reserve x,x1,x2,x3 for Real;

theorem
  cosh(x/2)<>0 implies coth(x/2)=(cosh(x)+1)/(sinh(x))
proof
  assume cosh(x/2)<>0;
  then
A1: 2*cosh.(x/2)<>0 by SIN_COS2:def 4;
  (cosh(x)+1)/(sinh(x))=(cosh.(2*(x/2))+1)/(sinh(2*(x/2))) by SIN_COS2:def 4
    .=(2*(cosh.(x/2))^2-1+1)/(sinh(2*(x/2))) by SIN_COS2:23
    .=(2*(cosh.(x/2))^2)/(sinh.(2*(x/2))) by SIN_COS2:def 2
    .=(2*(cosh.(x/2)*cosh.(x/2)))/(2*sinh.(x/2)*cosh.(x/2)) by SIN_COS2:23
    .=(2*cosh.(x/2)*cosh.(x/2))/(2*cosh.(x/2)*sinh.(x/2))
    .=cosh.(x/2)/sinh.(x/2) by A1,XCMPLX_1:91
    .=cosh(x/2)/sinh.(x/2) by SIN_COS2:def 4
    .=cosh(x/2)/sinh(x/2) by SIN_COS2:def 2;
  hence thesis;
end;
