@JeanAbouSamra No this is not stupid at all: constructively, “least upper bound” (l.u.b.) and “supremum” need not coincide: “supremum” is the notion I defined, and “l.u.b.” is (as name suggests) the one you wrote. A supremum is necessarily a l.u.b., but the converse may fail, as explained in the beginning of Lubarski & Richman, “Zero Sets of Univariate Polynomials”: let $p$ be a truth value and consider $S := \{-1\} \cup \{0 : p\lor\neg p\}$, then $\operatorname{lub} S = 0$ but if $\sup S$ exists then $p\lor\neg p$ (so if every lub is a sup then LEM holds).
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