reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th21:
  for R1 being RestFunc holds -R1 is RestFunc
proof
  let R1 be RestFunc;
A1: R1 is total by FDIFF_1:def 2;
  then
A2: dom R1 = REAL by PARTFUN1:def 2;
A3: for h being 0-convergent non-zero Real_Sequence
  holds -(R1/*h) = (-R1)/*h
  proof
    let h be 0-convergent non-zero Real_Sequence;
    rng h c= dom R1 by A2,VALUED_0:def 3;
    hence thesis by RFUNCT_2:10;
  end;
  now
    let h be 0-convergent non-zero Real_Sequence;
A4: (h")(#)(R1/*h) is convergent by FDIFF_1:def 2;
A5: (h")(#)((-R1)/*h) = (h")(#)(-(R1/*h)) by A3
      .= - ((h")(#)(R1/*h)) by SEQ_1:19;
    hence (h")(#)((-R1)/*h) is convergent by A4,SEQ_2:9;
A6: lim ((h")(#)(R1/*h)) = 0 by FDIFF_1:def 2;
    thus lim ((h")(#)((-R1)/*h)) = - (lim ((h")(#)(R1/*h))) by A4,A5,SEQ_2:10
      .= 0 by A6;
  end;
  hence thesis by A1,FDIFF_1:def 2;
end;
