  The long-range Froehlich electron-phonon interaction has been identified as
the most essential for pairing in high-temperature superconductors owing to
poor screening, as is now confirmed by optical, isotope substitution, recent
photoemission and some other measurements. I argue that low energy physics in
cuprate superconductors is that of superlight small bipolarons, which are
real-space hole pairs dressed by phonons in doped charge-transfer Mott
insulators. They are itinerant quasiparticles existing in the Bloch states at
low temperatures as also confirmed by continuous-time quantum Monte-Carlo
algorithm (CTQMC) fully taking into account realistic Coulomb and long-range
Froehlich interactions. Here I suggest that a parameter-free evaluation of Tc,
unusual upper critical fields, the normal state Nernst effect, diamagnetism,
the Hall-Lorenz numbers and giant proximity effects strongly support the
three-dimensional (3D) Bose-Einstein condensation of mobile small bipolarons
with zero off-diagonal order parameter above the resistive critical temperature
Tc at variance with phase fluctuation scenarios of cuprates.
