let C1, C2 be Coherence_Space; :: thesis: for f being U-continuous Function of C1,C2
for a being Element of C1 holds f . a = (graph f) .: (Fin a)
let f be U-continuous Function of C1,C2; :: thesis: for a being Element of C1 holds f . a = (graph f) .: (Fin a)
let a be Element of C1; :: thesis: f . a = (graph f) .: (Fin a)
set X = graph f;
A1: dom f = C1 by FUNCT_2:def 1;

A2: now

let x be set ; :: thesis: ( x in graph f implies x `1 is finite )
assume x in graph f ; :: thesis: x `1 is finite
then ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) by Def17;
hence x `1 is finite by MCART_1:7; :: thesis: verum

end;

A3: for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in graph f holds
[b,y] in graph f by A1, Th26;
for a being finite Element of C1
for y1, y2 being set st [a,y1] in graph f & [a,y2] in graph f holds
{y1,y2} in C2 by Th27;
then consider g being U-continuous Function of C1,C2 such that
A4: ( graph f = graph g & ( for a being Element of C1 holds g . a = (graph f) .: (Fin a) ) ) by A2, A3, Lm4;
g . a = (graph f) .: (Fin a) by A4;
hence f . a = (graph f) .: (Fin a) by A4, Th29; :: thesis: verum