reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem Th10:
  for A being set holds
  varcl union A = union the set of all varcl a where a is Element of A
proof
  let A be set;
  set X = the set of all varcl a where a is Element of A;
A1: union A c= union X
  proof
    let x be object;
    assume x in union A;
    then consider Y such that
A2: x in Y and
A3: Y in A by TARSKI:def 4;
    reconsider Y as Element of A by A3;
A4: Y c= varcl Y by Def1;
    varcl Y in X;
    hence thesis by A2,A4,TARSKI:def 4;
  end;
  now
    let x,y be set;
    assume [x,y] in union X;
    then consider Y being set such that
A5: [x,y] in Y and
A6: Y in X by TARSKI:def 4;
    ex a being Element of A st ( Y = varcl a) by A6;
    then
A7: x c= Y by A5,Def1;
    Y c= union X by A6,ZFMISC_1:74;
    hence x c= union X by A7;
  end;
  hence varcl union A c= union X by A1,Def1;
  let x be object;
  assume x in union X;
  then consider Y being set such that
A8: x in Y and
A9: Y in X by TARSKI:def 4;
  consider a being Element of A such that
A10: Y = varcl a by A9;
  A is empty or A is not empty;
  then a in A or a is empty by SUBSET_1:def 1;
  then a c= union A by ZFMISC_1:74;
  then Y c= varcl union A by A10,Th9;
  hence thesis by A8;
end;
