reserve x,y,t for Real;

theorem
  x>0 & x<=1 implies sech1"(x)=cosh1"(1/x)
proof
  assume that
A1: x>0 and
A2: x<=1;
A3: 1-x^2>=0 by A1,A2,Th22;
A4: x^2>0 by A1;
  cosh1"(1/x)=log(number_e,(1/x)+sqrt(1/x^2-1^2)) by XCMPLX_1:76
    .=log(number_e,(1/x)+sqrt((1-1*(x^2))/x^2)) by A4,XCMPLX_1:126
    .=log(number_e,(1/x)+sqrt(1-(x^2))/sqrt(x^2)) by A1,A3,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;
end;
