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

theorem Th42:
  (CircleMap(R^1(0)))" = Circle2IntervalR
proof
  reconsider A as non empty Subset of R^1;
  set f = CircleMap(R^1(0));
  set Y = the carrier of (R^1|A);
  reconsider f as Function of R^1|A, TOUC by Th32;
  reconsider r0 = In(0,REAL) as Point of R^1 by TOPMETR:17;
  set F = AffineMap(2*PI,0);
A1: dom id Y = Y by RELAT_1:45;
 CircleMap.r0 = c[10] by Th32;
  then
A2: rng f = X by FUNCT_2:def 3;
A3: Y = A by PRE_TOPC:8;
A4: now
    let a be object;
    assume
A5: a in dom (C*f);
    then reconsider b = a as Point of R^1|A;
    reconsider c = b as Element of REAL by XREAL_0:def 1;
    consider x, y being Real such that
A6: f.b = |[x,y]| and
A7: y >= 0 implies C.(f.b) = arccos x/P2 and
A8: y <= 0 implies C.(f.b) = 1-arccos x/P2 by Def13;
A9: f.b = CircleMap.b by A3,FUNCT_1:49
      .= |[ (cos*F).c, (sin*F).c ]| by Lm20;
    then y = (sin*F).c by A6,SPPOL_2:1;
    then
A10: y = sin.(F.c) by FUNCT_2:15;
    x = (cos*F).c by A6,A9,SPPOL_2:1;
    then x = cos.(F.c) by FUNCT_2:15;
    then
A11: x = cos(F.c) by SIN_COS:def 19;
A12: c < 1 by A3,XXREAL_1:4;
A13: F.c = 2*PI*c+0 by FCONT_1:def 4;
    then
A14: F.c/(2*PI*1) = c/1 by XCMPLX_1:91;
A15: F.(1-c) = 2*PI*(1-c)+0 by FCONT_1:def 4;
    then
A16: F.(1-c)/(2*PI*1) = (1-c)/1 by XCMPLX_1:91;
A17: now
      per cases;
      suppose
A18:    y >= 0;
        then not F.c in ].PI,2*PI.[ by A10,COMPTRIG:9;
        then F.c <= PI or F.c >= 2*PI by XXREAL_1:4;
        then F.c/P2 <= (1*PI)/P2 or F.c/P2 >= 2*PI/P2 by XREAL_1:72;
        then c <= 1/2 by A14,A12,XCMPLX_1:60,91;
        then
A19:    2*PI*c <= 2*PI*(1/2) by XREAL_1:64;
        0 <= c by A3,XXREAL_1:4;
        hence C.(f.b) = F.c/P2 by A7,A11,A13,A18,A19,SIN_COS6:92
          .= b by A13,XCMPLX_1:89;
      end;
      suppose
A20:    y < 0;
        then not F.c in [.0,PI.] by A10,COMPTRIG:8;
        then F.c < 0 or F.c > PI by XXREAL_1:1;
        then F.c/P2 < 0/P2 or F.c/P2 > (1*PI)/P2 by XREAL_1:74;
        then c < 0 or c > 1/2 by A14,XCMPLX_1:91;
        then 1-c < 1-1/2 by A3,XREAL_1:15,XXREAL_1:4;
        then
A21:    2*PI*(1-c) <= 2*PI*(1/2) by XREAL_1:64;
A22:    1-c > 1-1 by A12,XREAL_1:15;
        cos.(F.(1-c)) = cos(-2*PI*c+2*PI*1) by A15,SIN_COS:def 19
          .= cos(-2*PI*c) by COMPLEX2:9
          .= cos(2*PI*c) by SIN_COS:31;
        then arccos x = arccos cos F.(1-c) by A11,A13,SIN_COS:def 19
          .= F.(1-c) by A15,A22,A21,SIN_COS6:92;
        hence C.(f.b) = b by A8,A16,A20;
      end;
    end;
    thus (C*f).a = C.(f.b) by A5,FUNCT_1:12
      .= (id Y).a by A17;
  end;
  dom (C*f) = Y by FUNCT_2:def 1;
  then C = f qua Function" by A2,A1,A4,FUNCT_1:2,FUNCT_2:30;
  hence thesis by TOPS_2:def 4;
end;
