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;

theorem Th1:
  abs (x-y) = abs (y-x)
  proof
    now
      dom abs(x-y) = Seg n & dom abs(y-x) = Seg n by FINSEQ_2:124;
      hence dom abs(x-y) = dom abs(y-x);
      thus for i be object st i in dom abs(x-y) holds
        abs(x-y).i = abs(y-x).i
      proof
        let i be object;
        assume i in dom abs(x-y);
        reconsider fxy = x-y,fyx = y - x as complex-valued Function;
A1:     abs fxy.i = |.(x-y).i.| & abs fyx.i = |.(y-x).i.| by VALUED_1:18;
        reconsider j = i as set by TARSKI:1;
        reconsider rx = x,ry = y as Element of n-tuples_on REAL;
A2:     (rx-ry).j = rx.j - ry.j & -(ry-rx).j = -(ry.j-rx.j) by RVSUM_1:27;
        reconsider c1 = (x-y).i, c2 = (y-x).i as ExtReal;
A3:     |.c2.| = |.-c2.| & c1 = -c2 by A2,XXREAL_3:def 3,EXTREAL1:29;
        |.c1.| = |. (x-y).i qua Complex.| & |.c2.| = |.(y-x).i qua Complex.|
          by EXTREAL1:12;
        hence abs(x-y).i = abs(y-x).i by A1,A3;
      end;
    end;
    hence thesis by FUNCT_1:def 11;
  end;
