
theorem
  for G2 being _Graph, V being set, G1 being addLoops of G2, V
  holds G1.minDegree() c= G2.minDegree() +` 2
proof
  let G2 be _Graph, V be set, G1 be addLoops of G2, V;
  per cases;
  suppose A1: V c= the_Vertices_of G2;
    consider v2 being Vertex of G2 such that
      A2: G2.minDegree() = v2.degree() and
      A3: for w2 being Vertex of G2 holds v2.degree() c= w2.degree() by Th36;
    A4: the_Vertices_of G1 = the_Vertices_of G2 by A1, GLIB_012:def 5;
    then reconsider v1 = v2 as Vertex of G1;
    per cases;
    suppose not v2 in V;
      then A5: v1.degree() = v2.degree() by A1, GLIB_012:44;
      now
        let w1 be Vertex of G1;
        reconsider w2 = w1 as Vertex of G2 by A4;
        per cases;
        suppose w1 in V;
          then A6: w1.degree() = w2.degree() +` 2 by A1, GLIB_012:43;
          A7: v1.degree() c= w2.degree() by A3, A5;
          w2.degree() c= w1.degree() by A6, CARD_2:94;
          hence v1.degree() c= w1.degree() by A7, XBOOLE_1:1;
        end;
        suppose not w1 in V;
          then w1.degree() = w2.degree() by A1, GLIB_012:44;
          hence v1.degree() c= w1.degree() by A3, A5;
        end;
      end;
      then G1.minDegree() = v1.degree() by Th36;
      hence thesis by A2, A5, CARD_2:94;
    end;
    suppose A8: v2 in V & for w2 being Vertex of G2 st not w2 in V
        holds v2.degree() +` 2 c= w2.degree();
      then A9: v1.degree() = v2.degree() +` 2 by A1, GLIB_012:43;
      now
        let w1 be Vertex of G1;
        reconsider w2 = w1 as Vertex of G2 by A4;
        per cases;
        suppose w1 in V;
          then A10: w1.degree() = w2.degree() +` 2 by A1, GLIB_012:43;
          v2.degree() c= w2.degree() & 2 c= 2 by A3;
          hence v1.degree() c= w1.degree() by A9, A10, CARD_2:83;
        end;
        suppose A11: not w1 in V;
          then w1.degree() = w2.degree() by A1, GLIB_012:44;
          hence v1.degree() c= w1.degree() by A8, A9, A11;
        end;
      end;
      hence thesis by A2, A9, Th36;
    end;
    suppose v2 in V & ex w2 being Vertex of G2 st not w2 in V
        & not v2.degree() +` 2 c= w2.degree();
      then consider w2 being Vertex of G2 such that
        A12: not w2 in V & not v2.degree() +` 2 c= w2.degree();
      A13: w2.degree() in v2.degree() +` 2 by A12, ORDINAL1:16;
      reconsider w1 = w2 as Vertex of G1 by A4;
      A14: w1.degree() = w2.degree() by A1, A12, GLIB_012:44;
      per cases;
      suppose A15: for u2 being Vertex of G2 holds not u2 in V implies
          w2.degree() c= u2.degree();
        now
          let u1 be Vertex of G1;
          reconsider u2 = u1 as Vertex of G2 by A4;
          per cases;
          suppose u1 in V;
            then A16: u1.degree() = u2.degree() +` 2 by A1, GLIB_012:43;
            v2.degree() c= u2.degree() & 2 c= 2 by A3;
            then v2.degree() +` 2 c= u2.degree() +` 2 by CARD_2:83;
            hence w1.degree() c= u1.degree() by A13, A14, A16, ORDINAL1:def 2;
          end;
          suppose not u1 in V;
            then u1.degree() = u2.degree() & w2.degree() c= u2.degree()
              by A1, A15, GLIB_012:44;
            hence w1.degree() c= u1.degree() by A14;
          end;
        end;
        then G1.minDegree() = w1.degree() by Th36;
        hence thesis by A2, A13, A14, ORDINAL1:def 2;
      end;
      suppose ex u2 being Vertex of G2 st not u2 in V &
          not w2.degree() c= u2.degree();
        then consider u2 being Vertex of G2 such that
          A17: not u2 in V & not w2.degree() c= u2.degree();
        v2.degree() c= w2.degree() & v2.degree() c= u2.degree() by A3;
        then A18: u2.degree() = v2.degree() by A13, A17, Lm2;
        reconsider u1 = u2 as Vertex of G1 by A4;
        A19: u1.degree() = u2.degree() by A1, A17, GLIB_012:44;
        now
          let x1 be Vertex of G1;
          reconsider x2 = x1 as Vertex of G2 by A4;
          A20: u1.degree() c= x2.degree() by A3, A18, A19;
          per cases;
          suppose x1 in V;
            then x1.degree() = x2.degree() +` 2 by A1, GLIB_012:43;
            then x2.degree() c= x1.degree() by CARD_2:94;
            hence u1.degree() c= x1.degree() by A20, XBOOLE_1:1;
          end;
          suppose not x1 in V;
            hence u1.degree() c= x1.degree() by A1, A20, GLIB_012:44;
          end;
        end;
        then A21: G1.minDegree() = u1.degree() by Th36;
        u1.degree() in w2.degree() by A17, A19, ORDINAL1:16;
        hence thesis by A2, A13, A21;
      end;
    end;
  end;
  suppose not(V c= the_Vertices_of G2);
    then G1 == G2 by GLIB_012:def 5;
    then G1.minDegree() = G2.minDegree() by Th62;
    hence thesis by CARD_2:94;
  end;
end;
