reserve k,j,n for Nat,
  r for Real;
reserve x,x1,x2,y for Element of REAL n;
reserve f for real-valued FinSequence;

theorem Th2:
  for f being complex-valued Function holds abs -f = abs f
  proof
    let f be complex-valued Function;
A1: dom abs(-f) = dom(-f) by VALUED_1:def 11;
A2: dom abs(f) = dom f by VALUED_1:def 11;
A3: dom -f = dom f by VALUED_1:8;
    now
      let x be object;
      assume x in dom abs(-f);
      thus abs(-f).x = |.(-f).x.| by VALUED_1:18
      .= |.-f.x.| by VALUED_1:8
      .= |.f.x.| by COMPLEX1:52
      .= abs(f).x by VALUED_1:18;
    end;
    hence thesis by A1,A2,A3,FUNCT_1:2;
  end;
