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

theorem Th39:
  b-a <= 1 implies for d being set st d in IntIntervals(a,b) holds
  CircleMap|d is one-to-one
proof
  assume that
A1: b-a <= 1;
  let d be set;
  assume d in IntIntervals(a,b);
  then consider n being Element of INT such that
A2: d = ].a+n,b+n.[;
A3: CircleMap | [.a+n,a+n+1.[ is one-to-one;
  b-a+(a+n) <= 1+(a+n) by A1,XREAL_1:6;
  hence thesis by A2,A3,SIN_COS6:2,XXREAL_1:45;
end;
