In L-fuzzy topology, the theorem of existence of uniformly continuous mappings is very essential for the theory of uniform spaces and theory of metric spaces. In fact, the main and basic theorem "An L-fuzzy topological space is uniformizable if and only if it is completely regular" is just based on the theorem of existence of uniformly continuous mappings. In this paper, a complete and concrete proof for the existence of lattice-valued uniformly continuous mappings will be given, the errors appeared in some bypast proofs will be corrected; they seem to mean that the widely accepted skeleton of proof is not correct.
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