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

theorem
  for a,b being Function of Y,BOOLEAN holds a '<' (b 'imp' a) 'eqv' a
proof
  let a,b be Function of Y,BOOLEAN;
  let z be Element of Y;
  assume
    a.z=TRUE;
  then
A2: 'not' a.z=FALSE;
  ((b 'imp' a) 'eqv' a).z =(('not' b 'or' a) 'eqv' a).z by BVFUNC_4:8
    .=((('not' b 'or' a) 'imp' a) '&' (a 'imp' ('not' b 'or' a))).z by
BVFUNC_4:7
    .=(('not'( 'not' b 'or' a) 'or' a) '&' (a 'imp' ('not' b 'or' a))).z by
BVFUNC_4:8
    .=(('not'( 'not' b 'or' a) 'or' a) '&' ('not' a 'or' ('not' b 'or' a))).
  z by BVFUNC_4:8
    .=('not'( 'not' b 'or' a) 'or' a).z '&' ('not' a 'or' ('not' b 'or' a)).
  z by MARGREL1:def 20
    .=(('not'( 'not' b 'or' a)).z 'or' a.z) '&' ('not' a 'or' ('not' b
  'or' a)).z by BVFUNC_1:def 4
    .=('not' ('not' b 'or' a).z 'or' a.z) '&' ('not' a 'or' ('not' b 'or'
  a)).z by MARGREL1:def 19
    .=('not'( ('not' b).z 'or' a.z) 'or' a.z) '&' ('not' a 'or' ('not' b
  'or' a)).z by BVFUNC_1:def 4
    .=(('not' 'not' b.z '&' 'not' a.z) 'or' a.z) '&' ('not' a 'or' (
  'not' b 'or' a)).z by MARGREL1:def 19
    .=((b.z '&' 'not' a.z) 'or' a.z) '&' (('not' a).z 'or' ('not' b
  'or' a).z) by BVFUNC_1:def 4
    .=((b.z '&' 'not' a.z) 'or' a.z) '&' (('not' a).z 'or' (('not' b).
  z 'or' a.z)) by BVFUNC_1:def 4
    .=((b.z '&' 'not' a.z) 'or' a.z) '&' (('not' a).z 'or' ('not' b.
  z 'or' a.z)) by MARGREL1:def 19
    .=((b.z '&' 'not' a.z) 'or' a.z) '&' ('not' a.z 'or' ('not' b.
  z 'or' a.z)) by MARGREL1:def 19
    .=TRUE '&' (FALSE 'or' ('not' b.z 'or' TRUE)) by A2
    .=FALSE 'or' ('not' b.z 'or' TRUE) 
    .='not' b.z 'or' TRUE
    .=TRUE;
  hence thesis;
end;
