
theorem Th26:
for X be non empty set, S be Field_Subset of X, E being Set_Sequence of S
 for i be Nat holds (Partial_Union E).i in S
proof
    let X be non empty set, S be Field_Subset of X, E being Set_Sequence of S;
    defpred P[Nat] means (Partial_Union E).$1 in S;

    (Partial_Union E).0 = E.0 by PROB_3:def 2; then
A1: P[0];
A2: for n be Nat st P[n] holds P[n+1]
    proof
     let n be Nat;
     assume A3: P[n];
     (Partial_Union E).(n+1) = (Partial_Union E).n \/ E.(n+1) by PROB_3:def 2;
     hence thesis by A3,PROB_1:3;
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
end;
