
theorem Th68:
  for C being Coherence_Space, x,y being set st {x,y} c= union C &
  not {x,y} in C holds {x,y} in 'not' C
proof
  let C be Coherence_Space, x,y be set;
  assume that
A1: {x,y} c= union C and
A2: not {x,y} in C;
  now
    let a be Element of C;
    x in a or not x in a;
    then consider z being set such that
A3: x in a & z = x or not x in a & z = y;
    take z;
    thus {x,y} /\ a c= {z}
    proof
      let v be object;
      assume
A4:   v in {x,y} /\ a;
      then
A5:   v in {x,y} by XBOOLE_0:def 4;
A6:   v in a by A4,XBOOLE_0:def 4;
      per cases by A5,TARSKI:def 2;
      suppose
        v = x;
        hence thesis by A3,A4,TARSKI:def 1,XBOOLE_0:def 4;
      end;
      suppose
A7:     v = y;
        then x in a implies {x,y} c= a by A6,ZFMISC_1:32;
        hence thesis by A2,A3,A7,CLASSES1:def 1,TARSKI:def 1;
      end;
    end;
  end;
  hence thesis by A1;
end;
