defpred S_{1}[ 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 S_{1}[n,x,y]

A2: F . 0 = halfline_fin (a,(b + 1)) and

A3: for n being Nat holds S_{1}[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;

S_{1}[n,F . n,F . (n + 1)]
by A3;

hence F . (n + 1) = halfline_fin (a,(b + (1 / (n + 1)))) ; :: thesis: verum

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 S

proof

consider F being SetSequence of REAL such that
let n be Nat; :: thesis: for x being Subset of REAL ex y being Subset of REAL st S_{1}[n,x,y]

let x be Subset of REAL; :: thesis: ex y being Subset of REAL st S_{1}[n,x,y]

take halfline_fin (a,(b + (1 / (n + 1)))) ; :: thesis: S_{1}[n,x, halfline_fin (a,(b + (1 / (n + 1))))]

thus S_{1}[n,x, halfline_fin (a,(b + (1 / (n + 1))))]
; :: thesis: verum

end;let x be Subset of REAL; :: thesis: ex y being Subset of REAL st S

take halfline_fin (a,(b + (1 / (n + 1)))) ; :: thesis: S

thus S

A2: F . 0 = halfline_fin (a,(b + 1)) and

A3: for n being Nat holds S

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;

S

hence F . (n + 1) = halfline_fin (a,(b + (1 / (n + 1)))) ; :: thesis: verum