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 Th12:
  x in Union A1 iff ex n st x in A1.n
proof
  set DX = union rng A1;
  for x holds x in DX iff ex n st x in A1.n
  proof
    let x;
    thus x in DX implies ex n st x in A1.n
    proof
      assume x in DX;
      then consider Y such that
A1:   x in Y and
A2:   Y in rng A1 by TARSKI:def 4;
      consider p being object such that
A3:   p in dom A1 and
A4:   Y = A1.p by A2,FUNCT_1:def 3;
      p in NAT by A3,FUNCT_2:def 1;
      hence thesis by A1,A4;
    end;
    thus (ex n st x in A1.n) implies x in DX
    proof
      given n such that
A5:   x in A1.n;
      n in NAT by ORDINAL1:def 12;
      then n in dom A1 by FUNCT_2:def 1;
      then A1.n in rng A1 by FUNCT_1:def 3;
      hence thesis by A5,TARSKI:def 4;
    end;
  end;
  hence thesis;
end;
