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

theorem
  for a,b being Function of Y,BOOLEAN holds a '<' (a '&' b) 'eqv'
  (b '&' a) 'eqv' a
proof
  let a,b be Function of Y,BOOLEAN;
  let z be Element of Y;
  assume
A1: a.z=TRUE;
A2: ((a '&' b) 'eqv' (a '&' b)).z =(((a '&' b) 'imp' (a '&' b)) '&' ((a '&'
  b) 'imp' (a '&' b))).z by BVFUNC_4:7
    .=('not'( a '&' b) 'or' (a '&' b)).z by BVFUNC_4:8
    .=(I_el(Y)).z by BVFUNC_4:6
    .=TRUE by BVFUNC_1:def 11;
  ((a '&' b) 'eqv' (b '&' a) 'eqv' a).z =((((a '&' b) 'eqv' (a '&' b))
  'imp' a) '&' (a 'imp' ((a '&' b) 'eqv' (a '&' b)))).z by BVFUNC_4:7
    .=(((a '&' b) 'eqv' (a '&' b)) 'imp' a).z '&' (a 'imp' ((a '&' b) 'eqv'
  (a '&' b))).z by MARGREL1:def 20
    .=('not'( (a '&' b) 'eqv' (a '&' b)) 'or' a).z '&' (a 'imp' ((a '&' b)
  'eqv' (a '&' b))).z by BVFUNC_4:8
    .=('not'( (a '&' b) 'eqv' (a '&' b)) 'or' a).z '&' ('not' a 'or' ((a '&'
  b) 'eqv' (a '&' b))).z by BVFUNC_4:8
    .=(('not'( (a '&' b) 'eqv' (a '&' b))).z 'or' a.z) '&' ('not' a 'or' (
  (a '&' b) 'eqv' (a '&' b))).z by BVFUNC_1:def 4
    .=(('not'( (a '&' b) 'eqv' (a '&' b))).z 'or' a.z) '&' (('not' a).z
  'or' ((a '&' b) 'eqv' (a '&' b)).z) by BVFUNC_1:def 4
    .=('not' ((a '&' b) 'eqv' (a '&' b)).z 'or' a.z) '&' (('not' a).z 'or'
  ((a '&' b) 'eqv' (a '&' b)).z) by MARGREL1:def 19
    .=(FALSE 'or' a.z) '&' (('not' a).z 'or' TRUE) by A2
    .=a.z '&' (('not' a).z 'or' TRUE) 
    .=TRUE '&' a.z 
    .=TRUE by A1;
  hence thesis;
end;
