reserve Y for non empty set;

theorem :: Identity law
  for a being Function of Y,BOOLEAN holds a 'imp' a = I_el Y
proof
  let a be Function of Y,BOOLEAN;
    let x be Element of Y;
A1: (a 'imp' a).x = ('not' a.x) 'or' a.x & (I_el Y).x=TRUE by BVFUNC_1:def 8
,def 11;
A2: 'not' FALSE=TRUE by MARGREL1:11;
    now
      per cases by XBOOLEAN:def 3;
      case
        a.x=TRUE;
        hence thesis by A1,BINARITH:10;
      end;
      case
        a.x=FALSE;
        hence thesis by A2,A1,BINARITH:10;
      end;
    end;
    hence thesis;
end;
