Connectedness of the continuum in intuitionistic mathematics.

Working in INT (Intuitionistic analysis) we prove a strong, constructive connectedness property of the continuum: for any non-empty sets, A and B, if R=A boolean OR B then A boolean AND B is non-empty. It is well known that the intuitionistic continuum is indecomposable: if R=A boolean OR B and A boolean AND B= empty set then A=R or A=empty set, but this property is essentially negative-equivalent to if A, B are non-empty and R=A boolean OR B then not sign (A boolean AND B not equal empty set). Our connectedness property is positive; so, given a is an element of A, b is an element of B, and a witness to R=A boolean OR B, to prove our theorem we must construct a real number r is an element of A boolean AND B. We can construct the needed real number using only Bishop's constructive mathematics (BISH) and a weak form of Brouwer's continuity principle (and the choice principles that come from the Brouwer-Heyting-Kolmogorov interpretation of quantifiers in constructive type theory). We also replace indecomposability by connectedness in some results of van Dalen that use additional intuitionistic axioms.