
theorem
  for C being Coherence_Space for x,y being set holds [x,y] in Web 'not'
  C iff x in union C & y in union C & (x = y or not [x,y] in Web C)
proof
  let C be Coherence_Space, x,y be set;
A1: {x,y} c= union C & not {x,y} in C implies {x,y} in 'not' C by Th68;
A2: union 'not' C = union C by Th66;
  x <> y & {x,y} in C implies not {x,y} in 'not' C by Th67;
  hence [x,y] in Web 'not' C implies x in union C & y in union C & (x = y or
  not [x,y] in Web C) by A2,COH_SP:5,ZFMISC_1:87;
  assume that
A3: x in union C and
A4: y in union C and
A5: x = y or not [x,y] in Web C;
  x = y & {x} in 'not' C & {x} = {x,y} or not {x,y} in C by A2,A3,A5,COH_SP:4,5
,ENUMSET1:29;
  hence thesis by A1,A3,A4,COH_SP:5,ZFMISC_1:32;
end;
