reserve x,y,t for Real;

theorem
  x>0 implies csch"(x)=sinh"(1/x)
proof
  assume
A1: x>0;
  then
A2: x^2>0;
  sinh"(1/x)=log(number_e,(1/x+sqrt(1/x^2+1^2))) by XCMPLX_1:76
    .=log(number_e,(1/x+sqrt((1+x^2*1)/(x^2)))) by A2,XCMPLX_1:113
    .=log(number_e,(1/x+sqrt(1+x^2)/sqrt(x^2))) by A1,SQUARE_1:30
    .=log(number_e,(1/x+sqrt(1+x^2)/x)) by A1,SQUARE_1:22
    .=log(number_e,(1+sqrt(1+x^2))/x);
  hence thesis by A1,Def8;
end;
