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 ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;

theorem
  max+(f|Y)=max+f|Y & max-(f|Y)=max-f|Y
proof
A1: dom max+(f|Y) = dom(f|Y) by RFUNCT_3:def 10
    .= dom f /\ Y by RELAT_1:61
    .= dom max+f /\ Y by RFUNCT_3:def 10
    .= dom(max+f|Y) by RELAT_1:61;
  for x being Element of X st x in dom max+(f|Y) holds (max+(f|Y)).x = (
  max+f|Y).x
  proof
    let x be Element of X;
    assume
A2: x in dom max+(f|Y);
    then
A3: x in dom max+f /\ Y by A1,RELAT_1:61;
    then
A4: x in Y by XBOOLE_0:def 4;
A5: x in dom max+f by A3,XBOOLE_0:def 4;
A6: (max+(f|Y)).x = max+((f|Y).x) by A2,RFUNCT_3:def 10
      .= max(f.x,0) by A4,FUNCT_1:49;
    (max+f|Y).x = (max+f).x by A1,A2,FUNCT_1:47
      .= max+(f.x) by A5,RFUNCT_3:def 10;
    hence thesis by A6;
  end;
  hence max+(f|Y) = max+f|Y by A1,PARTFUN1:5;
A7: dom max-(f|Y) = dom(f|Y) by RFUNCT_3:def 11
    .= dom f /\ Y by RELAT_1:61
    .= dom max-f /\ Y by RFUNCT_3:def 11
    .= dom(max-f|Y) by RELAT_1:61;
  for x being Element of X st x in dom max-(f|Y) holds (max-(f|Y)).x = (
  max-f|Y).x
  proof
    let x be Element of X;
    assume
A8: x in dom max-(f|Y);
    then
A9: x in dom max-f /\ Y by A7,RELAT_1:61;
    then
A10: x in Y by XBOOLE_0:def 4;
A11: x in dom max-f by A9,XBOOLE_0:def 4;
A12: (max-(f|Y)).x = max-((f|Y).x) by A8,RFUNCT_3:def 11
      .= max(-f.x,0) by A10,FUNCT_1:49;
    (max-f|Y).x = (max-f).x by A7,A8,FUNCT_1:47
      .= max-(f.x) by A11,RFUNCT_3:def 11;
    hence thesis by A12;
  end;
  hence thesis by A7,PARTFUN1:5;
end;
