The article includes: theorems related to domains, theorems related to Cartesian products presented earlier in various articles and simplified here by substituting domains for sets and omitting theâ€¦ (More)

We deal with a nonâ€“empty set of functions and a nonâ€“empty set of functions from a set A to a nonâ€“empty set B. In the case when B is a nonâ€“empty set, BA is redefined. It yields a nonâ€“empty set ofâ€¦ (More)

We descr ibe the MIZAR Computer Assisted ReaÂ soning system. Several vers ions of MIZAR are c u r r e n t l y implemented w i t h vary ing powers of expressiveness and proof ana l ys i s . The mainâ€¦ (More)

We compare the styles of several proof assistants for mathematics. We present Pythagorasâ€™ proof of the irrationality of âˆš 2 both informal and formalized in (1) HOL, (2) Mizar, (3) PVS, (4) Coq, (5)â€¦ (More)

The purpose of this article is to define projections of ordered pairs, and to introduce triples and quadruples, and their projections. The theorems in this paper may be roughly divided into twoâ€¦ (More)

For simplicity we adopt the following rules: x is arbitrary, m, n are natural numbers, f , g are functions, and A, B are sets. We now state several propositions: (1) For every function f and forâ€¦ (More)

The information that all members of a set enjoy a property expressed by an adjective can be processed in a systematic way. The purpose of the work is to find out how to do that. If it works,â€¦ (More)

The article deals with parameterized families of sets. When treated in a similar way as sets (due to systematic overloading notation used for sets) they are called many sorted sets. For instance, ifâ€¦ (More)