reserve Omega for set;
reserve X, Y, Z, p,x,y,z for set;
reserve D, E for Subset of Omega;
reserve f for Function;
reserve m,n for Nat;
reserve r,r1 for Real;
reserve seq for Real_Sequence;
reserve F for Field_Subset of X;
reserve ASeq,BSeq for SetSequence of Omega;
reserve A1 for SetSequence of X;

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;
