theorem Th8: :: MESFUN17:8

for I, J, K being closed_interval Subset of REAL
for f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real
for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL st f is_continuous_on [:[:I,J:],K:] & f = g holds
for e being Real st 0 < e holds
ex r being Real st
( 0 < r & ( for x1, x2, y1, y2, z1, z2 being Real st x1 in I & x2 in I & y1 in J & y2 in J & z1 in K & z2 in K & |.(x2 - x1).| < r & |.(y2 - y1).| < r & |.(z2 - z1).| < r holds
|.((g . [x2,y2,z2]) - (g . [x1,y1,z1])).| < e ) )