Continua of constant distances in span theory

It is proved that, for each non-negative number not exceeding the span of a mapping /: X-* Y, where X and Y are compact metric spaces, there exists a non-empty continuum K c X x X with identical two projections and such that the distances between f(x) and f(x') are all equal to for (x, x') e K . Similar results hold for other types of spans.