reserve A,X for non empty set;
reserve f for PartFunc of [:X,X:],REAL;
reserve a for Real;

theorem Th23:
  for X being non empty set, f being PartFunc of [:X,X:],REAL st
  f is nonnegative Reflexive discerning holds
  low_toler(f,0)[*] = low_toler(f,0)
proof
  let X be non empty set, f be PartFunc of [:X,X:],REAL such that
A1: f is nonnegative Reflexive discerning;
  now
    let p be object such that
A2: p in low_toler(f,0)[*];
    consider x,y being object such that
A3: p = [x,y] by A2,RELAT_1:def 1;
    low_toler(f,0) reduces x,y by A2,A3,REWRITE1:20;
    then consider r being RedSequence of low_toler(f,0) such that
A4: r.1 = x & r.len r = y by REWRITE1:def 3;
A5: now
      let i be Nat;
      assume i in dom r & i+1 in dom r;
      then [r.i,r.(i+1)] in low_toler(f,0) by REWRITE1:def 2;
      hence r.i = r.(i+1) by A1,Th20;
    end;
A6: x is Element of X by A2,A3,ZFMISC_1:87;
    0 < len r by REWRITE1:def 2;
    then 0+1 <= len r by NAT_1:13;
    then 1 in dom r & len r in dom r by FINSEQ_3:25;
    then r.1 = r. len r by A5,Th2;
    hence p in low_toler(f,0) by A1,A3,A4,A6,Th21;
  end;
  then
  low_toler(f,0) c= low_toler(f,0)[*] & low_toler(f,0)[*] c= low_toler(f,
  0) by LANG1:18;
  hence thesis by XBOOLE_0:def 10;
end;
