reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  a,b,c,u for Function of Y,BOOLEAN,
  PA for a_partition of Y;

theorem
  All(a 'or' b,PA,G) '<' Ex(a,PA,G) 'or' All(b,PA,G)
proof
  let z be Element of Y;
  assume
A1: All(a 'or' b,PA,G).z=TRUE;
  per cases;
  suppose
    ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x=TRUE;
    then B_SUP(a,CompF(PA,G)).z = TRUE by BVFUNC_1:def 17;
    then Ex(a,PA,G).z=TRUE by BVFUNC_2:def 10;
    hence (Ex(a,PA,G) 'or' All(b,PA,G)) .z = TRUE 'or' All(b,PA,G).z by
BVFUNC_1:def 4
      .=TRUE by BINARITH:10;
  end;
  suppose
    (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b.x
=TRUE) & not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x=TRUE
    );
    then B_INF(b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 16;
    then All(b,PA,G).z=TRUE by BVFUNC_2:def 9;
    hence (Ex(a,PA,G) 'or' All(b,PA,G)) .z =Ex(a,PA,G).z 'or' TRUE by
BVFUNC_1:def 4
      .=TRUE by BINARITH:10;
  end;
  suppose
A2: not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds b.x=TRUE) & not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) &
    a.x=TRUE);
    then consider x1 being Element of Y such that
A3: x1 in EqClass(z,CompF(PA,G)) and
A4: b.x1<>TRUE;
A5: a.x1<>TRUE by A2,A3;
A6: b.x1=FALSE by A4,XBOOLEAN:def 3;
    (a 'or' b).x1 = a.x1 'or' b.x1 by BVFUNC_1:def 4
      .= FALSE 'or' FALSE by A5,A6,XBOOLEAN:def 3
      .= FALSE;
    hence thesis by A1,A3,Lm1;
  end;
end;
