While product integration with respect to matrix–valued semimartiangles has been studied reasonably well, no work to date deals with stochastic product integration involving (infinite dimensionals) cylindrical processes. The objective of this series of three papers is to develop a comprehensive theory of product integral using exponentials of infinite dimensional semimartinagles. After introducing the exponential of a finite rank Hilbert–Schmidt operator (K2-) valued Brownian motion, we construct the exponential of processes of the form β+V and M + V where β is a general K2-Brownian motion, M is a K2-valued martiangle and V is a K2-process of bounded variation. Finally, we show the existence of a process, called the negative exponential, which is the multiplicative inverse of the exponential of the process X