reserve x,x1,x2,x3 for Real;

theorem Th36:
  coth(x)=(exp_R(x)+exp_R(-x))/(exp_R(x)-exp_R(-x)) & sech(x)=2/(
  exp_R(x)+exp_R(-x)) & cosech(x)=2/(exp_R(x)-exp_R(-x))
proof
A1: sech(x)=1/cosh.x by SIN_COS2:def 4
    .=1/((exp_R.(x)+exp_R.(-x))/2) by SIN_COS2:def 3
    .=1*2/((exp_R.(x)+exp_R.(-x))/2*2) by XCMPLX_1:91
    .=2/(exp_R.(x)+exp_R(-x)) by SIN_COS:def 23
    .=2/(exp_R(x)+exp_R(-x)) by SIN_COS:def 23;
A2: cosech(x)=1/sinh.x by SIN_COS2:def 2
    .=1/((exp_R.(x)-exp_R.(-x))/2) by SIN_COS2:def 1
    .=1*2/((exp_R.(x)-exp_R.(-x))/2*2) by XCMPLX_1:91
    .=2/(exp_R.(x)-exp_R(-x)) by SIN_COS:def 23;
  coth(x)=cosh.x/sinh(x) by SIN_COS2:def 4
    .=cosh.x/sinh.x by SIN_COS2:def 2
    .=((exp_R.(x)+exp_R.(-x))/2)/sinh.x by SIN_COS2:def 3
    .=((exp_R.(x)+exp_R.(-x))/2)/((exp_R.(x)-exp_R.(-x))/2) by SIN_COS2:def 1
    .=((exp_R.(x)+exp_R.(-x))/2*2)/((exp_R.(x)-exp_R.(-x))/2*2) by XCMPLX_1:91
    .=(exp_R.(x)+exp_R.(-x))/(exp_R.(x)-exp_R(-x)) by SIN_COS:def 23
    .=(exp_R.(x)+exp_R.(-x))/(exp_R(x)-exp_R(-x)) by SIN_COS:def 23
    .=(exp_R.(x)+exp_R(-x))/(exp_R(x)-exp_R(-x)) by SIN_COS:def 23
    .=(exp_R(x)+exp_R(-x))/(exp_R(x)-exp_R(-x)) by SIN_COS:def 23;
  hence thesis by A1,A2,SIN_COS:def 23;
end;
