reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;

theorem
  abs f = max+f + max-f
proof
  abs f = R_EAL(abs f);
  then abs f = |. R_EAL f .| by Th44;
  then abs f = max+(R_EAL f) + max-(R_EAL f) by MESFUNC2:24;
  then abs f = R_EAL(max+f)+max-(R_EAL f) by Th30;
  then abs f = R_EAL(max+f)+R_EAL(max-f) by Th30;
  then abs f = R_EAL(max+f + max-f) by Th23;
  hence thesis;
end;
