reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem
  for f1 be PartFunc of W,REAL n holds (-1)(#)f1 = -f1
proof
  let f1 be PartFunc of W,REAL n;
A1:  dom((-1)(#)f1) = dom f1 by VALUED_2:def 39; then
A2: dom((-1)(#)f1) = dom(-f1) by Def3;
  now let x be Element of W;
    assume A3: x in dom((-1)(#)f1);
A4:  f1.x = f1/.x by A1,A3,PARTFUN1:def 6;
A5: (-f1)/.x = -(f1/.x) by A2,A3,Def3;
    (f1[#](-1)).x = (-1)(#)(f1.x) by A3,VALUED_2:def 39;
    hence ((-1)(#)f1).x = (-f1).x by A4,A2,A3,A5,PARTFUN1:def 6;
  end;
  hence thesis by A2,PARTFUN1:5;
end;
