defpred S₁[ set , set , set ] means for x, y being Subset of X
for s being Nat st s = $1 & x = $2 & y = $3 holds
y = x /\ (A1 . (s + 1));
A1: for n being Nat
for x being Subset of X ex y being Subset of X st S₁[n,x,y]

proof

let n be Nat; :: thesis: for x being Subset of X ex y being Subset of X st S₁[n,x,y]
let x be Subset of X; :: thesis: ex y being Subset of X st S₁[n,x,y]
take x /\ (A1 . (n + 1)) ; :: thesis: S₁[n,x,x /\ (A1 . (n + 1))]
thus S₁[n,x,x /\ (A1 . (n + 1))] ; :: thesis: verum

end;

consider F being SetSequence of X such that
A2: F . 0 = A1 . 0 and
A3: for n being Nat holds S₁[n,F . n,F . (n + 1)] from RECDEF_1:sch 2(A1);
take F ; :: thesis: ( F . 0 = A1 . 0 & ( for n being Nat holds F . (n + 1) = (F . n) /\ (A1 . (n + 1)) ) )
thus F . 0 = A1 . 0 by A2; :: thesis: for n being Nat holds F . (n + 1) = (F . n) /\ (A1 . (n + 1))
let n be Nat; :: thesis: F . (n + 1) = (F . n) /\ (A1 . (n + 1))
reconsider n = n as Element of NAT by ORDINAL1:def 12;
S₁[n,F . n,F . (n + 1)] by A3;
hence F . (n + 1) = (F . n) /\ (A1 . (n + 1)) ; :: thesis: verum