reserve Al for QC-alphabet;
reserve i,j,k for Nat,
  A,D for non empty set;
reserve f1,f2 for Element of Funcs(Valuations_in(Al,A),BOOLEAN),
  x,x1,y for bound_QC-variable of Al,
  v,v1 for Element of Valuations_in(Al,A);
reserve ll for CQC-variable_list of k,Al;
reserve p,q,s,t for Element of CQC-WFF(Al),
  J for interpretation of Al,A,
  P for QC-pred_symbol of k,Al,
  r for Element of relations_on A;

theorem Th24:
  J,v |= p => q iff (J,v |= p implies J,v |= q)
proof
  thus J,v |= p => q & J,v |= p implies J,v |= q by Th23;
  assume J,v |= p implies J,v |= q;
  then Valid(p, J).v = TRUE implies Valid(q, J).v = TRUE;
  then Valid(p, J).v = FALSE or Valid(q, J).v = TRUE by XBOOLEAN:def 3;
  hence thesis by Th23;
end;
