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 Th41:
  for Y being non empty Subset-Family of [:X,X:] st
  Y c= subbasis_Pervin_uniformity(SF) holds
  (meet Y)~ = meet (Y[~])
  proof
    let Y be non empty Subset-Family of [:X,X:];
    assume
A1: Y c= subbasis_Pervin_uniformity(SF);
    thus (meet Y)~ c= meet (Y[~])
    proof
      let x be object;
      assume
A3:   x in (meet Y)~;
      then consider a,b be object such that
      a in X and
      b in X and
A4:   [a,b] = x by ZFMISC_1:def 2;
A5:   [b,a] in meet Y by A3,A4,RELAT_1:def 7;
      Y is non empty;
      then consider y be object such that
A6:   y in Y;
      reconsider y as Element of Y by A6;
      reconsider z = y as Relation of X;
A7:   y~ in Y[~];
      now
        let Z be set;
        assume
A8:     Z in Y[~]; then
A9:     Z in Y by A1,Th40;
        reconsider T = Z as Relation of X by A8;
        T in subbasis_Pervin_uniformity(SF) by A9,A1;
        then consider A be Element of SF such that
A10:    T = block_Pervin_uniformity(A);
        T~ = T by A10,Th39; then
        [b,a] in T~ by A9,A5,SETFAM_1:def 1;
        hence [a,b] in Z by RELAT_1:def 7;
      end;
      hence thesis by A7,A4,SETFAM_1:def 1;
    end;
      let x be object;
      assume
A11:  x in meet (Y[~]);
      then consider a,b be object such that
      a in X and
      b in X and
A12:  [a,b] = x by ZFMISC_1:def 2;
      now
        let Z be set;
        assume
A13:    Z in Y;
        then reconsider T = Z as Relation of X;
        reconsider R = T as Element of Y by A13;
        R~ = T~;
        then T~ in Y[~];
        then [a,b] in T~ by A11,A12,SETFAM_1:def 1;
        hence [b,a] in Z by RELAT_1:def 7;
      end;
      then [b,a] in meet Y by SETFAM_1:def 1;
      hence thesis by A12,RELAT_1:def 7;
  end;
