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 Th20:
  for L1 being LinearFunc holds -L1 is LinearFunc
proof
  let L1 be LinearFunc;
  consider g1 being Real such that
A1: for p holds L1.p = g1*p by FDIFF_1:def 3;
A2: L1 is total by FDIFF_1:def 3;
  now
    let p;
    reconsider pp=p as Element of REAL by XREAL_0:def 1;
    thus (-L1).p = -L1.pp by A2,RFUNCT_1:58
      .= -g1*p by A1
      .= (-g1)*p;
  end;
  hence thesis by A2,FDIFF_1:def 3;
end;
