
theorem Th70:
  for C being Coherence_Space holds 'not' 'not' C = C
proof
  let C be Coherence_Space;
  thus 'not' 'not' C c= C by Lm7;
  let a be object;
    reconsider aa=a as set by TARSKI:1;
  assume
A1: a in C;
A2: union C = union 'not' C & union 'not' C = union 'not' 'not' C by Th66;
  now
    let x,y be set;
    assume that
A3: x in aa and
A4: y in aa;
A5: x in union C by A1,A3,TARSKI:def 4;
    {x,y} c= aa by A3,A4,ZFMISC_1:32;
    then {x,y} in C by A1,CLASSES1:def 1;
    then
A6: x <> y implies not {x,y} in 'not' C by Th67;
    y in union C by A1,A4,TARSKI:def 4;
    then
A7: {x,y} c= union C by A5,ZFMISC_1:32;
    {x,x} = {x} by ENUMSET1:29;
    hence {x,y} in 'not' 'not' C by A2,A5,A7,A6,Th68,COH_SP:4;
  end;
  hence thesis by COH_SP:6;
end;
