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 Th14:
  Valid(p '&' 'not' p,J).v = FALSE
proof
A1: now
    assume (Valid(p,J)).v = TRUE;
    then 'not'(Valid(p,J).v) = FALSE by MARGREL1:11;
    hence (Valid(p,J).v) '&' 'not'(Valid(p,J).v) = FALSE by MARGREL1:12;
  end;
A2: Valid(p,J).v = FALSE implies (Valid(p,J).v) '&' 'not' (Valid(p,J).v) =
  FALSE by MARGREL1:12;
  Valid(p '&' 'not' p,J).v = (Valid(p,J).v) '&' (Valid('not' p,J).v) by Th12
    .= (Valid(p,J).v) '&' 'not'(Valid(p,J).v) by Th10;
  hence thesis by A1,A2,XBOOLEAN:def 3;
end;
