reserve r,s,t,u for Real;

theorem Th25:
  for X being RealLinearSpace, A being symmetric Subset of X, x
  being Point of X st x in A holds -x in A
proof
  let X be RealLinearSpace, A be symmetric Subset of X, x be Point of X such
  that
A1: x in A;
  A = -A by Def5
    .= (-1)*A;
  then consider v being Point of X such that
A2: x = (-1)*v and
A3: v in A by A1;
  (-1)*x = (-1)*(-1)*v by A2,RLVECT_1:def 7
    .= v by RLVECT_1:def 8;
  hence thesis by A3,RLVECT_1:16;
end;
