theorem Th1:
  for A, B being set st for x being set st x in A ex K being set st
  K c= B & x c= union K holds union A c= union B
proof
  let A, B be set such that
A1: for x being set st x in A ex K being set st K c= B & x c= union K;
  let a be object;
  assume a in union A;
  then consider Z being set such that
A2: a in Z and
A3: Z in A by TARSKI:def 4;
  consider K being set such that
A4: K c= B and
A5: Z c= union K by A1,A3;
  ex S being set st a in S & S in K by A2,A5,TARSKI:def 4;
  hence thesis by A4,TARSKI:def 4;
end;
