Th2, Def2, Def4;
end;
