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A comment on Toward an API for the Real Numbers. Parent comment
Proof: 1) suppose you can decide whether "x<y" for all x,y. 2) Then, for all x,y you can decide whether x==y because that's just "!(x<y) && !(y<x)"
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Proof: 1) suppose you can decide whether "x<y" for all x,y. 2) Then, for all x,y you can decide whether x==y because that's just "!(x<y) && !(y<x)"