reserve X1,x,y,z for set,
  n,m for Nat,
  X for non empty set;
reserve A,B for Event of Borel_Sets,
  D for Subset of REAL;
reserve Q for QM_Str;
reserve A1 for Element of Obs Q;
reserve s for Element of Sts Q;
reserve E for Event of Borel_Sets;
reserve ASeq for SetSequence of Borel_Sets;
reserve Q for Quantum_Mechanics;
reserve s for Element of Sts Q;
reserve x1 for Element of X1;
reserve Inv for Function of X1,X1;
reserve p,q,r,p1,q1 for Element of Prop Q;
reserve B,C for Subset of Prop Q;

theorem Th11:
  for B,C st B in Class PropRel Q & C in Class PropRel Q for p1,q1
  holds p1 in B & q1 in B & 'not' p1 in C implies 'not' q1 in C
proof
  let B,C such that
A1: B in Class PropRel Q and
A2: C in Class PropRel Q;
  consider y being object such that
A3: y in Prop Q and
A4: C = Class(PropRel Q,y) by A2,EQREL_1:def 3;
  let p1,q1;
  assume that
A5: p1 in B & q1 in B and
A6: 'not' p1 in C;
  ex x being object st x in Prop Q & B = Class(PropRel Q,x)
by A1,EQREL_1:def 3;
  then [p1,q1] in PropRel Q by A5,EQREL_1:22;
  then
A7: p1 <==> q1 by Def12;
  now
    reconsider E1 = q1`2`, E = p1`2` as Event of Borel_Sets
    by PROB_1:20;
    let s;
    set r1 = Meas(p1`1,s).E, r2 = Meas(q1`1,s).E1;
    1 - r1 = Meas(p1`1,s).p1`2 by Th1
      .= Meas(q1`1,s).q1`2 by A7,Th2
      .= 1 - r2 by Th1;
    hence
    Meas(('not' p1)`1,s).('not' p1)`2 = Meas(('not' q1)`1,s).('not' q1)`2
    ;
  end;
  then
A10: 'not' p1 <==> 'not' q1 by Th2;
  reconsider q = y as Element of Prop Q by A3;
  ['not' p1,q] in PropRel Q by A4,A6,EQREL_1:19;
  then 'not' p1 <==> q by Def12;
  then q <==> 'not' q1 by A10,Th7;
  then ['not' q1,q] in PropRel Q by Def12;
  hence thesis by A4,EQREL_1:19;
end;
