theorem Th11:
  union rng A1 is Subset of X
proof
  for Y st Y in rng A1 holds Y c= X
  proof
    let Y;
    assume Y in rng A1;
    then consider y being object such that
A1: y in dom A1 and
A2: Y = A1.y by FUNCT_1:def 3;
    reconsider y as Element of NAT by A1,FUNCT_2:def 1;
    Y = A1.y by A2;
    hence thesis;
  end;
  hence thesis by ZFMISC_1:76;
end;
