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

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in union (LinCoh C1,C2) or x in [:(union C1),(union C2):] )
assume x in union (LinCoh C1,C2) ; :: thesis: x in [:(union C1),(union C2):]
then consider a being set such that
A1: x in a and
A2: a in LinCoh C1,C2 by TARSKI:def 4;
ex f being U-linear Function of C1,C2 st a = LinTrace f by A2, Def21;
hence x in [:(union C1),(union C2):] by A1; :: thesis: verum

end;

let x, y be set ; :: according to RELAT_1:def 3 :: thesis: ( not [x,y] in [:(union C1),(union C2):] or [x,y] in union (LinCoh C1,C2) )
assume A3: [x,y] in [:(union C1),(union C2):] ; :: thesis: [x,y] in union (LinCoh C1,C2)
then A4: y in union C2 by ZFMISC_1:106;
x in union C1 by A3, ZFMISC_1:106;
then ex f being U-linear Function of C1,C2 st LinTrace f = {[x,y]} by A4, Th60;
then ( [x,y] in {[x,y]} & {[x,y]} in LinCoh C1,C2 ) by Def21, TARSKI:def 1;
hence [x,y] in union (LinCoh C1,C2) by TARSKI:def 4; :: thesis: verum