reserve T,T1,T2,S for non empty TopSpace;
reserve GY for non empty TopSpace,
  r,s for Real;

theorem Th6:
  for T being non empty TopSpace, a being Point of T, P being
  constant Path of a, a holds - P = P
proof
  let T be non empty TopSpace, a be Point of T, P be constant Path of a, a;
A1: dom P = the carrier of I[01] by FUNCT_2:def 1;
A2: for x be object st x in the carrier of I[01] holds P.x = (-P).x
  proof
    let x be object;
    assume
A3: x in the carrier of I[01];
    then reconsider x2 = x as Real;
    reconsider x3 = 1 - x2 as Point of I[01] by A3,Lm5;
    (-P).x = P.x3 by A3,Def6
      .= P.x by A1,A3,FUNCT_1:def 10;
    hence thesis;
  end;
  dom (-P) = the carrier of I[01] by FUNCT_2:def 1;
  hence thesis by A1,A2,FUNCT_1:2;
end;
