reconsider fZ = f | Z as Element of PFuncs REAL ,REAL by PARTFUN1:119;
defpred S₁[ set , set , set ] means ex h being PartFunc of REAL ,REAL st
( $2 = h & $3 = h `| Z );
A1: for n being Element of NAT
for x being Element of PFuncs REAL ,REAL ex y being Element of PFuncs REAL ,REAL st S₁[n,x,y]

proof

let n be Element of NAT ; :: thesis: for x being Element of PFuncs REAL ,REAL ex y being Element of PFuncs REAL ,REAL st S₁[n,x,y]
let x be Element of PFuncs REAL ,REAL ; :: thesis: ex y being Element of PFuncs REAL ,REAL st S₁[n,x,y]
reconsider x9 = x as PartFunc of REAL ,REAL by PARTFUN1:120;
reconsider y = x9 `| Z as Element of PFuncs REAL ,REAL by PARTFUN1:119;
ex h being PartFunc of REAL ,REAL st
( x = h & y = h `| Z ) ;
hence ex y being Element of PFuncs REAL ,REAL st S₁[n,x,y] ; :: thesis: verum

end;

consider g being Function of NAT ,(PFuncs REAL ,REAL ) such that
A2: ( g . 0 = fZ & ( for n being Element of NAT holds S₁[n,g . n,g . (n + 1)] ) ) from RECDEF_1:sch 2(A1);
reconsider g = g as Functional_Sequence of REAL ,REAL ;
take g ; :: thesis: ( g . 0 = f | Z & ( for i being natural number holds g . (i + 1) = (g . i) `| Z ) )
thus g . 0 = f | Z by A2; :: thesis: for i being natural number holds g . (i + 1) = (g . i) `| Z
let i be natural number ; :: thesis: g . (i + 1) = (g . i) `| Z
i is Element of NAT by ORDINAL1:def 13;
then S<sub>1</sub>[i,g . i,g . (i + 1)] by A2;
hence g . (i + 1) = (g . i) `| Z ; :: thesis: verum