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 Th2:
  a '&' (a 'imp' b) '<' b
proof
  let z be Element of Y;
A1: (a '&' (a 'imp' b)).z =a.z '&' (a 'imp' b).z by MARGREL1:def 20
    .=a.z '&' ('not' a 'or' b).z by BVFUNC_4:8
    .=a.z '&' (('not' a).z 'or' b.z) by BVFUNC_1:def 4
    .=a.z '&' ('not' a).z 'or' a.z '&' b.z by XBOOLEAN:8
    .=a.z '&' 'not' a.z 'or' a.z '&' b.z by MARGREL1:def 19
    .=FALSE 'or' a.z '&' b.z by XBOOLEAN:138
    .=a.z '&' b.z;
  assume
A2: (a '&' (a 'imp' b)).z=TRUE;
  now
    assume b.z<>TRUE;
    then a.z '&' b.z = FALSE '&' a.z by XBOOLEAN:def 3
      .= FALSE;
    hence contradiction by A2,A1;
  end;
  hence thesis;
end;
