
theorem Th28:
  for X be non empty set, f be PartFunc of X,ExtREAL, A be set
  holds max+(f|A)=max+f|A & max-(f|A)=max-f|A
proof
  let X be non empty set;
  let f be PartFunc of X,ExtREAL;
  let A be set;
A1: dom max+(f|A) = dom(f|A) by MESFUNC2:def 2
    .= dom f /\ A by RELAT_1:61
    .= dom max+f /\ A by MESFUNC2:def 2
    .= dom(max+f|A) by RELAT_1:61;
  for x being Element of X st x in dom max+(f|A) holds (max+(f|A)).x = (
  max+f|A).x
  proof
    let x be Element of X;
    assume
A2: x in dom max+(f|A);
    then
A3: (max+f|A).x = (max+f).x by A1,FUNCT_1:47;
A4: x in dom max+f /\ A by A1,A2,RELAT_1:61;
    then
A5: x in dom max+f by XBOOLE_0:def 4;
A6: x in A by A4,XBOOLE_0:def 4;
    (max+(f|A)).x = max((f|A).x,0) by A2,MESFUNC2:def 2
      .= max(f.x,0) by A6,FUNCT_1:49;
    hence thesis by A5,A3,MESFUNC2:def 2;
  end;
  hence max+(f|A) = max+f|A by A1,PARTFUN1:5;
A7: dom max-(f|A) = dom(f|A) by MESFUNC2:def 3
    .= dom f /\ A by RELAT_1:61
    .= dom max-f /\ A by MESFUNC2:def 3
    .= dom(max-f|A) by RELAT_1:61;
  for x being Element of X st x in dom max-(f|A) holds (max-(f|A)).x = (
  max-f|A).x
  proof
    let x be Element of X;
    assume
A8: x in dom max-(f|A);
    then
A9: (max-f|A).x = (max-f).x by A7,FUNCT_1:47;
A10: x in dom max-f /\ A by A7,A8,RELAT_1:61;
    then
A11: x in dom max-f by XBOOLE_0:def 4;
A12: x in A by A10,XBOOLE_0:def 4;
    (max-(f|A)).x = max(-(f|A).x,0) by A8,MESFUNC2:def 3
      .= max(-f.x,0) by A12,FUNCT_1:49;
    hence thesis by A11,A9,MESFUNC2:def 3;
  end;
  hence thesis by A7,PARTFUN1:5;
end;
