reconsider fZ = f as Element of PFuncs ( the carrier of V, the carrier of W) by PARTFUN1:45;
defpred S₁[ set , set , set ] means ex g being PartFunc of V,W st
( $2 = g & $3 = fD (g,h) );
A1: for n being Nat
for x being Element of PFuncs ( the carrier of V, the carrier of W) ex y being Element of PFuncs ( the carrier of V, the carrier of W) st S₁[n,x,y]

proof

let n be Nat; :: thesis: for x being Element of PFuncs ( the carrier of V, the carrier of W) ex y being Element of PFuncs ( the carrier of V, the carrier of W) st S₁[n,x,y]
let x be Element of PFuncs ( the carrier of V, the carrier of W); :: thesis: ex y being Element of PFuncs ( the carrier of V, the carrier of W) st S₁[n,x,y]
reconsider x9 = x as PartFunc of the carrier of V, the carrier of W by PARTFUN1:46;
reconsider y = fD (x9,h) as Element of PFuncs ( the carrier of V, the carrier of W) by PARTFUN1:45;
ex w being PartFunc of the carrier of V, the carrier of W st
( x = w & y = fD (w,h) ) ;
hence ex y being Element of PFuncs ( the carrier of V, the carrier of W) st S₁[n,x,y] ; :: thesis: verum

end;

consider g being sequence of (PFuncs ( the carrier of V, the carrier of W)) 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 the carrier of V, the carrier of W ;
take g ; :: thesis: ( g . 0 = f & ( for n being Nat holds g . (n + 1) = fD ((g . n),h) ) )
thus g . 0 = f by A2; :: thesis: for n being Nat holds g . (n + 1) = fD ((g . n),h)
let i be Nat; :: thesis: g . (i + 1) = fD ((g . i),h)
S₁[i,g . i,g . (i + 1)] by A2;
hence g . (i + 1) = fD ((g . i),h) ; :: thesis: verum