reserve x for Real;

theorem Th17:
  0 <= x & x < 2*PI & sin x = 0 implies x = 0 or x = PI
proof
  assume that
A1: 0 <= x & x < 2*PI and
A2: sin x = 0;
  sin.x = 0 by A2,SIN_COS:def 17;
  then ( not x in ].0,PI.[)& not x in ].PI,2*PI.[ by Th7,Th9;
  then x = 0 or x >= PI & PI >= x by A1,XXREAL_1:4;
  hence thesis by XXREAL_0:1;
end;
