reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th2:
  for r be Real holds |.r.| = max+(r) + max-(r)
proof
  let r be Real;
  now
    per cases;
    case
A1:   0<=r;
      then max+(r) = r & max-(r) = 0 by XXREAL_0:def 10;
      hence thesis by A1,ABSVALUE:def 1;
    end;
    case
A2:   r<0;
      then max+(r) = 0 & max-(r) = -r by XXREAL_0:def 10;
      hence thesis by A2,ABSVALUE:def 1;
    end;
  end;
  hence thesis;
end;
