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

theorem Th23:
  sin r = -1 implies r = 3/2*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: sin r = -1;
  then (cos r)*(cos r)+(-1)*(-1) = 1 by SIN_COS:29;
  then
A4: cos r = 0;
  0+T(i) <= w+T(i) & w+T(i) < 2*PI+T(i) by A2,XREAL_1:6;
  then r = PI/2+T(i) or r = 3/2*PI+T(i) by A1,A4,Th22;
  hence thesis by A3,COMPLEX2:8,SIN_COS:77;
end;
