reserve Y for non empty set;
reserve Y for non empty set;
reserve Y for non empty set;

theorem
  for a being Function of Y,BOOLEAN holds a 'eqv' O_el(Y) = 'not' a
proof
  let a be Function of Y,BOOLEAN;
    let x be Element of Y;
    (a 'eqv' O_el(Y)).x =((a 'imp' O_el(Y)) '&' (O_el(Y) 'imp' a)).x by
BVFUNC_4:7
      .=(('not' a 'or' O_el(Y)) '&' (O_el(Y) 'imp' a)).x by BVFUNC_4:8
      .=(('not' a 'or' O_el(Y)) '&' ('not' O_el(Y) 'or' a)).x by BVFUNC_4:8
      .=('not' a '&' ('not' O_el(Y) 'or' a)).x by BVFUNC_1:9
      .=('not' a '&' (I_el(Y) 'or' a)).x by BVFUNC_1:2
      .=('not' a '&' I_el(Y)).x by BVFUNC_1:10
      .=('not' a).x by BVFUNC_1:6;
    hence thesis;
end;
