theorem
  for x being set st x in dom f holds (max+f.x = f.x or max+f.x = 0) &
  (max-f.x = -(f.x) or max-f.x = 0)
proof
  let x be set;
  assume
A1: x in dom f;
  then max+(R_EAL f).x = (R_EAL f).x or max+(R_EAL f).x = 0. by MESFUNC2:18;
  hence max+f.x = f.x or max+f.x = 0 by Th30;
A2: max+(R_EAL f) = max+f & max-(R_EAL f) = max-f by Th30;
  max-(R_EAL f).x = -((R_EAL f).x) or max-(R_EAL f).x = 0. by A1,MESFUNC2:18;
  hence thesis by A2,SUPINF_2:2;
end;
