In the following basic principles of algorithms computing guaranteed bounds are developed from a theoretical and a practical point of view. Some fundamental theoretical facts are repeated where, for more detailed information, the reader is referred to the literature (...) Furthermore practical aspects are discussed, especially how the process of computing a guaranteed result really works performed by means of checking assumptions of mathematical theorems. This checking process is performed automatically. The various steps from the mathematical theorem down to the practical verification are described in detail. In contrast, standard floating-point algorithms usually deliver good approximations to the solution of a given numerical problem but there is neither a verification or a solution to the given problem actually exist. There are simple examples where the floating-point approximation is drstically wrong. A programming environment has been developed which allows to specify commands to the computer in mathematical notation. Because the system (preliminary name CALCULUS) works interactively, no type specification is necessary at all allowing specifying algorithms like in a math book. CALCULUS works right now on IBM System /370 machines under VM operating system. It is planned to have a C version for IBM System/2, SUN work stations and others available early next year. Some examples demonstrating the system are presented.