  The breaking down of the equivalence principle, when discussed in the context
of Sikorski's differential spaces theory, leads to the definition of the
so-called differentially singular boundary (d-boundary) and to the concept of
differential space with singularity associated with a given space-time
differential manifold. This enables us to define the time orientability, the
beginning of the cosmological time and the smooth evolution for the flat
Friedmanian world model with the initial singularity. The simplest smoothly
evolved models are studied. It is shown, that the cosmological matter causing
such an evolution can be of three different types. One of them is the fluid
with dark energy properties, the second the fluid with attraction properties,
and the third a mixture of the other two. Among all investigated smoothly
evolved solutions, models qualitatively consistent with the observational data
of type Ia supernovae have been found.
