reserve x,x1,x2,x3 for Real;

theorem
  sin(x1)<>0 & sin(x2)<>0 & sin(x3)<>0 implies sin(x1+x2+x3)=sin(x1)*sin
  (x2)*sin(x3)*(cot(x2)*cot(x3)+ cot(x1)*cot(x3)+cot(x1)*cot(x2)-1)
proof
  assume that
A1: sin(x1)<>0 and
A2: sin(x2)<>0 and
A3: sin(x3)<>0;
  sin(x1+x2+x3) = sin(x1+(x2+x3))
    .=sin(x1)*cos(x2+x3)+cos(x1)*sin(x2+x3) by SIN_COS:75
    .=sin(x1)*(cos(x2)*cos(x3)-sin(x2)*sin(x3)) + cos(x1)*sin(x2+x3) by
SIN_COS:75
    .=sin(x1)*(cos(x2)*cos(x3))-sin(x1)*(sin(x2)*sin(x3)) +cos(x1)*(sin(x2)*
  cos(x3)+cos(x2)*sin(x3)) by SIN_COS:75
    .=sin(x1)*cos(x2)*cos(x3)-sin(x1)*sin(x2)*sin(x3) +(cos(x1)*sin(x2)*cos(
  x3)+cos(x1)*cos(x2)*sin(x3))
    .=sin(x1)*(sin(x2)*cot(x2))*cos(x3)-sin(x1)*sin(x2)*sin(x3) +(cos(x1)*
  sin(x2)*cos(x3)+cos(x1)*cos(x2)*sin(x3)) by A2,Th2
    .=sin(x1)*(sin(x2)*cot(x2))*(sin(x3)*cot(x3))-sin(x1)*sin(x2) *sin(x3)+(
  cos(x1)*sin(x2)*cos(x3)+cos(x1)*cos(x2)*sin(x3)) by A3,Th2
    .=(sin(x1)*sin(x2)*sin(x3))*(cot(x2)*cot(x3)-1)+((sin(x1)*cot(x1) )*sin(
  x2)*cos(x3)+cos(x1)*cos(x2)*sin(x3)) by A1,Th2
    .=(sin(x1)*sin(x2)*sin(x3))*(cot(x2)*cot(x3)-1)+((sin(x1)*cot(x1) )*sin(
  x2)*(sin(x3)*cot(x3))+cos(x1)*cos(x2)*sin(x3)) by A3,Th2
    .=(sin(x1)*sin(x2)*sin(x3))*(cot(x2)*cot(x3)-1)+sin(x1)*sin(x2 )*sin(x3)
  *cot(x1)*cot(x3)+cos(x1)*cos(x2)*sin(x3)
    .=(sin(x1)*sin(x2)*sin(x3))*((cot(x2)*cot(x3)-1)+cot(x1)*cot(x3)) +(sin(
  x1)*cot(x1))*cos(x2)*sin(x3) by A1,Th2
    .=(sin(x1)*sin(x2)*sin(x3))*((cot(x2)*cot(x3)-1)+cot(x1)*cot(x3)) +(sin(
  x1)*cot(x1))*(sin(x2)*cot(x2))*sin(x3) by A2,Th2
    .=(sin(x1)*sin(x2)*sin(x3))*(cot(x2)*cot(x3)+cot(x1)*cot(x3) +cot(x1)*
  cot(x2)-1);
  hence thesis;
end;
