reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;
reserve n for Nat,
        S for Subset-Family of REAL;
reserve n       for Nat,
        a,b,c,d for Element of REAL n;
reserve n for non zero Nat;
reserve n     for non zero Nat,
        x,y,z for Element of REAL n;

theorem Th56:
  (the set of all |. x.i - y.i .| where
                                  i is Element of Seg n) is real-membered &
  (the set of all |. x.i - y.i .| where i is Element of Seg n) = rng abs(x-y)
  proof
    set SA = the set of all |.x.i - y.i.| where i is Element of Seg n;
    now
      let t be object;
      assume t in SA;
      then ex i be Element of Seg n st t = |.x.i-y.i.|;
      hence t is real;
    end;
    hence SA is real-membered by MEMBERED:def 3;
A1: SA c= rng abs(x-y)
    proof
      let t be object;
      assume t in SA;
      then consider i be Element of Seg n such that
A2:   t = |.x.i-y.i.|;
A3:   t = |.(x-y).i.| & dom abs(x-y) = Seg n by A2,RVSUM_1:27,FINSEQ_2:124;
      reconsider f = (x-y) as complex-valued Function;
      t = |.f.|.i by A3,VALUED_1:18;
      hence t in rng abs(x-y) by A3,FUNCT_1:def 3;
    end;
      for t be object st t in rng abs (x-y) holds t in SA
      proof
        let t be object;
        assume t in rng abs (x-y);
        then consider i be object such that
A4:     i in dom abs (x-y) and
A5:     t = (abs (x-y)).i by FUNCT_1:def 3;
A6:     i is Element of Seg n by A4,FINSEQ_2:124;
        reconsider f = (x-y) as complex-valued Function;
        t = |.(x-y).i.| & (x-y).i = x.i - y.i by A4,A5,VALUED_1:18,RVSUM_1:27;
        hence thesis by A6;
      end;
      then rng abs (x-y) c= SA;
    hence thesis by A1;
  end;
