theorem
  x is Element of .: (G,X) iff x is Function of X, the carrier of G
proof
  x is Element of .:(G,X) implies x is Element of Funcs(X, carr(G)) by Th17;
  hence x is Element of .:(G,X) implies x is Function of X, carr(G);
  assume x is Function of X, carr(G);
  then reconsider f = x as Function of X, carr(G);
A1: rng f c= carr(G);
  carr(.:(G,X)) = Funcs(X, carr(G)) & dom f = X by Th17,FUNCT_2:def 1;
  hence thesis by A1,FUNCT_2:def 2;
end;
