reserve x,x1,x2,x3 for Real;

theorem
  sin(x)<>0 implies (cosec(x))^2 = 1 +(cot(x))^2
proof
  assume sin(x)<>0;
  then
A1: (sin(x))^2 <>0 by SQUARE_1:12;
  (cosec(x))^2=1^2/(sin(x))^2 by XCMPLX_1:76
    .=((sin(x))^2+(cos(x))^2)/(sin(x))^2 by SIN_COS:29
    .=(sin(x))^2/(sin(x))^2+(cos(x))^2/(sin(x))^2 by XCMPLX_1:62
    .=1+(cos(x))^2/(sin(x))^2by A1,XCMPLX_1:60
    .=1+(cos(x)/sin(x))^2 by XCMPLX_1:76;
  hence thesis;
end;
