reserve X, Y for non empty set;

theorem
  Zmf(X,X) is symmetric antisymmetric transitive
proof
  thus Zmf(X,X) is symmetric
  proof
    let x,y be Element of X;
    Zmf(X,X). [x,y] = 0 & Zmf(X,X). [y,x] = 0 by FUZZY_4:21;
    hence thesis;
  end;
  thus Zmf(X,X) is antisymmetric
  proof
    let x,y be Element of X;
    Zmf(X,X). [x,y] = 0 by FUZZY_4:21;
    hence thesis;
  end;
  thus Zmf(X,X) is transitive
  proof
    let x,y,z be Element of X;
A1: Zmf(X,X). [x,z] = 0 by FUZZY_4:20;
    Zmf(X,X). [x,y] "/\" Zmf(X,X). [y,z] = min(0, Zmf(X,X). [y,z]) by
FUZZY_4:20
      .= min(0,0) by FUZZY_4:20
      .= 0;
    hence thesis by A1,LFUZZY_0:3;
  end;
end;
