
theorem
  for G2 being _Graph, v being object
  for V being non empty finite set, G1 being addAdjVertexAll of G2, v, V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  ex p being non empty Graph-yielding FinSequence
  st p.1 = G2 & p.len p = G1 & len p = card V + 1 &
    (ex w being Vertex of G2, e being object
     st e in the_Edges_of G1 \ the_Edges_of G2 &
     (p.2 is addAdjVertex of G2,v,e,w or p.2 is addAdjVertex of G2,w,e,v)) &
    for n being Element of dom p st 2 <= n & n <= len p - 1 holds
    ex w being Vertex of G2, e being object
    st e in the_Edges_of G1 \ the_Edges_of p.n &
      (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v)
proof
  let G2 be _Graph, v be object, V be non empty finite set;
  let G1 be addAdjVertexAll of G2, v, V;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  set v0 = the Element of V;
  set V0 = V \ {v0};
  v0 in {v0} by TARSKI:def 1;
  then not v0 in V0 by XBOOLE_0:def 5;
  then V = V0 \/ {v0} & {v0} misses V0 by ZFMISC_1:50, ZFMISC_1:116;
  then consider p being non empty Graph-yielding FinSequence such that
    A2: p.1 = G2 & p.len p = G1 & len p = card V0 + 2 and
    A3: p.2 is addAdjVertexAll of G2, v, {v0} and
    A4: for n being Element of dom p st 2 <= n & n <= len p - 1 holds
      ex w being Vertex of G2, e being object
      st e in the_Edges_of G1 \ the_Edges_of p.n &
        (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v)
    by A1, Th76;
  take p;
  thus p.1 = G2 & p.len p = G1 by A2;
  thus len p = card V - card {v0} + 2 by A2, CARD_2:44
    .= card V - 1 + 2 by CARD_1:30
    .= card V + 1;
  hereby
    reconsider w = v0 as Vertex of G2 by A1, TARSKI:def 3;
    consider e being object such that
      A5: not e in the_Edges_of G2 and
      A6: p.2 is addAdjVertex of G2,v,e,w or p.2 is addAdjVertex of G2,w,e,v
      by A1, A3, GLIB_007:56;
    A7: e in the_Edges_of G1
    proof
      defpred P[Nat] means $1 + 2 <= len p implies ex k being Element of dom p
        st k = $1 + 2 & e in the_Edges_of p.k;
      A8: P[0]
      proof
        assume 0 + 2 <= len p;
        card V0 + 2 - card V0 <= len p - 0 by A2, XREAL_1:10;
        then reconsider k = 2 as Element of dom p by FINSEQ_3:25;
        take k;
        thus k = 0 + 2;
        per cases by A6;
        suppose p.k is addAdjVertex of G2,v,e,w;
          then e Joins v,w,p.k by A1, A5, GLIB_006:132;
          hence e in the_Edges_of p.k by GLIB_000:def 13;
        end;
        suppose p.k is addAdjVertex of G2,w,e,v;
          then e Joins w,v,p.k by A1, A5, GLIB_006:131;
          hence e in the_Edges_of p.k by GLIB_000:def 13;
        end;
      end;
      A9: for m being Nat st P[m] holds P[m+1]
      proof
        let m be Nat;
        assume A10: P[m];
        assume A11: m+1 + 2 <= len p;
        then m+1 + 2 - 1 <= len p - 0 by XREAL_1:13;
        then consider k0 being Element of dom p such that
          A12: k0 = m + 2 & e in the_Edges_of p.k0 by A10;
        1+0 <= m+1+2 by XREAL_1:7;
        then reconsider k = m+1 + 2 as Element of dom p by A11, FINSEQ_3:25;
        take k;
        thus k = m+1 + 2;
        A13: 0+(1+1) <= m+(1+1) by XREAL_1:7;
        m+1 + 2 - 1 <= len p - 1 by A11, XREAL_1:9;
        then consider u being Vertex of G2, f being object
          such that f in the_Edges_of G1 \ the_Edges_of p.k0 and
          A14: (p.(k0+1) is addEdge of p.k0,v,f,u or
            p.(k0+1) is addEdge of p.k0,u,f,v) by A4, A12, A13;
        the_Edges_of p.k0 c= the_Edges_of p.k by A12, A14, GLIB_006:def 9;
        hence thesis by A12;
      end;
      A15: for m being Nat holds P[m] from NAT_1:sch 2(A8,A9);
      reconsider m = len p - 2 as Nat by A2;
      consider k being Element of dom p such that
        A16: k = m + 2 & e in the_Edges_of p.k by A15;
      thus thesis by A2, A16;
    end;
    take w,e;
    thus e in the_Edges_of G1 \ the_Edges_of G2 &
      (p.2 is addAdjVertex of G2,v,e,w or p.2 is addAdjVertex of G2,w,e,v)
      by A5, A6, A7, XBOOLE_0:def 5;
  end;
  thus thesis by A4;
end;
