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 & a in LinCoh C1,C2 ) by TARSKI:def 4;
ex f being U-linear Function of C1,C2 st a = LinTrace f by A1, Def21;
hence x in [:(union C1),(union C2):] by A1; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in [:(union C1),(union C2):] or x in union (LinCoh C1,C2) )
assume x in [:(union C1),(union C2):] ; :: thesis: x in union (LinCoh C1,C2)
then ( x = [(x `1 ),(x `2 )] & x `1 in union C1 & x `2 in union C2 ) by MCART_1:10, MCART_1:23;
then ex f being U-linear Function of C1,C2 st LinTrace f = {x} by Th60;
then ( x in {x} & {x} in LinCoh C1,C2 ) by Def21, TARSKI:def 1;
hence x in union (LinCoh C1,C2) by TARSKI:def 4; :: thesis: verum