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

theorem Th41:
  CircleMap(R^1(a+i)) = CircleMap(R^1(a)) * (AffineMap(1,-i) | ].a +i,a+i+1.[)
proof
  set W = ].a,a+1.[;
  set Q = ].a+i,a+i+1.[;
  set h = CircleMap(R^1(a+i));
  set g = CircleMap(R^1(a));
  set F = AffineMap(1,-i);
  set f = F|Q;
A1: dom h = Q by Lm18,RELAT_1:62;
  dom F = REAL by FUNCT_2:def 1;
  then
A2: dom f = Q by RELAT_1:62;
A3: for x being object st x in dom h holds h.x = (g*f).x
  proof
    let x be object;
    assume
A4: x in dom h;
    then reconsider y = x as Real;
    y < a+i+1 by A1,A4,XXREAL_1:4;
    then
A5: y-i < a+i+1-i by XREAL_1:9;
    a+i < y by A1,A4,XXREAL_1:4;
    then a+i-i < y-i by XREAL_1:9;
    then
A6: y-i in W by A5,XXREAL_1:4;
    thus (g*f).x = g.(f.x) by A1,A2,A4,FUNCT_1:13
      .= g.(F.x) by A1,A4,FUNCT_1:49
      .= g.(1*y+-i) by FCONT_1:def 4
      .= CircleMap.(y+-i) by A6,FUNCT_1:49
      .= CircleMap.y by Th31
      .= h.x by A1,A4,FUNCT_1:49;
  end;
A7: rng f = ].a,a+1.[ by Lm24;
  dom g = W by Lm18,RELAT_1:62;
  then dom (g*f) = dom f by A7,RELAT_1:27;
  hence thesis by A2,A3,Lm18,RELAT_1:62;
end;
