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ΔW ≥ ΔF

Where W is the work done and F is the Helmholtz Free Energy, F = E - TS.

Then it defines the Gibbs free Energy;

G = F + PV

"For a system at constant temperature and pressure, G never increases";

So ΔG = ΔF + Δ(PV) = ΔF + PΔV = ΔF + ΔW

Then it says "We already know ΔW ≤ -ΔF and therefore ΔW + ΔF = ΔG ≤ 0".

But how can we assume ΔW ≤ -ΔF from the fact that ΔW ≥ ΔF? If ΔW is positive and |ΔW| ≥ |ΔF| then this assumption does not work, and also I have nothing that indicates to me the change in the work done is either positive or negative. I can't seem to show how this proof was meant to work otherwise though, any help?