reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;
reserve g for EColoring of G;

theorem Th122:
  (G is c-ecolorable & for d being Cardinal st G is d-ecolorable holds c c= d)
  iff G.eChromaticNum() = c
proof
  G is G.size()-ecolorable & G.size() c= G.size() by Th100;
  then A1: G.size() in EColSet(G);
  hereby
    assume G is c-ecolorable;
    then A2: G.eChromaticNum() c= c by Lm15;
    assume A3: for d being Cardinal st G is d-ecolorable holds c c= d;
    now
      let x be set;
      assume x in EColSet(G);
      then consider d being cardinal Subset of G.size() such that
        A4: x = d & G is d-ecolorable;
      thus c c= x by A3, A4;
    end;
    then c c= G.eChromaticNum() by A1, SETFAM_1:5;
    hence G.eChromaticNum() = c by A2, XBOOLE_0:def 10;
  end;
  thus thesis by Lm14, Lm15;
end;
