theorem Th3: :: UNIFORM1:3

for N, M being non empty MetrSpace
for f being Function of (TopSpaceMetr N),(TopSpaceMetr M) st ( for r being Real
for u being Element of N
for u1 being Element of M st r > 0 & u1 = f . u holds
ex s being Real st
( s > 0 & ( for w being Element of N
for w1 being Element of M st w1 = f . w & dist (u,w) < s holds
dist (u1,w1) < r ) ) ) holds
f is continuous