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 Th7:
 for r,s,x holds Al2-Cast(r 'or' s) = Al2-Cast(r) 'or' Al2-Cast(s) &
 Al2-Cast(Ex(x,r)) = Ex(Al2-Cast(x),Al2-Cast(r))
proof
  let r,s,x;
A1: Al2-Cast('not' r) = 'not' Al2-Cast(r) &
   Al2-Cast('not' s) = 'not' Al2-Cast(s) by QC_TRANS:8;
  thus Al2-Cast(r 'or' s)
    = Al2-Cast('not' ('not' r '&' 'not' s)) by QC_LANG2:def 3
   .= 'not' Al2-Cast('not' r '&' 'not' s) by QC_TRANS:8
   .= 'not' ('not' Al2-Cast(r) '&' 'not' Al2-Cast(s)) by A1,QC_TRANS:8
   .= Al2-Cast(r) 'or' Al2-Cast(s) by QC_LANG2:def 3;
  thus Al2-Cast(Ex(x,r)) = Al2-Cast('not' All(x,'not' r)) by QC_LANG2:def 5
   .= 'not' Al2-Cast(All(x,'not' r)) by QC_TRANS:8
   .= 'not' All(Al2-Cast(x),Al2-Cast('not' r)) by QC_TRANS:8
   .= 'not' All(Al2-Cast(x),'not' Al2-Cast(r)) by QC_TRANS:8
   .= Ex(Al2-Cast(x),Al2-Cast(r)) by QC_LANG2:def 5;
end;
