reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem Th38:
  for G2, v, V for G1 being addAdjVertexToAll of G2,v,V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  for e1, u being object
  holds not e1 DJoins u,v,G1 & (not u in V implies not e1 DJoins v,u,G1) &
    for e2 being object
    holds e1 DJoins v,u,G1 & e2 DJoins v,u,G1 implies e1 = e2
proof
  let G2, v, V;
  let G1 be addAdjVertexToAll of G2,v,V;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  then A2: the_Vertices_of G1 = the_Vertices_of G2 \/ {v} &
    the_Edges_of G1 = the_Edges_of G2 \/ (V --> the_Edges_of G2) &
    the_Source_of G1 = the_Source_of G2 +* ((V --> the_Edges_of G2) --> v) &
    the_Target_of G1 = the_Target_of G2 +* pr1(V,{the_Edges_of G2})
    by Def2;
  let e1, u be object;
  thus not e1 DJoins u,v,G1
  proof
    assume e1 DJoins u,v,G1;
    then A3: e1 in the_Edges_of G1 & (the_Source_of G1).e1 = u &
      (the_Target_of G1).e1 = v by GLIB_000:def 14;
    not e1 in the_Edges_of G2
    proof
      reconsider e3=e1 as set by TARSKI:1;
      assume A4: e1 in the_Edges_of G2;
      then (the_Target_of G2).e3 = v by A3, GLIB_006:def 9;
      hence contradiction by A1, A4, FUNCT_2:5;
    end;
    then e1 in V --> the_Edges_of G2 by A2, A3, XBOOLE_0:def 3;
    then A5: e1 in [:V,{the_Edges_of G2}:];
    then A6: e1 in dom pr1(V,{the_Edges_of G2}) by FUNCT_3:def 4;
    consider x1,y1 being object such that
      A7: x1 in V & y1 in {the_Edges_of G2} and
      A8: e1 = [x1,y1] by A5, ZFMISC_1:def 2;
    v = pr1(V,{the_Edges_of G2}).e1 by A6, A2, A3, FUNCT_4:13
      .= pr1(V,{the_Edges_of G2}).(x1,y1) by A8, BINOP_1:def 1
      .= x1 by A7, FUNCT_3:def 4;
    hence contradiction by A1,A7;
  end;
  thus not u in V implies not e1 DJoins v,u,G1
  proof
    assume A9: not u in V;
    assume e1 DJoins v,u,G1;
    then A10: e1 in the_Edges_of G1 &
      (the_Source_of G1).e1 = v & (the_Target_of G1).e1 = u by GLIB_000:def 14;
    not e1 in dom pr1(V, {the_Edges_of G2})
    proof
      assume A11: e1 in dom pr1(V, {the_Edges_of G2});
      then consider x,y being object such that
        A12: x in V & y in {the_Edges_of G2} & e1=[x,y] by ZFMISC_1:def 2;
      u = pr1(V, {the_Edges_of G2}).e1 by A2, A10, A11, FUNCT_4:13
        .= pr1(V, {the_Edges_of G2}).(x,y) by A12, BINOP_1:def 1
        .= x by A12, FUNCT_3:def 4;
      hence contradiction by A9, A12;
    end;
    then not e1 in [: V, {the_Edges_of G2} :] by FUNCT_3:def 4;
    then A13: not e1 in V --> the_Edges_of G2;
    then not e1 in dom ((V --> the_Edges_of G2) --> v);
    then A14: (the_Source_of G1).e1 = (the_Source_of G2).e1
      by A2, FUNCT_4:11;
    e1 in the_Edges_of G2 by A2, A10, A13, XBOOLE_0:def 3;
    hence contradiction by A14, A10, A1, FUNCT_2:5;
  end;
  let e2 be object;
  assume e1 DJoins v,u,G1 & e2 DJoins v,u,G1;
  then A15: e1 in the_Edges_of G1 & e2 in the_Edges_of G1 &
    (the_Source_of G1).e1 = v & (the_Source_of G1).e2 = v &
    (the_Target_of G1).e1 = u & (the_Target_of G1).e2 = u by GLIB_000:def 14;
  not e1 in the_Edges_of G2 & not e2 in the_Edges_of G2
  proof
    assume e1 in the_Edges_of G2 or e2 in the_Edges_of G2;
    then per cases;
    suppose A16: e1 in the_Edges_of G2;
      reconsider e3=e1 as set by TARSKI:1;
      (the_Source_of G2).e3 = v by A15, A16, GLIB_006:def 9;
      hence contradiction by A1,A16, FUNCT_2:5;
    end;
    suppose A17: e2 in the_Edges_of G2;
      reconsider e3=e2 as set by TARSKI:1;
      (the_Source_of G2).e3 = v by A15, A17, GLIB_006:def 9;
      hence contradiction by A1, A17, FUNCT_2:5;
    end;
  end;
  then e1 in V --> the_Edges_of G2 & e2 in V --> the_Edges_of G2
    by A2, A15, XBOOLE_0:def 3;
  then A18: e1 in [:V,{the_Edges_of G2}:] & e2 in [:V,{the_Edges_of G2}:];
  then A19: e1 in dom pr1(V,{the_Edges_of G2}) &
    e2 in dom pr1(V,{the_Edges_of G2}) by FUNCT_3:def 4;
  consider x1,y1 being object such that
    A20: x1 in V & y1 in {the_Edges_of G2} and
    A21: e1 = [x1,y1] by A18, ZFMISC_1:def 2;
  consider x2,y2 being object such that
    A22: x2 in V & y2 in {the_Edges_of G2} and
    A23: e2 = [x2,y2] by A18, ZFMISC_1:def 2;
  A24: u = pr1(V,{the_Edges_of G2}).e1 by A2, A15, A19, FUNCT_4:13
    .= pr1(V,{the_Edges_of G2}).(x1,y1) by A21, BINOP_1:def 1
    .= x1 by A20, FUNCT_3:def 4;
  A25: u = pr1(V,{the_Edges_of G2}).e2 by A2, A15, A19, FUNCT_4:13
    .= pr1(V,{the_Edges_of G2}).(x2,y2) by A23, BINOP_1:def 1
    .= x2 by A22, FUNCT_3:def 4;
  y1 = the_Edges_of G2 & y2 = the_Edges_of G2 by A20, A22, TARSKI:def 1;
  hence thesis by A21, A23, A24, A25;
end;
