reserve x,x1,x2,x3 for Real;

theorem Th11:
  cos(x)<>0 implies (sec x)^2 = 1 + (tan x)^2
proof
  assume cos(x)<>0;
  then
A1: (cos(x))^2 <>0 by SQUARE_1:12;
  (sec(x))^2=1^2/(cos(x))^2 by XCMPLX_1:76
    .=((sin(x))^2+(cos(x))^2)/(cos(x))^2 by SIN_COS:29
    .=(sin(x))^2/(cos(x))^2+(cos(x))^2/(cos(x))^2 by XCMPLX_1:62
    .=(sin(x))^2/(cos(x))^2+1 by A1,XCMPLX_1:60
    .=(sin(x)/cos(x))^2+1 by XCMPLX_1:76;
  hence thesis;
end;
