reserve x,y for set;
reserve s,s1,s2,s4,r,r1,r2 for Real;
reserve n,m,i,j for Element of NAT;
reserve p for Element of NAT;

theorem Th11:
  for r,s being Real holds |.r-s.|=|.s-r.|
proof
  let r,s be Real;
  per cases by XXREAL_0:1;
  suppose
    r>s;
    then s-r<0 by XREAL_1:49;
    then |.s-r.|=-(s-r) by ABSVALUE:def 1
      .=r-s;
    hence thesis;
  end;
  suppose
    r=s;
    hence thesis;
  end;
  suppose
    r<s;
    then r-s<0 by XREAL_1:49;
    then |.r-s.|=-(r-s) by ABSVALUE:def 1
      .=s-r;
    hence thesis;
  end;
end;
