reserve th, th1, th2, th3 for Real;

theorem
  cos(th1)<>0 & cos(th2)<>0 implies tan(th1-th2)=(tan(th1)-tan(th2))/(1+
  tan(th1)*tan(th2))
proof
  assume that
A1: cos(th1)<>0 and
A2: cos(th2)<>0;
  tan(th1-th2)= (sin(th1+(-th2))/(cos(th1)*cos(th2))) /(cos(th1+(-th2))/(
  cos(th1)*cos(th2))) by A1,A2,XCMPLX_1:55
    .= ((sin(th1)*cos(-th2)+cos(th1)*sin(-th2))/(cos(th1)*cos(th2))) /((cos(
  th1+(-th2)))/(cos(th1)*cos(th2))) by SIN_COS:75
    .= ((sin(th1)*cos(th2)+cos(th1)*sin(-th2))/(cos(th1)*cos(th2))) /((cos(
  th1+(-th2)))/(cos(th1)*cos(th2))) by SIN_COS:31
    .= ((sin(th1)*cos(th2)+cos(th1)*(-sin(th2)))/(cos(th1)*cos(th2))) /((cos
  (th1+(-th2)))/(cos(th1)*cos(th2))) by SIN_COS:31
    .= ((sin(th1)*cos(th2)-cos(th1)*sin(th2))/(cos(th1)*cos(th2))) /((cos(
  th1)*cos(-th2)-sin(th1)*sin(-th2))/(cos(th1)*cos(th2))) by SIN_COS:75
    .= ((sin(th1)*cos(th2)-cos(th1)*sin(th2))/(cos(th1)*cos(th2))) /((cos(
  th1)*cos(th2)-sin(th1)*sin(-th2))/(cos(th1)*cos(th2))) by SIN_COS:31
    .= ((sin(th1)*cos(th2)-cos(th1)*sin(th2))/(cos(th1)*cos(th2))) /((cos(
  th1)*cos(th2)-sin(th1)*(-sin(th2)))/(cos(th1)*cos(th2))) by SIN_COS:31
    .= (sin(th1)*cos(th2)/(cos(th1)*cos(th2)) - cos(th1)*sin(th2)/(cos(th1)*
  cos(th2))) /((cos(th1)*cos(th2)+sin(th1)*sin(th2))/(cos(th1)*cos(th2))) by
XCMPLX_1:120
    .= (sin(th1)*cos(th2)/(cos(th1)*cos(th2)) -cos(th1)*sin(th2)/(cos(th1)*
cos(th2))) /(cos(th1)*cos(th2)/(cos(th1)*cos(th2)) +sin(th1)*sin(th2)/(cos(th1)
  *cos(th2))) by XCMPLX_1:62
    .= (sin(th1)/cos(th1)*(cos(th2)/cos(th2)) -cos(th1)*sin(th2)/(cos(th1)*
  cos(th2))) /(cos(th1)*cos(th2)/(cos(th1)*cos(th2)) +(sin(th1)*sin(th2))/(cos(
  th1)*cos(th2))) by XCMPLX_1:76
    .= (sin(th1)/cos(th1)*(cos(th2)/cos(th2)) -sin(th2)/cos(th2)*(cos(th1)/
  cos(th1))) /(cos(th1)*cos(th2)/(cos(th1)*cos(th2)) +(sin(th1)*sin(th2))/(cos(
  th1)*cos(th2))) by XCMPLX_1:76
    .= (sin(th1)/cos(th1)*(cos(th2)/cos(th2)) -sin(th2)/cos(th2)*(cos(th1)/
cos(th1))) /((cos(th1)/cos(th1))*(cos(th2)/cos(th2)) +(sin(th1)*sin(th2))/(cos(
  th1)*cos(th2))) by XCMPLX_1:76
    .= (sin(th1)/cos(th1)*(cos(th2)/cos(th2)) -sin(th2)/cos(th2)*(cos(th1)/
cos(th1))) /((cos(th1)/cos(th1))*(cos(th2)/cos(th2)) +(sin(th1)/cos(th1))*(sin(
  th2)/cos(th2))) by XCMPLX_1:76
    .= (sin(th1)/cos(th1)-sin(th2)/cos(th2)*(cos(th1)/cos(th1))) /((cos(th1)
/cos(th1))*(cos(th2)/cos(th2)) +(sin(th1)/cos(th1))*(sin(th2)/cos(th2))) by A2,
XCMPLX_1:88
    .= (sin(th1)/cos(th1)-sin(th2)/cos(th2)) /((cos(th1)/cos(th1))*(cos(th2)
  /cos(th2)) +(sin(th1)/cos(th1))*(sin(th2)/cos(th2))) by A1,XCMPLX_1:88
    .= (sin(th1)/cos(th1)-sin(th2)/cos(th2)) /((cos(th1)/cos(th1))+(sin(th1)
  /cos(th1))*(sin(th2)/cos(th2))) by A2,XCMPLX_1:88
    .= (tan(th1)-tan(th2))/(1+tan(th1)*tan(th2)) by A1,XCMPLX_1:60;
  hence thesis;
end;
