reserve x, r for Real;
reserve A for symmetrical Subset of COMPLEX;
reserve F,G for PartFunc of REAL, REAL;

theorem
  F is odd implies r (#) F is odd
proof
A1: dom F=dom( r (#) F) by VALUED_1:def 5;
  assume
A2: F is odd;
  for x st x in dom( r (#) F) & -x in dom( r (#) F) holds ( r (#) F).(-x)=
  -( r (#) F).x
  proof
    let x;
    assume that
A3: x in dom( r (#) F) and
A4: -x in dom( r (#) F);
    ( r (#) F).(-x)=r * F.(-x) by A4,VALUED_1:def 5
      .=r * (-F.x) by A2,A1,A3,A4,Def6
      .=-(r * F.x)
      .=-(r (#) F).x by A3,VALUED_1:def 5;
    hence thesis;
  end;
  then r (#) F is with_symmetrical_domain quasi_odd by A2,A1;
  hence thesis;
end;
