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 '&' b) 'or' (c '&' d) '<' a 'or' c
proof
  let z be Element of Y;
A1: ((a '&' b) 'or' (c '&' d)).z =(a '&' b).z 'or' (c '&' d).z by
BVFUNC_1:def 4
    .=(a.z '&' b.z) 'or' (c '&' d).z by MARGREL1:def 20
    .=(a.z '&' b.z) 'or' (c.z '&' (d).z) by MARGREL1:def 20;
  assume
A2: ((a '&' b) 'or' (c '&' d)).z=TRUE;
  now
    assume (a 'or' c).z<>TRUE;
    then (a 'or' c).z=FALSE by XBOOLEAN:def 3;
    then
A3: a.z 'or' c.z=FALSE by BVFUNC_1:def 4;
    (a.z '&' b.z) 'or' (c.z '&' (d).z) =(c.z 'or' (a.z '&' b.z)) '&' ((a.z
    '&' b.z) 'or' (d).z) by XBOOLEAN:9
      .=((a.z 'or' c.z) '&' (c.z 'or' b.z)) '&' ((a.z '&' b.z) 'or' (d).z)
    by XBOOLEAN:9
      .=FALSE by A3;
    hence contradiction by A2,A1;
  end;
  hence thesis;
end;
