
theorem Th66:
  for C being Coherence_Space holds union 'not' C = union C
proof
  let C be Coherence_Space;
  hereby
    let x be object;
    assume x in union 'not' C;
    then consider a being set such that
A1: x in a and
A2: a in 'not' C by TARSKI:def 4;
    a c= union C by A2,Th65;
    hence x in union C by A1;
  end;
  let x be object;
  assume x in union C;
  then
A3: {x} c= union C by ZFMISC_1:31;
  for a being Element of C
   ex z being set st {x} /\ a c= {z}
  proof let a be Element of C;
    consider z being object such that
A4:  {x} /\ a c= {z} by XBOOLE_1:17;
   reconsider z as set by TARSKI:1;
   take z;
   thus thesis by A4;
  end;
  then x in {x} & {x} in 'not' C by A3,ZFMISC_1:31;
  hence thesis by TARSKI:def 4;
end;
