reconsider fZ = f | Z as Element of PFuncs (REAL,REAL) by PARTFUN1:45;

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

A2: ( g . 0 = fZ & ( for n being Nat holds S_{1}[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_{1}[i,g . i,g . (i + 1)]
by A2;

hence g . (i + 1) = (g . i) `| Z ; :: thesis: verum

defpred S

( $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 S

proof

consider g being sequence of (PFuncs (REAL,REAL)) such that
let n be Nat; :: thesis: for x being Element of PFuncs (REAL,REAL) ex y being Element of PFuncs (REAL,REAL) st S_{1}[n,x,y]

let x be Element of PFuncs (REAL,REAL); :: thesis: ex y being Element of PFuncs (REAL,REAL) st S_{1}[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 S_{1}[n,x,y]
; :: thesis: verum

end;let x be Element of PFuncs (REAL,REAL); :: thesis: ex y being Element of PFuncs (REAL,REAL) st S

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 S

A2: ( g . 0 = fZ & ( for n being Nat holds S

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

hence g . (i + 1) = (g . i) `| Z ; :: thesis: verum