reserve N,M,K for ExtNat;
reserve X for ext-natural-membered set;

theorem
  <% 0, 1, 2 %> = id 3
proof
  A1: dom <% 0, 1, 2 %> = dom id 3 by Th8, CARD_1:51;
  now
    let x be object;
    assume x in dom <% 0, 1, 2 %>;
    then A2: x in 3 & x in {0,1,2} by A1, Th8;
    then per cases by ENUMSET1:def 1;
    suppose x = 0;
      hence <% 0, 1, 2 %>.x = (id 3).x by A2, FUNCT_1:18;
    end;
    suppose x = 1;
      hence <% 0, 1, 2 %>.x = (id 3).x by A2, FUNCT_1:18;
    end;
    suppose x = 2;
      hence <% 0, 1, 2 %>.x = (id 3).x by A2, FUNCT_1:18;
    end;
  end;
  hence thesis by A1, FUNCT_1:2;
end;
