  Using the rigorous path integral formalism of Feynman and Kac we prove
London's eighty years old conjecture that during the superfluid transition in
liquid helium Bose-Einstein condensation (BEC) takes place. The result is
obtained by proving first that at low enough temperatures macroscopic
permutation cycles appear in the system, and then showing that this implies
BEC. We find also that in the limit of zero temperature the infinite cycles
cover the whole system, while BEC remains partial. For the Bose-condensed fluid
at rest we define a macroscopic wave function. Via the equivalence of 1/2 spins
and hard-core bosons the method extends to lattice models. We show that at low
enough temperatures the spin-1/2 axially anisotropic Heisenberg models,
including the isotropic ferro- and antiferromagnet and the XY model, undergo
magnetic ordering.
