reserve c,c1,c2 for Cardinal, G,G1,G2 for _Graph, v for Vertex of G;

theorem Th45:
  G.minInDegree() = c & G.minOutDegree() = c &
    G.supInDegree() = c & G.supOutDegree() = c implies G is c-Dregular
proof
  assume A1: G.minInDegree() = c & G.minOutDegree() = c &
    G.supInDegree() = c & G.supOutDegree() = c;
  let v;
  G.minInDegree() c= v.inDegree() & v.inDegree() c= G.supInDegree()
    by GLIB_013:35;
  hence v.inDegree() = c by A1, XBOOLE_0:def 10;
  G.minOutDegree() c= v.outDegree() & v.outDegree() c= G.supOutDegree()
    by GLIB_013:35;
  hence v.outDegree() = c by A1, XBOOLE_0:def 10;
end;
