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

theorem
  for x, y, z being object holds <% x, y, z %> * <% 2, 1, 0 %> = <% z, y, x %>
proof
  let x,y,z be object;
  rng <% 2, 1, 0 %> = {2,1,0} by Th8
    .= {0,1,2} by ENUMSET1:60
    .= dom <% x, y, z %> by Th8;
  then A1: dom(<% x, y, z %> * <% 2, 1, 0 %>) = dom <% 2, 1, 0 %> by RELAT_1:27
    .= {0,1,2} by Th8;
  then A2: dom(<% x, y, z %> * <% 2, 1, 0 %>) = dom <% z, y, x %> by Th8;
  now
    let a be object;
    assume A3: a in dom(<% x, y, z %> * <% 2, 1, 0 %>);
    then per cases by A1, ENUMSET1:def 1;
    suppose A4: a = 0;
      thus (<% x, y, z %> * <% 2, 1, 0 %>).a
         = <% x, y, z %>.(<% 2, 1, 0 %>.a) by A3, FUNCT_1:12
        .= <% z, y, x %>.a by A4;
    end;
    suppose A5: a = 1;
      thus (<% x, y, z %> * <% 2, 1, 0 %>).a
         = <% x, y, z %>.(<% 2, 1, 0 %>.a) by A3, FUNCT_1:12
        .= <% z, y, x %>.a by A5;
    end;
    suppose A6: a = 2;
      thus (<% x, y, z %> * <% 2, 1, 0 %>).a
         = <% x, y, z %>.(<% 2, 1, 0 %>.a) by A3, FUNCT_1:12
        .= <% z, y, x %>.a by A6;
    end;
  end;
  hence thesis by A2, FUNCT_1:2;
end;
