reserve x,x1,x2,x3 for Real;

theorem
  sinh(x/2)<>0 implies tanh(x/2)=(cosh(x)-1)/(sinh(x))
proof
  assume sinh(x/2)<>0;
  then
A1: 2*sinh.(x/2)<>0 by SIN_COS2:def 2;
  (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))^2-1))/(sinh(2*(x/2)))
    .=(2*(sinh.(x/2))^2)/(sinh(2*(x/2))) by Lm3
    .=(2*(sinh.(x/2))^2)/(sinh.(2*(x/2))) by SIN_COS2:def 2
    .=(2*sinh.(x/2)*sinh.(x/2))/(2*sinh.(x/2)*cosh.(x/2)) by SIN_COS2:23
    .=sinh.(x/2)/cosh.(x/2) by A1,XCMPLX_1:91
    .=tanh.(x/2) by SIN_COS2:17
    .=tanh(x/2) by SIN_COS2:def 6;
  hence thesis;
end;
