reserve X, Y for non empty set;

theorem
  Umf(X,X) is symmetric transitive reflexive
proof
  thus Umf(X,X) is symmetric
  proof
    let x,y be Element of X;
    thus Umf(X,X).(x,y) = Umf(X,X). [x,y] .= 1 by FUZZY_4:21
      .= Umf(X,X). [y,x] by FUZZY_4:21
      .= Umf(X,X).(y,x);
  end;
  thus Umf(X,X) is transitive
  proof
    let x,y,z be Element of X;
    Umf(X,X). [x,y] "/\" Umf(X,X). [y,z] = min(1,Umf(X,X). [y,z]) by FUZZY_4:20
      .= min(1,1) by FUZZY_4:20
      .= 1;
    then Umf(X,X). [x,y] "/\" Umf(X,X). [y,z] <= Umf(X,X). [x,z] by FUZZY_4:20;
    hence thesis by LFUZZY_0:3;
  end;
  thus Umf(X,X) is reflexive
  proof
    let x be Element of X;
    Umf(X,X). [x,x] = 1 by FUZZY_4:21;
    hence thesis;
  end;
end;
