reserve X for non empty set;
reserve e for set;
reserve x for Element of X;
reserve f,g for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve F for Function of RAT,S;
reserve p,q for Rational;
reserve r for Real;
reserve n,m for Nat;
reserve A,B for Element of S;

theorem Th9:
  for C being non empty set, f being PartFunc of C,ExtREAL holds -f = (-1)(#)f
proof
  let C be non empty set;
  let f be PartFunc of C,ExtREAL;
A1: dom (-f) = dom f by MESFUNC1:def 7;
A2: dom ((-1)(#)f) = dom f by MESFUNC1:def 6;
 for x being Element of C st x in dom f holds (-f).x = ((-1)(#)f).x
  proof
    let x be Element of C;
    assume
A3: x in dom f;
then
A4:((-1)(#) f).x=( -1)*(f.x) by A2,MESFUNC1:def 6;
((-1)(#)f).x = ((-jj)(#)f).x
      .= (-( 1.))*(f.x) by SUPINF_2:2,A4
      .= -( 1.)*(f.x) by XXREAL_3:92
      .= -( 1)*(f.x)
      .= -(f.x) by XXREAL_3:81;
    hence thesis by A1,A3,MESFUNC1:def 7;
  end;
  hence thesis by A1,A2,PARTFUN1:5;
end;
