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

theorem
  for T being non empty TopSpace, a being Point of T, P being constant
  Path of a, a holds P = I[01] --> a
proof
  let T be non empty TopSpace, a be Point of T, P be constant Path of a, a;
  set IT = I[01] --> a;
A1: dom P = the carrier of I[01] by FUNCT_2:def 1;
A2: a,a are_connected;
A3: for x be object st x in the carrier of I[01] holds P.x = IT.x
  proof
    0 in { r : 0 <= r & r <= 1 };
    then
A4: 0 in the carrier of I[01] by BORSUK_1:40,RCOMP_1:def 1;
    let x be object;
    assume
A5: x in the carrier of I[01];
    P.0 = a by A2,Def2;
    then P.x = a by A1,A5,A4,FUNCT_1:def 10
      .= IT.x by A5,FUNCOP_1:7;
    hence thesis;
  end;
  dom IT = the carrier of I[01] by FUNCT_2:def 1;
  hence thesis by A1,A3,FUNCT_1:2;
end;
