Relations between invariants of formally real fields

For every formally real field F the following holds: l(F) < infinity iff ud(F) < infinity iff (u) over tilde d(F) < infinity iff c(F) < infinity iff there exists i is an element of N\{0} such that B-F(i) < infinity iff for every i is an element of N\{0} we have that B-F(i) < infinity iff there exists alpha is an element of Sigma(F) such that c(F, alpha) < infinity. The conditions 'F satisfies B-3' and '2 less than or equal to c(F, alpha) < infinity for some alpha is an element of Sigma(F)' imply that the following holds: c(F, beta) less than or equal to c(F) less than or equal to n less than or equal to c(F, beta) + 2 less than or equal to c(F) + 2 for all n is an element of {l(F) = 1 + B-F(1),...,i + B-F(i),...,l(F) - 1 + B-F(l(F) - 1), (u) over tilde d(F), ud(F)} and for all beta is an element of Sigma(F).