reserve n for Element of NAT,
  i for Integer,
  a, b, r for Real,
  x for Point of TOP-REAL n;

theorem Th43:
  (CircleMap(R^1(1/2)))" = Circle2IntervalL
proof
  reconsider A1 as non empty Subset of R^1;
  set f = CircleMap(R^1(1/2));
  set Y = the carrier of (R^1|A1);
  reconsider f as Function of R^1|A1, TOUCm by Lm19;
  set G = AffineMap(2*PI,0);
A1: dom id Ym = Ym by RELAT_1:45;
A2: rng f = Xm by Lm19,FUNCT_2:def 3;
A3: Y = A1 by PRE_TOPC:8;
A4: now
    let a be object;
    assume
A5: a in dom (Cm*f);
    then reconsider b = a as Point of R^1|A1;
    reconsider c = b as Real;
    consider x, y being Real such that
A6: f.b = |[x,y]| and
A7: y >= 0 implies Cm.(f.b) = 1+arccos x/P2 and
A8: y <= 0 implies Cm.(f.b) = 1-arccos x/P2 by Def14;
A9: f.b = CircleMap.b by A3,FUNCT_1:49
      .= |[ cos(2*PI*c), sin(2*PI*c) ]| by Def11;
    then
A10: y = sin(2*PI*c) by A6,SPPOL_2:1;
A11: 1/2 < c by A3,XXREAL_1:4;
    then 2*PI*(1/2) < 2*PI*c by XREAL_1:68;
    then
A12: PI+2*PI*0 < 2*PI*c;
A13: c < 3/2 by A3,XXREAL_1:4;
    then c-1 < 3/2-1 by XREAL_1:9;
    then
A14: 2*PI*(c-1) <= 2*PI*(1/2) by XREAL_1:64;
    2*PI*c <= 2*PI*(1/2+1) by A13,XREAL_1:64;
    then
A15: 2*PI*c <= PI+2*PI*1;
A16: G.(1-c) = 2*PI*(1-c)+0 by FCONT_1:def 4;
    then
A17: G.(1-c)/(2*PI*1) = (1-c)/1 by XCMPLX_1:91;
A18: x = cos(2*PI*c) by A6,A9,SPPOL_2:1
      .= cos(2*PI*c+2*PI*(-1)) by COMPLEX2:9
      .= cos(2*PI*(c-1));
A19: now
      per cases;
      suppose
A20:    c >= 1;
        then
A21:    1-1 <= c-1 by XREAL_1:9;
        2*PI*c >= 2*PI*1 by A20,XREAL_1:64;
        hence Cm.(f.b) = 1+(2*PI*(c-1))/P2 by A7,A18,A10,A14,A15,A21,
SIN_COS6:16,92
          .= 1+(c-1) by XCMPLX_1:89
          .= b;
      end;
      suppose
A22:    c < 1;
        then 2*PI*c < 2*PI*1 by XREAL_1:68;
        then
A23:    2*PI*c < 2*PI+2*PI*0;
        1-c < 1-1/2 by A11,XREAL_1:15;
        then
A24:    2*PI*(1-c) <= 2*PI*(1/2) by XREAL_1:64;
A25:    1-1 <= 1-c by A22,XREAL_1:15;
        arccos x = arccos cos(2*PI*c) by A6,A9,SPPOL_2:1
          .= arccos cos(-2*PI*c) by SIN_COS:31
          .= arccos cos(2*PI*(-c)+2*PI*1) by COMPLEX2:9
          .= G.(1-c) by A16,A25,A24,SIN_COS6:92;
        hence Cm.(f.b) = b by A8,A10,A17,A12,A23,SIN_COS6:12;
      end;
    end;
    thus (Cm*f).a = Cm.(f.b) by A5,FUNCT_1:12
      .= (id Ym).a by A19;
  end;
  dom (Cm*f) = Ym by FUNCT_2:def 1;
  then Cm = f qua Function" by A2,A1,A4,FUNCT_1:2,FUNCT_2:30;
  hence thesis by TOPS_2:def 4;
end;
