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All the major dietetics and health organizations in the world agree that vegan and vegetarian diets are just as healthy as, and probably healthier than omnivorous diets. Here are links to what some of them have to say on the subject:
Academy of Nutrition and Dietetics
The British National Health Service
The British Nutrition Foundation
The Dietitians Association of Australia
The United States Department of Agriculture
The National Health and Medical Research Council
The Heart and Stroke Foundation of Canada