theorem Th39: :: JGRAPH_1:40

for f being FinSequence of (TOP-REAL 2)
for a, c, d being Real st 1 <= len f & Y_axis f lies_between (Y_axis f) . 1,(Y_axis f) . (len f) & X_axis f lies_between c,d & a > 0 & ( for i being Nat st 1 <= i & i + 1 <= len f holds
|.((f /. i) - (f /. (i + 1))).| < a ) holds
ex g being FinSequence of (TOP-REAL 2) st
( g is special & g . 1 = f . 1 & g . (len g) = f . (len f) & len g >= len f & Y_axis g lies_between (Y_axis f) . 1,(Y_axis f) . (len f) & X_axis g lies_between c,d & ( for j being Nat st j in dom g holds
ex k being Nat st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Nat st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) )