reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem
  cos(2*PI*r) = 1 implies r in INT
proof
  reconsider d = 2 as positive Real;
  assume
A1: cos(2*PI*r) = 1;
  assume not r in INT;
  then r is not integer by INT_1:def 2;
  then reconsider t = frac r as positive Real by INT_1:46;
  set s = [\r/];
A2: r = s+t & d*PI*t < d*PI*1 by INT_1:42,43,XREAL_1:68;
  cos(2*PI*(s+t)) = cos(2*PI*s+2*PI*t) .= cos(2*PI*t) by COMPLEX2:9;
  hence contradiction by A1,A2,Th34;
end;
