reserve x,x1,x2,x3 for Real;

theorem Th6:
  cos(x)<>0 implies sin(2*x)=(2*tan(x))/(1+(tan(x))^2)
proof
  assume
A1: cos(x)<>0;
  then
A2: (cos(x))^2<>0 by SQUARE_1:12;
  sin(2*x)=2*sin(x)*cos(x)*1 by Th5
    .=2*sin(x)*cos(x)*(cos(x)/cos(x)) by A1,XCMPLX_1:60
    .=2*sin(x)*cos(x)*cos(x)/cos(x) by XCMPLX_1:74
    .=2*(cos(x))^2*sin(x)/cos(x)
    .=2*(cos(x))^2*tan(x)/1 by XCMPLX_1:74
    .=2*tan(x)*(cos(x))^2/((sin(x))^2+(cos(x))^2) by SIN_COS:29
    .=2*tan(x)/(((sin(x))^2+(cos(x))^2)/(cos(x))^2) by XCMPLX_1:77
    .=2*tan(x)/((sin(x))^2/(cos(x))^2+(cos(x))^2/(cos(x))^2) by XCMPLX_1:62
    .=2*tan(x)/((sin(x))^2/(cos(x))^2+1) by A2,XCMPLX_1:60
    .=2*tan(x)/((tan x)^2+1) by XCMPLX_1:76;
  hence thesis;
end;
