
theorem Th86:
  for C1,C2 being Coherence_Space for x,y being set holds x U+ y
  in C1 "\/" C2 iff x in C1 & y = {} or x = {} & y in C2
proof
  let C1,C2 be Coherence_Space, x,y be set;
A1: now
    given a being Element of C1 such that
A2: x U+ y = a U+ {};
    x = a by A2,Th80;
    hence x in C1 & y = {} by A2,Th80;
  end;
A3: now
    given a being Element of C2 such that
A4: x U+ y = {} U+ a;
    y = a by A4,Th80;
    hence y in C2 & x = {} by A4,Th80;
  end;
  x U+ y in C1 "\/" C2 iff x U+ y in the set of all
a U+ {} where a is Element of C1 or x U+ y in the set of all
{} U+ b where b is Element of C2 by XBOOLE_0:def 3;
hence thesis by A1,A3;
end;
