set 00 = the Element of F₁();
defpred S₁[ Nat, set , set ] means ( ( 0 <= $1 & $1 <= F₃() - 2 implies P₁[$1 + 1,$2,$3] ) & ( ( not 0 <= $1 or not $1 <= F₃() - 2 ) implies $3 = the Element of F₁() ) );
A2: for n being Nat
for x being Element of F₁() ex y being Element of F₁() st S₁[n,x,y]

proof

let n be Nat; :: thesis: for x being Element of F₁() ex y being Element of F₁() st S₁[n,x,y]
let x be Element of F₁(); :: thesis: ex y being Element of F₁() st S₁[n,x,y]
( 0 <= n & n <= F₃() - 2 implies ex y being Element of F₁() st S₁[n,x,y] )

proof

A3: 0 + 1 <= n + 1 by XREAL_1:6;
assume that
0 <= n and
A4: n <= F₃() - 2 ; :: thesis: ex y being Element of F₁() st S₁[n,x,y]
n + 1 <= (F₃() - 2) + 1 by A4, XREAL_1:6;
then consider y being Element of F₁() such that
A5: P₁[n + 1,x,y] by A1, A3;
take y ; :: thesis: S₁[n,x,y]
thus ( 0 <= n & n <= F₃() - 2 implies P₁[n + 1,x,y] ) by A5; :: thesis: ( ( not 0 <= n or not n <= F₃() - 2 ) implies y = the Element of F₁() )
thus ( ( not 0 <= n or not n <= F₃() - 2 ) implies y = the Element of F₁() ) by A4; :: thesis: verum

end;

hence ex y being Element of F₁() st S₁[n,x,y] ; :: thesis: verum

end;

consider f being sequence of F₁() such that
A6: ( f . 0 = F₂() & ( for n being Nat holds S₁[n,f . n,f . (n + 1)] ) ) from RECDEF_1:sch 2(A2);
defpred S₂[ object , object ] means for r being Real st r = $1 holds
$2 = f . (r - 1);
A7: for x being object st x in REAL holds
ex y being object st S₂[x,y]

proof

let x be object ; :: thesis: ( x in REAL implies ex y being object st S₂[x,y] )
assume x in REAL ; :: thesis: ex y being object st S₂[x,y]
then reconsider r = x as Real ;
take f . (r - 1) ; :: thesis: S₂[x,f . (r - 1)]
thus S₂[x,f . (r - 1)] ; :: thesis: verum

end;

consider g being Function such that
A8: ( dom g = REAL & ( for x being object st x in REAL holds
S₂[x,g . x] ) ) from CLASSES1:sch 1(A7);
Seg F₃() c= REAL by NUMBERS:19;
then A9: dom (g | (Seg F₃())) = Seg F₃() by A8, RELAT_1:62;
then reconsider p = g | (Seg F₃()) as FinSequence by FINSEQ_1:def 2;

now :: thesis: for x being object st x in rng p holds
x in F₁()

let x be object ; :: thesis: ( x in rng p implies x in F₁() )
assume x in rng p ; :: thesis: x in F₁()
then consider y being object such that
A10: y in dom p and
A11: x = p . y by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A10;
A12: f . (y - 1) in F₁()

proof

y <> 0 by A9, A10, FINSEQ_1:1;
then consider k being Nat such that
A13: y = k + 1 by NAT_1:6;
reconsider k = k as Element of NAT by ORDINAL1:def 12;
f . k in F₁() ;
hence f . (y - 1) in F₁() by A13; :: thesis: verum

end;

A14: y in REAL by XREAL_0:def 1;
p . y = g . y by A10, FUNCT_1:47;
hence x in F₁() by A8, A11, A12, A14; :: thesis: verum

end;

then rng p c= F₁() ;
then reconsider p = p as FinSequence of F₁() by FINSEQ_1:def 4;
take p ; :: thesis: ( len p = F₃() & ( p /. 1 = F₂() or F₃() = 0 ) & ( for n being Nat st 1 <= n & n <= F₃() - 1 holds
P₁[n,p /. n,p /. (n + 1)] ) )
thus len p = F₃() by A9, FINSEQ_1:def 3; :: thesis: ( ( p /. 1 = F₂() or F₃() = 0 ) & ( for n being Nat st 1 <= n & n <= F₃() - 1 holds
P₁[n,p /. n,p /. (n + 1)] ) )
( F₃() <> 0 implies p /. 1 = F₂() )

proof

assume F₃() <> 0 ; :: thesis: p /. 1 = F₂()
then consider k being Nat such that
A15: F₃() = k + 1 by NAT_1:6;
0 + 1 <= k + 1 by XREAL_1:7;
then A16: 1 in Seg F₃() by A15, FINSEQ_1:1;
then p /. 1 = p . 1 by A9, PARTFUN1:def 6
.= g . 1 by A16, FUNCT_1:49
.= f . (1 - 1) by A8, Lm2
.= F₂() by A6 ;
hence p /. 1 = F₂() ; :: thesis: verum

end;

hence ( p /. 1 = F₂() or F₃() = 0 ) ; :: thesis: for n being Nat st 1 <= n & n <= F₃() - 1 holds
P₁[n,p /. n,p /. (n + 1)]
let n be Nat; :: thesis: ( 1 <= n & n <= F₃() - 1 implies P₁[n,p /. n,p /. (n + 1)] )
assume that
A17: 1 <= n and
A18: n <= F₃() - 1 ; :: thesis: P₁[n,p /. n,p /. (n + 1)]
consider k being Nat such that
A19: n = k + 1 by A17, NAT_1:6;
A20: for n being Nat st n <= F₃() - 1 holds
p /. (n + 1) = f . n

proof

let n be Nat; :: thesis: ( n <= F₃() - 1 implies p /. (n + 1) = f . n )
assume n <= F₃() - 1 ; :: thesis: p /. (n + 1) = f . n
then A21: n + 1 <= (F₃() - 1) + 1 by XREAL_1:6;
n + 1 in REAL by XREAL_0:def 1;
then A22: g . (n + 1) = f . ((n + 1) - 1) by A8
.= f . n ;
1 <= n + 1 by NAT_1:11;
then A23: n + 1 in Seg F₃() by A21, FINSEQ_1:1;
then p /. (n + 1) = p . (n + 1) by A9, PARTFUN1:def 6;
hence p /. (n + 1) = f . n by A23, A22, FUNCT_1:49; :: thesis: verum

end;

A24: k <= k + 1 by NAT_1:11;
k <= (F₃() - 1) - 1 by A18, A19, XREAL_1:19;
then P₁[k + 1,f . k,f . (k + 1)] by A6;
then P₁[k + 1,f . k,p /. ((k + 1) + 1)] by A20, A18, A19;
hence P₁[n,p /. n,p /. (n + 1)] by A20, A18, A19, A24, XXREAL_0:2; :: thesis: verum