Suppose X is a space that is the union of the closed subspaces X_1, ., X_n; assume there is a point p of X such that X_i (intersection) X_j = {p} for...

Suppose X is a space that is the union of the closed subspaces X_1, ..., X_n; assume there is a point p of X such that X_i (intersection) X_j = {p} for all i not equal to j.Show that if for each i, the point p is a deformation retract of an open set W_i of X_i, then the fundamental group of X is the external free product of the groups pi_1(X_i, p) relative to the monomorphisms induced by inclusion.