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 Th42:
  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);
    then meet (Y[~]) = meet Y by Th40;
    hence thesis by A1,Th41;
  end;
