let X be set ; :: thesis: for A1 being SetSequence of X holds Union (Partial_Union A1) = Union A1
let A1 be SetSequence of X; :: thesis: Union (Partial_Union A1) = Union A1
thus Union (Partial_Union A1) c= Union A1 :: according to XBOOLE_0:def 10 :: thesis: Union A1 c= Union (Partial_Union A1)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Union (Partial_Union A1) or x in Union A1 )
assume x in Union (Partial_Union A1) ; :: thesis: x in Union A1
then consider n being Element of NAT such that
A1: x in (Partial_Union A1) . n by PROB_1:25;
consider k being Nat such that
A2: ( k <= n & x in A1 . k ) by A1, Th15;
k in NAT by ORDINAL1:def 13;
hence x in Union A1 by A2, PROB_1:25; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Union A1 or x in Union (Partial_Union A1) )
assume x in Union A1 ; :: thesis: x in Union (Partial_Union A1)
then consider n being Element of NAT such that
A3: x in A1 . n by PROB_1:25;
x in (Partial_Union A1) . n by A3, Th15;
hence x in Union (Partial_Union A1) by PROB_1:25; :: thesis: verum