
theorem Th70:
  for G2 being _Graph, v,w being Vertex of G2, e being object
  for G1 being addEdge of G2,v,e,w st v <> w holds
    G1.minDegree() = G2.minDegree() or
    G1.minDegree() = (v.degree() /\ w.degree()) +` 1
proof
  let G2 be _Graph, v,w be Vertex of G2, e be object;
  let G1 be addEdge of G2,v,e,w;
  assume A1: v <> w;
  per cases;
  suppose A2: not e in the_Edges_of G2;
    then A3: the_Vertices_of G1 = the_Vertices_of G2 by GLIB_006:def 11;
    then reconsider v9 = v, w9 = w as Vertex of G1;
    consider v1 being Vertex of G1 such that
      A4: v1.degree() = G1.minDegree() and
      A5: for w1 being Vertex of G1 holds v1.degree() c= w1.degree() by Th36;
    reconsider v4 = v1 as Vertex of G2 by A3;
    consider v2 being Vertex of G2 such that
      A6: v2.degree() = G2.minDegree() and
      A7: for w2 being Vertex of G2 holds v2.degree() c= w2.degree() by Th36;
    reconsider v3 = v2 as Vertex of G1 by A3;
    A8: v2.degree() c= v4.degree() & v1.degree() c= v3.degree() by A5, A7;
    A9: G2 is Subgraph of G1 by GLIB_006:57;
    then A10: v4.inDegree() c= v1.inDegree() & v2.inDegree() c= v3.inDegree()
      by GLIB_000:78, CARD_1:11;
    v4.outDegree() c= v1.outDegree() & v2.outDegree() c= v3.outDegree()
      by A9, GLIB_000:78, CARD_1:11;
    then v4.degree() c= v1.degree() & v2.degree() c= v3.degree()
      by A10, CARD_2:83;
    then A11: v2.degree() c= v1.degree() & v4.degree() c= v3.degree()
      by A8, XBOOLE_1:1;
    assume G1.minDegree() <> G2.minDegree();
    then A12: v1.degree() <> v2.degree() by A4, A6;
    then A13: v2.degree() +` 1 c= v1.degree() by A11, Lm1;
    A14: v2 = v or v2 = w
    proof
      assume v2 <> v & v2 <> w;
      then A15: v2.degree() = v3.degree() by GLIBPRE0:46;
      then A16: v3.degree() +` 1 c= v3.degree() by A8, A13, XBOOLE_1:1;
      v3.degree() c= v3.degree() +` 1 by CARD_2:94;
      then v3.degree() = v3.degree() +` 1 by A16, XBOOLE_0:def 10;
      hence contradiction by A8, A12, A13, A15, XBOOLE_0:def 10;
    end;
    then v3.degree() = v2.degree() +` 1 by A1, A2, GLIBPRE0:48,47;
    then v1.degree() = v2.degree() +` 1 by A8, A13, XBOOLE_0:def 10;
    hence thesis by A4, A7, A14, XBOOLE_1:28;
  end;
  suppose e in the_Edges_of G2;
    then G1 == G2 by GLIB_006:def 11;
    hence thesis by Th62;
  end;
end;
