
theorem Th76:
  for G2 being _Graph, v being object
  for G1 being addAdjVertexAll of G2,v st not v in the_Vertices_of G2
  holds G1.minDegree() = (G2.minDegree() +` 1) /\ G2.order()
proof
  let G2 be _Graph, v be object;
  let G1 be addAdjVertexAll of G2, v;
  assume A1: not v in the_Vertices_of G2;
  then reconsider v9 = v as Vertex of G1 by GLIB_007:50;
  consider v0 being Vertex of G2 such that
    A2: G2.minDegree() = v0.degree() and
    A3: for w0 being Vertex of G2 holds v0.degree() c= w0.degree() by Th36;
  A4: the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by A1, GLIB_007:def 4;
  then reconsider v1 = v0 as Vertex of G1 by XBOOLE_0:def 3;
  A5: the_Vertices_of G2 c= the_Vertices_of G2;
  A6: v9.degree() = G2.order() by A1, GLIBPRE0:55;
  A7: v1.degree() = v0.degree() +` 1 by A1, A5, GLIBPRE0:57;
  per cases;
  suppose A8: v0.degree() +` 1 c= v9.degree();
    then A9: (G2.minDegree() +` 1) /\ G2.order() = v0.degree() +` 1
      by A2, A6, XBOOLE_1:28;
    now
      let w be Vertex of G1;
      per cases by A4, XBOOLE_0:def 3;
      suppose w in the_Vertices_of G2;
        then reconsider w0 = w as Vertex of G2;
        v0.degree() c= w0.degree() & 1 c= 1 by A3;
        then v1.degree() c= w0.degree() +` 1 by A7, CARD_2:83;
        hence v1.degree() c= w.degree() by A1, A5, GLIBPRE0:57;
      end;
      suppose w in {v};
        hence v1.degree() c= w.degree() by A7, A8, TARSKI:def 1;
      end;
    end;
    hence thesis by A7, A9, Th36;
  end;
  suppose A10: v9.degree() c= v0.degree() +` 1;
    then A11: (G2.minDegree() +` 1) /\ G2.order() = v9.degree()
      by A2, A6, XBOOLE_1:28;
    now
      let w be Vertex of G1;
      per cases by A4, XBOOLE_0:def 3;
      suppose w in the_Vertices_of G2;
        then reconsider w0 = w as Vertex of G2;
        v0.degree() c= w0.degree() & 1 c= 1 by A3;
        then v0.degree() +` 1 c= w0.degree() +` 1 by CARD_2:83;
        then v9.degree() c= w0.degree() +` 1 by A10, XBOOLE_1:1;
        hence v9.degree() c= w.degree() by A1, A5, GLIBPRE0:57;
      end;
      suppose w in {v};
        hence v9.degree() c= w.degree() by TARSKI:def 1;
      end;
    end;
    hence thesis by A11, Th36;
  end;
end;
