reserve Y for non empty set;

theorem
  for a,b being Function of Y,BOOLEAN holds b 'imp' ((b 'imp' a)
  'imp' a)=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
        b.x=TRUE;
        hence ('not' b.x 'or' b.x)=TRUE by BINARITH:10;
      end;
      case
        b.x=FALSE;
        then 'not' b.x 'or' b.x =TRUE 'or' FALSE by MARGREL1:11
          .=TRUE by BINARITH:10;
        hence ('not' b.x 'or' b.x)=TRUE;
      end;
    end;
A2: 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;
    (b 'imp' ((b 'imp' a) 'imp' a)).x = 'not' b.x 'or' ((b 'imp' a) 'imp'
    a).x by BVFUNC_1:def 8
      .= 'not' b.x 'or' ('not' (b 'imp' a).x 'or' a.x) by BVFUNC_1:def 8
      .= 'not' b.x 'or' ('not' ('not' b.x 'or' a.x) 'or' a.x) by BVFUNC_1:def 8
      .= 'not' b.x 'or' ((a.x 'or' b.x) '&' TRUE) by A2,XBOOLEAN:9
      .= 'not' b.x 'or' (a.x 'or' b.x) by MARGREL1:14
      .= 'not' b.x 'or' b.x 'or' a.x by BINARITH:11
      .=TRUE by A1,BINARITH:10;
    hence thesis by BVFUNC_1:def 11;
end;
