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

theorem Th25:
  cos r = -1 implies r = PI + 2*PI*[\r/(2*PI)/]
proof
  set i = [\r/(2*PI)/];
  consider w being Real such that
A1: w = (2*PI)*-i+r and
A2: 0 <= w & w < 2*PI by COMPLEX2:1;
  assume
A3: cos r = -1;
  then (sin r)*(sin r)+(-1)*(-1) = 1 by SIN_COS:29;
  then
A4: sin r = 0;
  0+T(i) <= w+T(i) & w+T(i) < 2*PI+T(i) by A2,XREAL_1:6;
  then r = 0+T(i) or r = PI+T(i) by A1,A4,Th21;
  hence thesis by A3,COMPLEX2:9,SIN_COS:31;
end;
