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 Th1:
  (a 'or' b) '&' (b 'imp' c) '<' a 'or' c
proof
  let z be Element of Y;
A1: ((a 'or' b) '&' (b 'imp' c)).z =(a 'or' b).z '&' (b 'imp' c).z by
MARGREL1:def 20
    .=(a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8
    .=(a 'or' b).z '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4
    .=(a.z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4;
  assume
A2: ((a 'or' b) '&' (b 'imp' 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;
    c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3;
    then (a.z 'or' b.z) '&' (('not' b).z 'or' c.z) =b.z '&' 'not' b.z by A3,
MARGREL1:def 19
      .=FALSE by XBOOLEAN:138;
    hence contradiction by A2,A1;
  end;
  hence thesis;
end;
