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

theorem Th5:
  for T being non empty TopSpace, a being Point of T, P being
  constant Path of a, a holds P + P = P
proof
  let T be non empty TopSpace, a be Point of T, P be constant Path of a, a;
A1: the carrier of I[01] = dom P by FUNCT_2:def 1;
A2: for x be object st x in the carrier of I[01] holds P.x = (P+P).x
  proof
    let x be object;
    assume
A3: x in the carrier of I[01];
    then reconsider p = x as Point of I[01];
    x in { r : 0 <= r & r <= 1 } by A3,BORSUK_1:40,RCOMP_1:def 1;
    then consider r being Real such that
A4: r = x and
A5: 0 <= r and
A6: r <= 1;
    per cases;
    suppose
A7:   r < 1/2;
      then
A8:   r * 2 < 1/2 * 2 by XREAL_1:68;
      2 * r >= 0 by A5,XREAL_1:127;
      then 2 * r in { e where e is Real: 0 <= e & e <= 1 } by A8;
      then 2 * r in the carrier of I[01] by BORSUK_1:40,RCOMP_1:def 1;
      then P.(2*r) = P.p by A1,FUNCT_1:def 10;
      hence thesis by A4,A7,Def5;
    end;
    suppose
A9:   r >= 1/2;
      then r * 2 >= 1/2 * 2 by XREAL_1:64;
      then 2 * r >= 1 + 0;
      then
A10:  2 * r - 1 >= 0 by XREAL_1:19;
      r * 2 <= 1 * 2 by A6,XREAL_1:64;
      then 2 * r - 1 <= 2 - 1 by XREAL_1:13;
      then 2 * r - 1 in { e where e is Real : 0 <= e & e <= 1 } by A10;
      then 2 * r - 1 in the carrier of I[01] by BORSUK_1:40,RCOMP_1:def 1;
      then P.(2*r-1) = P.p by A1,FUNCT_1:def 10;
      hence thesis by A4,A9,Def5;
    end;
  end;
  dom (P + P) = the carrier of I[01] by FUNCT_2:def 1;
  hence thesis by A1,A2,FUNCT_1:2;
end;
