reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem Th24:
  for V being add-associative right_zeroed right_complementable
    non empty addLoopStr
  for f being PartFunc of C,V holds
  -(-f) = f
proof
  let V be add-associative right_zeroed right_complementable
  non empty addLoopStr;
  let f be PartFunc of C,V;
A1: dom(--f) = dom(-f) by Def5;
A2: dom(-f) = dom(f) by Def5;
  now
    let c;
    assume
A3: c in dom(f);
    hence (--f)/.c = -((-f)/.c) by A1,A2,Def5
    .= --(f/.c) by A2,A3,Def5
    .= f/.c;
  end;
  hence thesis by A1,A2,PARTFUN2:1;
end;
