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 Th10:
  p |- q iff [Class(PropRel Q,p),Class(PropRel Q,q)] in OrdRel Q
proof
  [p,p] in PropRel Q by Def12;
  then
A1: p in Class(PropRel Q,p) by EQREL_1:19;
  [q,q] in PropRel Q by Def12;
  then
A2: q in Class(PropRel Q,q) by EQREL_1:19;
A3: Class(PropRel Q,p) in Class PropRel Q & Class(PropRel Q,q) in Class
  PropRel Q by EQREL_1:def 3;
  thus p |- q implies [Class(PropRel Q,p),Class(PropRel Q,q)] in OrdRel Q
  proof
    assume p |- q;
    then for p1,q1 holds p1 in Class(PropRel Q,p) & q1 in Class(PropRel Q,q)
    implies p1 |- q1 by A1,A2,A3,Th9;
    hence thesis by A3,Def13;
  end;
  thus thesis by A1,A2,Def13;
end;
