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;
reserve t for TColoring of G;

theorem Th195:
  G is edgeless iff G.tChromaticNum() = 1
proof
  hereby
    assume A1: G is edgeless;
    now
      let c;
      assume A2: G is c-tcolorable;
      assume not 1 c= c;
      then c in 1 by ORDINAL1:16;
      then c = 0 by CARD_1:49, TARSKI:def 1;
      hence contradiction by A2;
    end;
    hence G.tChromaticNum() = 1 by A1, Th190;
  end;
  assume G.tChromaticNum() = 1;
  then G is 1-tcolorable by Th186, Th187;
  hence thesis;
end;
