A generalization of Borsukʹs pasting theorem and its application

Let A, B and X be finite-dimensional ANR compacta and let α :A→X and f :A→B be maps such that α restricted to the set Sf={x∈A∣f−1f(x)≠x} is one-to-one. Then the pushout Y of the diagram Xα←Af→B is ANR. We apply this result to a construction of ANRs Mp, p is prime, for which dim(Mp×Mq)≠dimMp+dimMq.