reserve Y for non empty set;

theorem
  for a,b being Function of Y,BOOLEAN holds 'not' a 'imp' (a 'imp'
  b)=I_el(Y)
proof
  let a,b be Function of Y,BOOLEAN;
    let x be Element of Y;
A1: now
      per cases by XBOOLEAN:def 3;
      case
        a.x=TRUE;
        hence ('not' a.x 'or' a.x)=TRUE by BINARITH:10;
      end;
      case
        a.x=FALSE;
        then 'not' a.x 'or' a.x =TRUE 'or' FALSE by MARGREL1:11
          .=TRUE by BINARITH:10;
        hence ('not' a.x 'or' a.x)=TRUE;
      end;
    end;
    ('not' a 'imp' (a 'imp' b)).x ='not' ('not' a).x 'or' (a 'imp' b).x by
BVFUNC_1:def 8
      .='not' ('not' a).x 'or' ('not' a.x 'or' b.x) by BVFUNC_1:def 8
      .=a.x 'or' ('not' a.x 'or' b.x) by MARGREL1:def 19
      .=TRUE 'or' b.x by A1,BINARITH:11
      .=TRUE by BINARITH:10;
    hence thesis by BVFUNC_1:def 11;
end;
