let C1, C2 be Coherence_Space; :: thesis: union (StabCoh C1,C2) = [:(Sub_of_Fin C1),(union C2):]
thus union (StabCoh C1,C2) c= [:(Sub_of_Fin C1),(union C2):] :: according to XBOOLE_0:def 10 :: thesis: [:(Sub_of_Fin C1),(union C2):] c= union (StabCoh C1,C2)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in union (StabCoh C1,C2) or x in [:(Sub_of_Fin C1),(union C2):] )
assume x in union (StabCoh C1,C2) ; :: thesis: x in [:(Sub_of_Fin C1),(union C2):]
then consider a being set such that
A1: x in a and
A2: a in StabCoh C1,C2 by TARSKI:def 4;
ex f being U-stable Function of C1,C2 st a = Trace f by A2, Def19;
then a c= [:(Sub_of_Fin C1),(union C2):] by Th47;
hence x in [:(Sub_of_Fin C1),(union C2):] by A1; :: thesis: verum

end;

let x, y be set ; :: according to RELAT_1:def 3 :: thesis: ( not [x,y] in [:(Sub_of_Fin C1),(union C2):] or [x,y] in union (StabCoh C1,C2) )
assume A3: [x,y] in [:(Sub_of_Fin C1),(union C2):] ; :: thesis: [x,y] in union (StabCoh C1,C2)
then A4: y in union C2 by ZFMISC_1:106;
A5: x in Sub_of_Fin C1 by A3, ZFMISC_1:106;
then x is finite by Def3;
then ex f being U-stable Function of C1,C2 st Trace f = {[x,y]} by A5, A4, Th43;
then ( [x,y] in {[x,y]} & {[x,y]} in StabCoh C1,C2 ) by Def19, TARSKI:def 1;
hence [x,y] in union (StabCoh C1,C2) by TARSKI:def 4; :: thesis: verum