reserve Al for QC-alphabet;
reserve i,j,n,k,l for Nat;
reserve a for set;
reserve T,S,X,Y for Subset of CQC-WFF(Al);
reserve p,q,r,t,F,H,G for Element of CQC-WFF(Al);
reserve s for QC-formula of Al;
reserve x,y for bound_QC-variable of Al;
reserve f,g for FinSequence of [:CQC-WFF(Al),Proof_Step_Kinds:];

theorem Th32:
  p in Cn(X) iff ex f st f is_a_proof_wrt X & Effect(f) = p
proof
  thus p in Cn(X) implies ex f st f is_a_proof_wrt X & Effect(f) = p
  proof
    assume p in Cn(X);
then  p in {F: ex f st f is_a_proof_wrt X & Effect(f) = F} by Th31;
then  ex F st F=p & ex f st f is_a_proof_wrt X & Effect(f) = F;
    hence thesis;
  end;
  thus (ex f st f is_a_proof_wrt X & Effect(f) = p) implies p in Cn(X)
  proof
    given f such that
A1: f is_a_proof_wrt X & Effect(f) = p;
 p in {F: ex f st f is_a_proof_wrt X & Effect(f) = F} by A1;
    hence thesis by Th31;
  end;
end;
