reserve G for _Graph;

theorem
  for v being object, V being Subset of the_Vertices_of G
  for H being addAdjVertexAll of G,v,V st not v in the_Vertices_of G
  holds VertexAdjSymRel(H) = VertexAdjSymRel(G) \/ [: {v},V :] \/ [: V,{v} :]
proof
  let v be object, V be Subset of the_Vertices_of G;
  let H be addAdjVertexAll of G,v,V;
  assume A1: not v in the_Vertices_of G;
  then consider E being set such that
    A2: card V = card E & E misses the_Edges_of G and
    A3: the_Edges_of H = the_Edges_of G \/ E and
    A4: for v1 being object st v1 in V ex e1 being object st e1 in E &
      e1 Joins v1,v,H & for e2 being object st e2 Joins v1,v,H holds e1 = e2
    by GLIB_007:def 4;
  set F = [: {v},V :], L = [: V,{v} :];
  now
    let v1,v2 be object;
    hereby
      assume [v1,v2] in VertexAdjSymRel(H);
      then consider e being object such that
        A5: e Joins v1,v2,H by Th32;
      per cases by A5, GLIB_006:72;
      suppose e Joins v1,v2,G;
        then [v1,v2] in VertexAdjSymRel(G) by Th32;
        then [v1,v2] in VertexAdjSymRel(G) \/ F by XBOOLE_0:def 3;
        hence [v1,v2] in VertexAdjSymRel(G) \/ F \/ L by XBOOLE_0:def 3;
      end;
      suppose not e in the_Edges_of G;
        then per cases by A1, A2, A3, A5, GLIB_007:51;
        suppose v1 = v & v2 in V;
          then v1 in {v} & v2 in V by TARSKI:def 1;
          then [v1,v2] in F by ZFMISC_1:87;
          then [v1,v2] in VertexAdjSymRel(G) \/ F by XBOOLE_0:def 3;
          hence [v1,v2] in VertexAdjSymRel(G) \/ F \/ L by XBOOLE_0:def 3;
        end;
        suppose v2 = v & v1 in V;
          then v2 in {v} & v1 in V by TARSKI:def 1;
          then [v1,v2] in L by ZFMISC_1:87;
          hence [v1,v2] in VertexAdjSymRel(G) \/ F \/ L by XBOOLE_0:def 3;
        end;
      end;
    end;
    assume [v1,v2] in VertexAdjSymRel(G) \/ F \/ L;
    then [v1,v2] in VertexAdjSymRel(G) \/ F or [v1,v2] in L by XBOOLE_0:def 3;
    then per cases by XBOOLE_0:def 3;
    suppose A6: [v1,v2] in VertexAdjSymRel(G);
      G is Subgraph of H by GLIB_006:57;
      hence [v1,v2] in VertexAdjSymRel(H) by A6, Th45, TARSKI:def 3;
    end;
    suppose [v1,v2] in F;
      then A7: v1 in {v} & v2 in V by ZFMISC_1:87;
      then consider e1 being object such that
        A8: e1 in E & e1 Joins v2,v,H and
        for e2 being object st e2 Joins v2,v,H holds e1 = e2 by A4;
      e1 Joins v2,v1,H by A7, A8, TARSKI:def 1;
      then e1 Joins v1,v2,H by GLIB_000:14;
      hence [v1,v2] in VertexAdjSymRel(H) by Th32;
    end;
    suppose [v1,v2] in L;
      then A9: v2 in {v} & v1 in V by ZFMISC_1:87;
      then consider e1 being object such that
        A10: e1 in E & e1 Joins v1,v,H and
        for e2 being object st e2 Joins v1,v,H holds e1 = e2 by A4;
      e1 Joins v1,v2,H by A9, A10, TARSKI:def 1;
      hence [v1,v2] in VertexAdjSymRel(H) by Th32;
    end;
  end;
  hence thesis by RELAT_1:def 2;
end;
