reserve x,x1,x2,x3 for Real;

theorem
  (sech(x))^2+(tanh(x))^2=1
proof
  cosh.x <>0 by SIN_COS2:15;
  then
A1: (cosh.x)^2<>0 by SQUARE_1:12;
  (sech(x))^2+(tanh(x))^2 =(1/cosh(x))^2 + (tanh.x)^2 by SIN_COS2:def 6
    .=(1/cosh.(x))^2+(tanh.(x))^2 by SIN_COS2:def 4
    .=(1^2/(cosh.(x))^2)+(tanh.(x))^2 by XCMPLX_1:76
    .=(1^2/(cosh.(x))^2)+(sinh.(x)/cosh.(x))^2 by SIN_COS2:17
    .=(1/(cosh.(x))^2)+((sinh.(x))^2/(cosh.(x))^2) by XCMPLX_1:76
    .=(1+(sinh.(x))^2)/(cosh.(x))^2 by XCMPLX_1:62
    .=((cosh.x)^2-(sinh.x)^2+(sinh.(x))^2)/(cosh.(x))^2 by SIN_COS2:14
    .=(cosh.x)^2/(cosh.(x))^2;
  hence thesis by A1,XCMPLX_1:60;
end;
