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

proof

let n be Nat; :: thesis: for x being Subset of REAL ex y being Subset of REAL st S₁[n,x,y]
let x be Subset of REAL; :: thesis: ex y being Subset of REAL st S₁[n,x,y]
take halfline_fin (a,(b + (1 / (n + 1)))) ; :: thesis: S₁[n,x, halfline_fin (a,(b + (1 / (n + 1))))]
thus S₁[n,x, halfline_fin (a,(b + (1 / (n + 1))))] ; :: thesis: verum

end;

consider F being SetSequence of REAL such that
A2: F . 0 = halfline_fin (a,(b + 1)) 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 = halfline_fin (a,(b + 1)) & ( for n being Nat holds F . (n + 1) = halfline_fin (a,(b + (1 / (n + 1)))) ) )
thus F . 0 = halfline_fin (a,(b + 1)) by A2; :: thesis: for n being Nat holds F . (n + 1) = halfline_fin (a,(b + (1 / (n + 1))))
let n be Nat; :: thesis: F . (n + 1) = halfline_fin (a,(b + (1 / (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) = halfline_fin (a,(b + (1 / (n + 1)))) ; :: thesis: verum