reserve T for non empty TopSpace,
  a, b, c, d for Point of T;

theorem Th50:
  for P being Path of a, b, Q being constant Path of b, b st a,b
  are_connected holds RePar (P, 1RP) = P + Q
proof
  let P be Path of a, b, Q be constant Path of b, b;
  set f = RePar (P, 1RP), g = P + Q;
  assume
A1: a, b are_connected;
A2: b, b are_connected;
  for p being Element of I[01] holds f.p = g.p
  proof
    0 in the carrier of I[01] by BORSUK_1:43;
    then
A3: 0 in dom Q by FUNCT_2:def 1;
    let p be Element of I[01];
    p in the carrier of I[01];
    then
A4: p in dom 1RP by FUNCT_2:def 1;
A5: f.p = (P * 1RP).p by A1,Def4,Th47
      .= P. (1RP.p) by A4,FUNCT_1:13;
    per cases;
    suppose
A6:   p <= 1/2;
      then f.p = P.(2*p) by A5,Def5
        .= g.p by A1,A6,BORSUK_2:def 5;
      hence thesis;
    end;
    suppose
A7:   p > 1/2;
      then 2*p - 1 is Point of I[01] by Th4;
      then 2*p - 1 in the carrier of I[01];
      then
A8:   2*p - 1 in dom Q by FUNCT_2:def 1;
      f.p = P.1 by A5,A7,Def5
        .= b by A1,BORSUK_2:def 2
        .= Q.0 by A2,BORSUK_2:def 2
        .= Q.(2*p - 1) by A3,A8,FUNCT_1:def 10
        .= g.p by A1,A7,BORSUK_2:def 5;
      hence thesis;
    end;
  end;
  hence thesis by FUNCT_2:63;
end;
