reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;
reserve US for UniformSpace;
reserve R for Relation of X;
reserve SF for Subset-Family of X, A for Element of SF;

theorem
  X is non empty implies block_Pervin_uniformity(A) =
  {[x,y] where x,y is Element of X: x in A iff y in A}
  proof
    assume
A1: X is non empty;
    set S = {[x,y] where x,y is Element of X: x in A iff y in A};
A2: block_Pervin_uniformity(A) c= S
    proof
      let x be object;
      assume
A3:   x in block_Pervin_uniformity(A); then
A4:   x in [: A, A :] or x in [: X \ A, X \ A :] by XBOOLE_0:def 3;
      consider a,b be object such that
A9:   a in X and
A10:  b in X and
A11:  x = [a,b] by A3,ZFMISC_1:def 2;
      (a in A & b in A) or (a in X \ A & b in X \ A) by A4,A11,ZFMISC_1:87;
      then (a in A & b in A) or ((a in X & not a in A) &
        (b in X & not b in A)) by XBOOLE_0:def 5;
      hence thesis by A9,A10,A11;
    end;
    S c= block_Pervin_uniformity(A)
    proof
      let x be object;
      assume x in S;
      then consider a,b be Element of X such that
A12:  x = [a,b] and
A13:  a in A iff b in A;
      x in [:A,A:] or (a in X\A & b in X\A)
        by A1,A13,A12,ZFMISC_1:87,XBOOLE_0:def 5;
      then x in [: A, A:] or x in [: X\A, X\A :] by A12,ZFMISC_1:87;
      hence thesis by XBOOLE_0:def 3;
    end;
    hence thesis by A2;
  end;
