( ( ( Then ( the distance ) ) ( between ( two points ) ) ) ( is ( defined ( as ( ( the infimum ) ( of ( ( the lengths ) ( of ( the curves ) ) ) ) ) ) ) ) )
(ROOT (S (NP (NP (ADVP (RB Then)) (DT the) (NN distance)) (PP (IN between) (NP (CD two) (NNS points)))) (VP (VBZ is) (VP (VBN defined) (PP (IN as) (NP (NP (DT the) (NN infimum)) (PP (IN of) (NP (NP (DT the) (NNS lengths)) (PP (IN of) (NP (DT the) (NNS curves)))))))))))
Then the distance between two points is defined as the infimum of the lengths of the curves
(ROOT (S (NP (NNP Distance)) (VP (VBZ is) (NP (NP (DT the) (NN term)) (PP (IN for) (NP (NP (DT the) (NN length)) (PP (IN of) (NP (NP (DT the) (NN route)) (PP (IN between) (NP (CD two) (NNS points))))))))) (. .)))