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doi: 10.1016/0022-4804(90)90061-6

These references make clear that B12 therapy is primarily about treating deficiency, not directly burning fat

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[6] A comprehensive meta-analysis of all studies on factors that could prevent Alzheimers by one of our Scientific Advisory Board members, Professor Jin Tai Yu of Fudan University in Shanghai, China, shows that either a high vitamin E or C intake showed a trend of attenuating risk by about 26 per cent, making these nutrients grade 1 top level prevention factors.[7] Inadequate vitamin C status in otherwise healthy young adults is related to a low level of mental vitality