reconsider fZ = f | Z as Element of PFuncs (REAL,REAL) by PARTFUN1:45;
defpred S₁[ set , set , set ] means ex h being PartFunc of REAL,REAL st
( $2 = h & $3 = h `| Z );
A1: for n being 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 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 PARTFUN1:46;
reconsider y = x9 `| Z as Element of PFuncs (REAL,REAL) by PARTFUN1:45;
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 sequence of (PFuncs (REAL,REAL)) such that
A2: ( g . 0 = fZ & ( for n being 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 Nat holds g . (i + 1) = (g . i) `| Z ) )
thus g . 0 = f | Z by A2; :: thesis: for i being Nat holds g . (i + 1) = (g . i) `| Z
let i be Nat; :: thesis: g . (i + 1) = (g . i) `| Z
S<sub>1</sub>[i,g . i,g . (i + 1)] by A2;
hence g . (i + 1) = (g . i) `| Z ; :: thesis: verum