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
  'not' (Ex(a,PA,G) '&' Ex(b,PA,G)) = All('not' a,PA,G) 'or' All('not' b ,PA,G)
proof
A1: All('not' b,PA,G) = B_INF('not' b,CompF(PA,G)) by BVFUNC_2:def 9;
A2: Ex(b,PA,G) = B_SUP(b,CompF(PA,G)) by BVFUNC_2:def 10;
A3: Ex(a,PA,G) = B_SUP(a,CompF(PA,G)) by BVFUNC_2:def 10;
A4: All('not' a,PA,G) 'or' All('not' b,PA,G) '<' 'not' (Ex(a,PA,G) '&' Ex(b
  ,PA,G))
  proof
    let z be Element of Y;
A5: All('not' b,PA,G).z=TRUE or All('not' b,PA,G).z=FALSE by XBOOLEAN:def 3;
    assume (All('not' a,PA,G) 'or' All('not' b,PA,G)) .z=TRUE;
    then
A6: All('not' a,PA,G).z 'or' All('not' b,PA,G).z=TRUE by BVFUNC_1:def 4;
    per cases by A6,A5,BINARITH:3;
    suppose
A7:   All('not' a,PA,G).z=TRUE;
A8:   now
        assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
        holds ('not' a).x=TRUE);
        then B_INF('not' a,CompF(PA,G)).z = FALSE by BVFUNC_1:def 16;
        hence contradiction by A7,BVFUNC_2:def 9;
      end;
A9:   now
        let x be Element of Y;
        assume x in EqClass(z,CompF(PA,G));
        then ('not' a).x=TRUE by A8;
        then 'not' a.x=TRUE by MARGREL1:def 19;
        hence a.x<>TRUE by MARGREL1:11;
      end;
      thus ('not' (Ex(a,PA,G) '&' Ex(b,PA,G))).z ='not' (Ex(a,PA,G) '&' Ex(b,
      PA,G)).z by MARGREL1:def 19
        .='not' (Ex(a,PA,G).z '&' Ex(b,PA,G).z) by MARGREL1:def 20
        .='not' (FALSE '&' Ex(b,PA,G).z) by A3,A9,BVFUNC_1:def 17
        .='not' (FALSE) by MARGREL1:12
        .=TRUE by MARGREL1:11;
    end;
    suppose
A10:  All('not' b,PA,G).z=TRUE;
A11:  now
        assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
        holds ('not' b).x=TRUE);
        then B_INF('not' b,CompF(PA,G)).z = FALSE by BVFUNC_1:def 16;
        hence contradiction by A10,BVFUNC_2:def 9;
      end;
A12:  now
        let x be Element of Y;
        assume x in EqClass(z,CompF(PA,G));
        then ('not' b).x=TRUE by A11;
        then 'not' b.x=TRUE by MARGREL1:def 19;
        hence b.x<>TRUE by MARGREL1:11;
      end;
      thus ('not' (Ex(a,PA,G) '&' Ex(b,PA,G))).z ='not' (Ex(a,PA,G) '&' Ex(b,
      PA,G)).z by MARGREL1:def 19
        .='not' (Ex(a,PA,G).z '&' Ex(b,PA,G).z) by MARGREL1:def 20
        .='not' (Ex(a,PA,G).z '&' FALSE) by A2,A12,BVFUNC_1:def 17
        .='not' (FALSE) by MARGREL1:12
        .=TRUE by MARGREL1:11;
    end;
  end;
A13: All('not' a,PA,G) = B_INF('not' a,CompF(PA,G)) by BVFUNC_2:def 9;
  'not' (Ex(a,PA,G) '&' Ex(b,PA,G)) '<' (All('not' a,PA,G) 'or' All('not'
  b,PA,G))
  proof
    let z be Element of Y;
    assume ('not' (Ex(a,PA,G) '&' Ex(b,PA,G))).z=TRUE;
    then 'not' (Ex(a,PA,G) '&' Ex(b,PA,G)).z=TRUE by MARGREL1:def 19;
    then (Ex(a,PA,G) '&' Ex(b,PA,G)).z=FALSE by MARGREL1:11;
    then
A14: Ex(a,PA,G).z '&' Ex(b,PA,G).z=FALSE by MARGREL1:def 20;
    per cases by A14,MARGREL1:12;
    suppose
A15:  Ex(a,PA,G).z=FALSE;
A16:  now
        let x be Element of Y;
        assume x in EqClass(z,CompF(PA,G));
        then a.x<>TRUE by A3,A15,BVFUNC_1:def 17;
        then a.x=FALSE by XBOOLEAN:def 3;
        then 'not' a.x=TRUE by MARGREL1:11;
        hence ('not' a).x=TRUE by MARGREL1:def 19;
      end;
      thus (All('not' a,PA,G) 'or' All('not' b,PA,G)) .z = All('not' a,PA,G).z
      'or' All('not' b,PA,G).z by BVFUNC_1:def 4
        .= TRUE 'or' All('not' b,PA,G).z by A13,A16,BVFUNC_1:def 16
        .= TRUE by BINARITH:10;
    end;
    suppose
A17:  Ex(b,PA,G).z=FALSE;
A18:  now
        let x be Element of Y;
        assume x in EqClass(z,CompF(PA,G));
        then b.x<>TRUE by A2,A17,BVFUNC_1:def 17;
        then b.x=FALSE by XBOOLEAN:def 3;
        then 'not' b.x=TRUE by MARGREL1:11;
        hence ('not' b).x=TRUE by MARGREL1:def 19;
      end;
      thus (All('not' a,PA,G) 'or' All('not' b,PA,G)) .z = All('not' a,PA,G).z
      'or' All('not' b,PA,G).z by BVFUNC_1:def 4
        .= All('not' a,PA,G).z 'or' TRUE by A1,A18,BVFUNC_1:def 16
        .= TRUE by BINARITH:10;
    end;
  end;
  hence thesis by A4,BVFUNC_1:15;
end;
