
theorem Th72:
  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.minOutDegree() = G2.minOutDegree() or
    G1.minOutDegree() = v.outDegree() +` 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.outDegree() = G1.minOutDegree() and
      A5: for w1 being Vertex of G1 holds v1.outDegree() c= w1.outDegree()
      by Th38;
    reconsider v4 = v1 as Vertex of G2 by A3;
    consider v2 being Vertex of G2 such that
      A6: v2.outDegree() = G2.minOutDegree() and
      A7: for w2 being Vertex of G2 holds v2.outDegree() c= w2.outDegree()
      by Th38;
    reconsider v3 = v2 as Vertex of G1 by A3;
    A8: v2.outDegree() c= v4.outDegree() & v1.outDegree() c= v3.outDegree()
      by A5, A7;
    G2 is Subgraph of G1 by GLIB_006:57;
    then v4.outDegree() c= v1.outDegree() & v2.outDegree()c= v3.outDegree()
      by CARD_1:11, GLIB_000:78;
    then A9: v2.outDegree() c= v1.outDegree() & v4.outDegree()c= v3.outDegree()
      by A8, XBOOLE_1:1;
    assume G1.minOutDegree() <> G2.minOutDegree();
    then A10: v1.outDegree() <> v2.outDegree() by A4, A6;
    then A11: v2.outDegree() +` 1 c= v1.outDegree() by A9, Lm1;
    A12: v2 = v
    proof
      assume A13: v2 <> v;
      A14: v2.outDegree() = v3.outDegree()
      proof
        per cases;
        suppose v2 = w;
          hence thesis by A1, A2, GLIBPRE0:48;
        end;
        suppose v2 <> w;
          hence thesis by A13, GLIBPRE0:46;
        end;
      end;
      then A15: v3.outDegree() +` 1 c= v3.outDegree() by A8, A11, XBOOLE_1:1;
      v3.outDegree() c= v3.outDegree() +` 1 by CARD_2:94;
      then v3.outDegree() = v3.outDegree() +` 1 by A15, XBOOLE_0:def 10;
      hence contradiction by A8, A10, A11, A14, XBOOLE_0:def 10;
    end;
    then v3.outDegree() = v2.outDegree() +` 1 by A1, A2, GLIBPRE0:47;
    hence thesis by A4, A8, A11, A12, XBOOLE_0:def 10;
  end;
  suppose e in the_Edges_of G2;
    then G1 == G2 by GLIB_006:def 11;
    hence thesis by Th62;
  end;
end;
