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

theorem Th44:
  for D be non empty set, F be PartFunc of D,REAL, X be set holds
  (max+ F)|X = max+ (F|X)
proof
  let D be non empty set, F be PartFunc of D,REAL, X be set;
A1: dom((max+ F)|X) = dom(max+ F) /\ X by RELAT_1:61;
A2: dom(max+ F) /\ X= dom F /\ X by Def10
    .= dom(F|X) by RELAT_1:61;
A3: dom(F|X) = dom(max+ (F|X)) by Def10;
  now
    let d be Element of D;
    assume
A4: d in dom((max+ F)|X);
    then
A5: d in dom(max+ F) by A1,XBOOLE_0:def 4;
    thus ((max+ F)|X).d = (max+ F).d by A4,FUNCT_1:47
      .=max+(F.d) by A5,Def10
      .=max+((F|X).d) by A1,A2,A4,FUNCT_1:47
      .=(max+ (F|X)).d by A1,A2,A3,A4,Def10;
  end;
  hence thesis by A2,A3,PARTFUN1:5,RELAT_1:61;
end;
