reserve X for set,
        A for Subset of X,
        R,S for Relation of X;
reserve QUS for Quasi-UniformSpace;
reserve SUS for Semi-UniformSpace;
reserve T for TopSpace;

theorem Th31:
  for UT being non empty strict Quasi-UniformSpace,
  FMT being non empty strict FMT_Space_Str, x being Element of  FMT st
  FMT = FMT_Space_Str(#the carrier of UT,Neighborhood(UT) #) holds
  ex y being Element of UT st x = y &
  U_FMT x = Neighborhood y
  proof
    let UT be non empty strict Quasi-UniformSpace,
        FMT be non empty strict FMT_Space_Str,
        x be Element of FMT;
    assume
A1: FMT = FMT_Space_Str(#the carrier of UT,Neighborhood(UT) #);
    U_FMT x = (the BNbd of FMT).x by FINTOPO2:def 6;
    then consider y be Element of UT such that
A2: x = y and
A3: U_FMT x = (Neighborhood(UT)).y by A1;
    (Neighborhood(UT)).y = Neighborhood(y) by Def5;
    hence thesis by A2,A3;
  end;
