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

theorem Th33:
  CircleMap"{c[10]} = INT
proof
  hereby
    let i be object;
    assume
A1: i in CircleMap"{c[10]};
    then reconsider x = i as Real;
    CircleMap.i in {c[10]} by A1,FUNCT_2:38;
    then CircleMap.i = c[10] by TARSKI:def 1;
    then |[ cos(2*PI*x), sin(2*PI*x) ]| = |[1,0]| by Def11;
    then cos(2*PI*x) = 1 by SPPOL_2:1;
    hence i in INT by SIN_COS6:44;
  end;
  let i be object;
  assume i in INT;
  then reconsider i as Integer;
  CircleMap.i = c[10] by Th32;
  then
A2: CircleMap.i in {c[10]} by TARSKI:def 1;
  i in R by TOPMETR:17,XREAL_0:def 1;
  hence thesis by A2,FUNCT_2:38;
end;
