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 Th9:
  for B,C st B in Class PropRel Q & C in Class PropRel Q for a,b,c
  ,d being Element of Prop Q holds a in B & b in B & c in C & d in C & a |- c
  implies b |- d
proof
  let B,C such that
A1: B in Class PropRel Q and
A2: C in Class PropRel Q;
  let a,b,c,d be Element of Prop Q;
  assume that
A3: a in B & b in B and
A4: c in C & d in C;
  assume
A5: a |- c;
  let s;
  ex y being object st y in Prop Q & C = Class(PropRel Q,y)
by A2,EQREL_1:def 3;
  then [c,d] in PropRel Q by A4,EQREL_1:22;
  then c <==> d by Def12;
  then
A6: Meas(c`1,s).c`2 = Meas(d`1,s).d`2 by Th2;
  ex x being object st x in Prop Q & B = Class(PropRel Q,x)
by A1,EQREL_1:def 3;
  then [a,b] in PropRel Q by A3,EQREL_1:22;
  then a <==> b by Def12;
  then Meas(a`1,s).a`2 = Meas(b`1,s).b`2 by Th2;
  hence thesis by A5,A6;
end;
