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

theorem Th31:
  CircleMap.r = CircleMap.(r+i)
proof
  defpred P[Integer] means CircleMap.r = CircleMap.(r+$1);
A1: for i holds P[i] implies P[i-1] & P[i+1]
  proof
    let i such that
A2: P[i];
    thus CircleMap.(r+(i-1)) = |[ cos(2*PI*(r+i-1)), sin(2*PI*(r+i-1)) ]| by
Def11
      .= |[ cos(2*PI*(r+i)), sin(2*PI*(r+i)+2*PI*(-1)) ]| by COMPLEX2:9
      .= |[ cos(2*PI*(r+i)), sin(2*PI*(r+i)) ]| by COMPLEX2:8
      .= CircleMap.r by A2,Def11;
    thus CircleMap.(r+(i+1)) = |[ cos(2*PI*(r+i+1)), sin(2*PI*(r+i+1)) ]| by
Def11
      .= |[ cos(2*PI*(r+i)), sin(2*PI*(r+i)+2*PI*1) ]| by COMPLEX2:9
      .= |[ cos(2*PI*(r+i)), sin(2*PI*(r+i)) ]| by COMPLEX2:8
      .= CircleMap.r by A2,Def11;
  end;
A3: P[0];
  for i holds P[i] from INT_1:sch 4(A3,A1);
  hence thesis;
end;
