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 = (A1 . (s + 1)) \ ((Partial_Union A1) . s);
set A = A1 . 0;
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:
let x be Subset of X; :: thesis:
take (A1 . (n + 1)) \ ((Partial_Union A1) . n) ; :: thesis:
thus S₁[n,x,(A1 . (n + 1)) \ ((Partial_Union A1) . n)] ; :: thesis:

end;

consider F being SetSequence of X such that
A2: ( F . 0 = A1 . 0 & ( for n being Nat holds S₁[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch 2(A1);
for n being Nat holds F . (n + 1) = (A1 . (n + 1)) \ ((Partial_Union A1) . n)

proof

let n be Nat; :: thesis:
reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
for x, y being Subset of X
for s being Nat st s = n1 & x = F . n1 & y = F . (n1 + 1) holds
y = (A1 . (s + 1)) \ ((Partial_Union A1) . s) by A2;
hence F . (n + 1) = (A1 . (n + 1)) \ ((Partial_Union A1) . n) ; :: thesis:

end;

hence ex b₁ being SetSequence of X st
( b₁ . 0 = A1 . 0 & ( for n being Nat holds b₁ . (n + 1) = (A1 . (n + 1)) \ ((Partial_Union A1) . n) ) ) by A2; :: thesis: