let C1, C2 be Coherence_Space; :: thesis: for X being Subset of [:C1,(union C2):] st ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being object st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being object st [a,y] in X & [b,y] in X holds
a = b ) holds
ex f being U-stable Function of C1,C2 st X = Trace f
let X be Subset of [:C1,(union C2):]; :: thesis: ( ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being object st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being object st [a,y] in X & [b,y] in X holds
a = b ) implies ex f being U-stable Function of C1,C2 st X = Trace f )
assume A1: ( ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being object st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being object st [a,y] in X & [b,y] in X holds
a = b ) & ( for f being U-stable Function of C1,C2 holds not X = Trace f ) ) ; :: thesis: contradiction
then ex f being U-stable Function of C1,C2 st
( X = Trace f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) ) by Lm5;
hence contradiction by A1; :: thesis: verum