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;

theorem Th8:
  p |- q implies 'not' q |- 'not' p
proof
  assume
A1: p |- q;
  let s;
  reconsider E1 = q`2` as Event of Borel_Sets by PROB_1:20;
  reconsider E = p`2` as Event of Borel_Sets by PROB_1:20;
  set p1 = Meas(p`1,s).E, p2 = Meas(q`1,s).E1;
A2: -(1-p1) = p1 -1 & -(1-p2) = p2 -1;
A4: Meas(q`1,s).q`2 = 1 - p2 & Meas(p`1,s).p`2 = 1 - p1 by Th1;
  Meas(p`1,s).p`2 <= Meas(q`1,s).q`2 by A1;
  then p2 -1 <= p1 - 1 by A4,A2,XREAL_1:24;
  hence thesis by XREAL_1:9;
end;
