reserve Al for QC-alphabet,
     PHI for Consistent Subset of CQC-WFF(Al),
     PSI for Subset of CQC-WFF(Al),
     p,q,r,s for Element of CQC-WFF(Al),
     A for non empty set,
     J for interpretation of Al,A,
     v for Element of Valuations_in(Al,A),
     m,n,i,j,k for Element of NAT,
     l for CQC-variable_list of k,Al,
     P for QC-pred_symbol of k,Al,
     x,y for bound_QC-variable of Al,
     z for QC-symbol of Al,
     Al2 for Al-expanding QC-alphabet;
reserve J2 for interpretation of Al2,A,
        Jp for interpretation of Al,A,
        v2 for Element of Valuations_in(Al2,A),
        vp for Element of Valuations_in(Al,A);

theorem Th13:
  for THETA being Consistent Subset of CQC-WFF(Al)
  st PHI c= THETA & PHI is with_examples holds THETA is with_examples
proof
  let THETA be Consistent Subset of CQC-WFF(Al) such that
A1:  PHI c= THETA & PHI is with_examples;
  now
    let x be bound_QC-variable of Al, p be Element of CQC-WFF(Al);
    consider y being bound_QC-variable of Al such that
A2: PHI |- ('not' Ex(x,p) 'or' (p.(x,y))) by A1,GOEDELCP:def 2;
    consider f being FinSequence of CQC-WFF(Al) such that
A3: rng f c= PHI & |- f^<*('not' Ex(x,p) 'or' (p.(x,y)))*>
    by A2, HENMODEL:def 1;
    take y;
    thus THETA |- ('not' Ex(x,p) 'or' (p.(x,y)))
     by A1,A3,HENMODEL:def 1,XBOOLE_1:1;
  end;
  hence thesis by GOEDELCP:def 2;
end;
