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