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 Th26:
  All(a 'imp' b,PA,G) = All('not' a 'or' b,PA,G)
proof
A1: All('not' a 'or' b,PA,G) '<' All(a 'imp' b,PA,G)
  proof
    let z be Element of Y;
    assume
A2: All('not' a 'or' b,PA,G).z=TRUE;
A3: now
      assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
      holds ('not' a 'or' b).x=TRUE);
      then B_INF('not' a 'or' b,CompF(PA,G)).z = FALSE by BVFUNC_1:def 16;
      hence contradiction by A2,BVFUNC_2:def 9;
    end;
    for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds (a
    'imp' b).x=TRUE
    proof
      let x be Element of Y;
A4:   ('not' a).x=TRUE or ('not' a).x=FALSE by XBOOLEAN:def 3;
      assume x in EqClass(z,CompF(PA,G));
      then ('not' a 'or' b).x=TRUE by A3;
      then
A5:   ('not' a).x 'or' b.x=TRUE by BVFUNC_1:def 4;
      per cases by A5,A4,BINARITH:3;
      suppose
        ('not' a).x=TRUE;
        then 'not' a.x=TRUE by MARGREL1:def 19;
        hence (a 'imp' b).x = TRUE 'or' b.x by BVFUNC_1:def 8
          .=TRUE by BINARITH:10;
      end;
      suppose
        b.x=TRUE;
        hence (a 'imp' b).x =('not' a.x) 'or' TRUE by BVFUNC_1:def 8
          .=TRUE by BINARITH:10;
      end;
    end;
    then B_INF(a 'imp' b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 16;
    hence thesis by BVFUNC_2:def 9;
  end;
  All(a 'imp' b,PA,G) '<' All('not' a 'or' b,PA,G)
  proof
    let z be Element of Y;
    assume
A6: All(a 'imp' b,PA,G).z=TRUE;
A7: now
      assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
      holds (a 'imp' b).x=TRUE);
      then B_INF(a 'imp' b,CompF(PA,G)).z = FALSE by BVFUNC_1:def 16;
      hence contradiction by A6,BVFUNC_2:def 9;
    end;
    now
      let x be Element of Y;
A8:   ('not' a.x)=TRUE or ('not' a.x)=FALSE by XBOOLEAN:def 3;
      assume x in EqClass(z,CompF(PA,G));
      then (a 'imp' b).x=TRUE by A7;
      then
A9:   ('not' a.x) 'or' b.x=TRUE by BVFUNC_1:def 8;
      per cases by A9,A8,BINARITH:3;
      suppose
        ('not' a.x)=TRUE;
        then ('not' a).x=TRUE by MARGREL1:def 19;
        hence ('not' a 'or' b).x =TRUE 'or' b.x by BVFUNC_1:def 4
          .=TRUE by BINARITH:10;
      end;
      suppose
        b.x=TRUE;
        hence ('not' a 'or' b).x=('not' a).x 'or' TRUE by BVFUNC_1:def 4
          .=TRUE by BINARITH:10;
      end;
    end;
    then B_INF('not' a 'or' b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 16;
    hence thesis by BVFUNC_2:def 9;
  end;
  hence thesis by A1,BVFUNC_1:15;
end;
