Packing and covering with matroid circuits

We prove a conjecture of Welsh, that for every matroid M without coloops, ν(M) + θ(M) ≤ ρ{variant}(M) + κ(M) where ν(M) is the maximum number of pairwise disjoint circuits, θ(M) is the minimum number of circuits whose union is E(M), ρ{variant}(M) is the corank of M, and κ(M) is the number of connected components of M. For binary matroids the result was previously proved by Oxley.