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
  J,v |= 'not' p iff not J,v |= p
proof
A1: now
    assume not J,v |= p;
    then not Valid(p,J).v = TRUE;
    then Valid(p,J).v = FALSE by XBOOLEAN:def 3;
    then 'not'(Valid(p,J).v) = TRUE by MARGREL1:11;
    then ('not' Valid(p,J)).v = TRUE by MARGREL1:def 19;
    then Valid('not' p,J).v = TRUE by Lm1;
    hence J,v |= 'not' p;
  end;
  now
    assume J,v |= 'not' p;
    then Valid('not' p,J).v = TRUE;
    then ('not' Valid(p,J)).v = TRUE by Lm1;
    then 'not'(Valid(p,J).v) = TRUE by MARGREL1:def 19;
    then Valid(p,J).v = FALSE by MARGREL1:11;
    hence not J,v |= p;
  end;
  hence thesis by A1;
end;
