let X be set ; :: thesis: for F being Field_Subset of X
for FSets being Set_Sequence of F holds Partial_Union FSets is Set_Sequence of F
let F be Field_Subset of X; :: thesis: for FSets being Set_Sequence of F holds Partial_Union FSets is Set_Sequence of F
let FSets be Set_Sequence of F; :: thesis: Partial_Union FSets is Set_Sequence of F
defpred S₁[ Nat] means (Partial_Union FSets) . $1 in F;
(Partial_Union FSets) . 0 = FSets . 0 by PROB_3:def 2;
then A1: S₁[ 0 ] ;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
(Partial_Union FSets) . (k + 1) = ((Partial_Union FSets) . k) \/ (FSets . (k + 1)) by PROB_3:def 2;
hence S₁[k + 1] by A3, PROB_1:9; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A1, A2);
hence Partial_Union FSets is Set_Sequence of F by DDef4; :: thesis: verum