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;

theorem
  a 'imp' b '<' (a '&' c) 'imp' b
proof
  let z be Element of Y;
A1: (a 'imp' b).z =('not' a 'or' b).z by BVFUNC_4:8
    .=('not' a).z 'or' b.z by BVFUNC_1:def 4;
  assume
A2: (a 'imp' b).z=TRUE;
  now
    assume ((a '&' c) 'imp' b).z<>TRUE;
    then ((a '&' c) 'imp' b).z=FALSE by XBOOLEAN:def 3;
    then ('not'( a '&' c) 'or' b).z=FALSE by BVFUNC_4:8;
    then ('not'( a '&' c)).z 'or' b.z=FALSE by BVFUNC_1:def 4;
    then ('not' a 'or' 'not' c) .z 'or' b.z=FALSE by BVFUNC_1:14;
    then (('not' c).z 'or' ('not' a).z) 'or' b.z=FALSE by BVFUNC_1:def 4;
    then ('not' c).z 'or' (('not' a).z 'or' b.z)=FALSE;
    hence contradiction by A2,A1;
  end;
  hence thesis;
end;
