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

theorem
  for x,y,z being object st x <> y & x <> z & y <> z
  holds <% x, y, z %> is one-to-one
proof
  let x,y,z be object;
  assume A1: x <> y & x <> z & y <> z;
  now
    let x1, x2 be object;
    assume A2: x1 in dom <%x,y,z%> & x2 in dom <%x,y,z%> &
      <%x,y,z%>.x1 = <%x,y,z%>.x2;
    then x1 in {0,1,2} & x2 in {0,1,2} by Th8;
    then per cases by ENUMSET1:def 1;
    suppose x1 = 0 & (x2 = 0 or x2 = 1 or x2 = 2);
      hence x1 = x2 by A1, A2; :: by contradiction
    end;
    suppose x1 = 1 & (x2 = 0 or x2 = 1 or x2 = 2);
      hence x1 = x2 by A1, A2; :: by contradiction
    end;
    suppose x1 = 2 & (x2 = 0 or x2 = 1 or x2 = 2);
      hence x1 = x2 by A1, A2; :: by contradiction
    end;
  end;
  hence thesis by FUNCT_1:def 4;
end;
