reserve Y for non empty set;
reserve Y for non empty set;
reserve Y for non empty set;
reserve Y for non empty set,
  a,b,c,d,e,f,g for Function of Y,BOOLEAN;
reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
  'imp' c) '<' (a 'imp' (b '&' c))
proof
  let a,b,c be Function of Y,BOOLEAN;
  let z be Element of Y;
A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by
MARGREL1:def 20
    .=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8
    .=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8
    .=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4
    .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4;
  assume
A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE;
  now
A3: (a 'imp' (b '&' c)).z =('not' a 'or' (b '&' c)).z by BVFUNC_4:8
      .=('not' a).z 'or' (b '&' c).z by BVFUNC_1:def 4
      .=('not' a).z 'or' (b.z '&' c.z) by MARGREL1:def 20
      .='not' a.z 'or' (b.z '&' c.z) by MARGREL1:def 19;
    assume
A4: (a 'imp' (b '&' c)).z<>TRUE;
    'not' a.z=TRUE or 'not' a.z=FALSE by XBOOLEAN:def 3;
    then
A5: ('not' a).z=FALSE by A4,A3,MARGREL1:def 19;
A6: (b.z '&' c.z)=TRUE or (b.z '&' c.z)=FALSE by XBOOLEAN:def 3;
    now
      per cases by A4,A3,A6,MARGREL1:12;
      case
        b.z=FALSE;
        hence thesis by A2,A1,A5;
      end;
      case
        c.z=FALSE;
        then (('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) =b.z '&'
        'not' b.z by A5,MARGREL1:def 19
          .=FALSE by XBOOLEAN:138;
        hence thesis by A2,A1;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
