reserve x,x1,x2,x3 for Real;

theorem
  cosh(x/2)<>0 implies tanh(x/2)=(sinh(x))/(cosh(x)+1)
proof
  assume cosh(x/2)<>0;
  then
A1: 2*cosh.(x/2)<>0 by SIN_COS2:def 4;
  (sinh(x))/(cosh(x)+1)=(sinh.(2*(x/2)))/(cosh(2*(x/2))+1) by SIN_COS2:def 2
    .=(sinh.(2*(x/2)))/(cosh.(2*(x/2))+1) by SIN_COS2:def 4
    .=(2*sinh.(x/2)*cosh.(x/2))/(cosh.(2*(x/2))+1) by SIN_COS2:23
    .=(2*sinh.(x/2)*cosh.(x/2))/(2*(cosh.(x/2))^2-1+1) by SIN_COS2:23
    .=(2*cosh.(x/2)*sinh.(x/2))/(2*cosh.(x/2)*cosh.(x/2))
    .=sinh.(x/2)/cosh.(x/2) by A1,XCMPLX_1:91
    .=tanh.(x/2) by SIN_COS2:17;
  hence thesis by SIN_COS2:def 6;
end;
