reserve i,j,k,n,n1,n2,m for Nat;
reserve a,r,x,y for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve C for non empty set;
reserve X for set;

theorem Th21:
  for f being PartFunc of C,REAL holds max-(f)=max+(-f)
proof
  let f be PartFunc of C,REAL;
  dom max+(-f)= dom -f by RFUNCT_3:def 10;
  then
A1: dom max+(-f)= dom f by VALUED_1:8;
A2: dom max-(f) = dom f by RFUNCT_3:def 11;
  for x1 being Element of C st x1 in dom f holds max-(f).x1=max+(-f).x1
  proof
    let x1 be Element of C;
    assume
A3: x1 in dom f;
    then
A4: max+(-f).x1 = max+((-f).x1) by A1,RFUNCT_3:def 10
      .= max((-f).x1,0 ) by RFUNCT_3:def 1
      .= max(-(f.x1),0) by VALUED_1:8;
    max-(f).x1 = max-(f.x1) by A2,A3,RFUNCT_3:def 11
      .= max(-(f.x1),0) by RFUNCT_3:def 2;
    hence thesis by A4;
  end;
  hence thesis by A2,A1,PARTFUN1:5;
end;
