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 Th2:
  p <==> q iff for s holds Meas(p`1,s).p`2 = Meas(q`1,s).q`2
proof
  thus p <==> q implies for s holds Meas(p`1,s).p`2 = Meas(q`1,s).q`2
  proof
    assume
A1: p <==> q;
    let s;
    q |- p by A1; then
A2: Meas(q`1,s).q`2 <= Meas(p`1,s).p`2;
    p |- q by A1;
    then Meas(p`1,s).p`2 <= Meas(q`1,s).q`2;
    hence thesis by A2,XXREAL_0:1;
  end;
  assume
A3: for s holds Meas(p`1,s).p`2 = Meas(q`1,s).q`2;
  thus p |- q
  by A3;
  let s;
  thus thesis by A3;
end;
