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 Th8:
  for f1 be PartFunc of W,REAL-NS n,
      g1 be PartFunc of W,REAL n st f1=g1 holds
    -f1 = -g1
proof
  let f1 be PartFunc of W,REAL-NS n,
      g1 be PartFunc of W,REAL n;
  assume A1: f1=g1;
  dom(-f1) = dom f1 by VFUNCT_1:def 5; then
A2: dom(-f1) = dom(-g1) by A1,Def3;
A3: now
    let x be Element of W;
    assume A4: x in dom(-f1);
    f1/.x=g1/.x by A1,REAL_NS1:def 4; then
A5: -(f1/.x)=-(g1/.x) by REAL_NS1:4;
A6: (-f1)/.x = -(f1/.x) by A4,VFUNCT_1:def 5;
    (-g1)/.x = -(g1/.x) by A2,A4,Def3;
    then (-f1).x = (-g1)/.x by A4,A5,A6,PARTFUN1:def 6;
    hence (-f1).x = (-g1).x by A2,A4,PARTFUN1:def 6;
  end;
  -f1 is PartFunc of W,REAL n by REAL_NS1:def 4;
  hence -f1 = -g1 by A2,A3,PARTFUN1:5;
end;
