reserve a, r, s for Real;

theorem Th25:
  for i being Integer, f being Path of R^1(0),R^1(i) holds Ciso.i
  = Class(EqRel(Tunit_circle(2),c[10]),CircleMap*f)
proof
  let i be Integer;
  let f be Path of R^1(0),R^1(i);
  set P = CircleMap*f;
A1: P.0 = CircleMap.(f.j0) by FUNCT_2:15
    .= CircleMap.R^1(0) by BORSUK_2:def 4
    .= CircleMap.0 by TOPREALB:def 2
    .= |[ cos(2*PI*0), sin(2*PI*0) ]| by TOPREALB:def 11
    .= c[10] by SIN_COS:31,TOPREALB:def 8;
  P.1 = CircleMap.(f.j1) by FUNCT_2:15
    .= CircleMap.R^1(i) by BORSUK_2:def 4
    .= CircleMap.i by TOPREALB:def 2
    .= |[ cos(2*PI*i+0), sin(2*PI*i+0) ]| by TOPREALB:def 11
    .= |[ cos(0), sin(2*PI*i+0) ]| by COMPLEX2:9
    .= c[10] by COMPLEX2:8,SIN_COS:31,TOPREALB:def 8;
  then reconsider P as Loop of c[10] by A1,BORSUK_2:def 4;
A2: cLoop(i) = CircleMap * ExtendInt(i) by Th20;
A3: cLoop(i),P are_homotopic
  proof
    reconsider J = R^1 as non empty interval SubSpace of R^1;
    reconsider r0 = R^1(0), ri = R^1(i) as Point of J;
    reconsider O = ExtendInt(i), ff = f as Path of r0,ri;
    reconsider G = R1Homotopy(O,ff) as Function of [:I[01],I[01]:],R^1;
    take F = CircleMap*G;
    thus F is continuous;
    let s be Point of I[01];
    thus F.(s,0) = CircleMap.(G.(s,j0)) by Lm5,BINOP_1:18
      .= CircleMap.((1-j0) * (ExtendInt(i)).s + j0 * f.s) by TOPALG_2:def 4
      .= (cLoop(i)).s by A2,FUNCT_2:15;
    thus F.(s,1) = CircleMap.(G.(s,j1)) by Lm5,BINOP_1:18
      .= CircleMap.((1-j1) * (ExtendInt(i)).s + j1 * f.s) by TOPALG_2:def 4
      .= P.s by FUNCT_2:15;
    thus F.(0,s) = CircleMap.(G.(j0,s)) by Lm5,BINOP_1:18
      .= CircleMap.((1-s) * (ExtendInt(i)).j0 + s * f.j0) by TOPALG_2:def 4
      .= CircleMap.((1-s) * R^1(0) + s * f.j0) by BORSUK_2:def 4
      .= CircleMap.((1-s) * R^1(0) + s * R^1(0)) by BORSUK_2:def 4
      .= CircleMap.((1-s) * 0 + s * 0) by TOPREALB:def 2
      .= c[10] by TOPREALB:32;
    thus F.(1,s) = CircleMap.(G.(j1,s)) by Lm5,BINOP_1:18
      .= CircleMap.((1-s) * (ExtendInt(i)).j1 + s * f.j1) by TOPALG_2:def 4
      .= CircleMap.((1-s) * R^1(i) + s * f.j1) by BORSUK_2:def 4
      .= CircleMap.((1-s) * R^1(i) + s * R^1(i)) by BORSUK_2:def 4
      .= CircleMap.i by TOPREALB:def 2
      .= c[10] by TOPREALB:32;
  end;
  thus Ciso.i = Class(EqRel(TUC,c[10]),cLoop(i)) by Def5
    .= Class(EqRel(TUC,c[10]),CircleMap*f) by A3,TOPALG_1:46;
end;
