defpred S₂[ set ] means f orbit $1 c= A;
consider Z being set such that
A9: for a being set holds
( a in Z iff ( a in dom f & S₂[a] ) ) from XBOOLE_0:sch 1();
A10: Z c= dom f

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( a nin Z or not a nin dom f )
thus ( a nin Z or not a nin dom f ) by A9; :: thesis: verum

end;

defpred S₃[ set , set ] means ex n being Nat st
( $2 = (iter (f,n)) . $1 & $2 nin A & ( for i being Nat st i < n holds
(iter (f,i)) . $1 in A ) );
A11: for a being set st a in (dom f) \ Z holds
ex b being set st S₃[a,b]

proof

let a be set ; :: thesis: ( a in (dom f) \ Z implies ex b being set st S₃[a,b] )
assume A12: a in (dom f) \ Z ; :: thesis: ex b being set st S₃[a,b]
then a nin Z by XBOOLE_0:def 5;
then not f orbit a c= A by A9, A12;
then consider y being set such that
A13: y in f orbit a and
A14: y nin A by TARSKI:def 3;
A15: ex n1 being Element of NAT st
( y = (iter (f,n1)) . a & a in dom (iter (f,n1)) ) by A13;
defpred S₄[ Nat] means (iter (f,$1)) . a nin A;
A16: ex n being Nat st S₄[n] by A14, A15;
consider n being Nat such that
A17: S₄[n] and
A18: for m being Nat st S₄[m] holds
n <= m from NAT_1:sch 5(A16);
take b = (iter (f,n)) . a; :: thesis: S₃[a,b]
take n ; :: thesis: ( b = (iter (f,n)) . a & b nin A & ( for i being Nat st i < n holds
(iter (f,i)) . a in A ) )
thus ( b = (iter (f,n)) . a & b nin A ) by A17; :: thesis: for i being Nat st i < n holds
(iter (f,i)) . a in A
let i be Nat; :: thesis: ( i < n implies (iter (f,i)) . a in A )
thus ( i < n implies (iter (f,i)) . a in A ) by A18; :: thesis: verum

end;

consider h being Function such that
A19: dom h = (dom f) \ Z and
A20: for a being set st a in (dom f) \ Z holds
S₃[a,h . a] from CLASSES1:sch 1(A11);
take i = ((Z --> 0) +* (g | Z)) +* h; :: thesis: ( dom i = dom f & ( for a being set st a in dom f holds
( ( f orbit a c= A implies i . a = g . a ) & ( for n being Nat st (iter (f,n)) . a nin A & ( for i being Nat st i < n holds
(iter (f,i)) . a in A ) holds
i . a = (iter (f,n)) . a ) ) ) )
dom (Z --> 0) = Z by FUNCOP_1:13;
then dom ((Z --> 0) +* (g | Z)) = Z \/ (dom (g | Z)) by FUNCT_4:def 1
.= Z by RELAT_1:58, XBOOLE_1:12 ;
hence dom i = Z \/ ((dom f) \ Z) by A19, FUNCT_4:def 1
.= dom f by A10, XBOOLE_1:45 ;
:: thesis: for a being set st a in dom f holds
( ( f orbit a c= A implies i . a = g . a ) & ( for n being Nat st (iter (f,n)) . a nin A & ( for i being Nat st i < n holds
(iter (f,i)) . a in A ) holds
i . a = (iter (f,n)) . a ) )
let a be set ; :: thesis: ( a in dom f implies ( ( f orbit a c= A implies i . a = g . a ) & ( for n being Nat st (iter (f,n)) . a nin A & ( for i being Nat st i < n holds
(iter (f,i)) . a in A ) holds
i . a = (iter (f,n)) . a ) ) )
assume A21: a in dom f ; :: thesis: ( ( f orbit a c= A implies i . a = g . a ) & ( for n being Nat st (iter (f,n)) . a nin A & ( for i being Nat st i < n holds
(iter (f,i)) . a in A ) holds
i . a = (iter (f,n)) . a ) )

hereby :: thesis: for n being Nat st (iter (f,n)) . a nin A & ( for i being Nat st i < n holds
(iter (f,i)) . a in A ) holds
i . a = (iter (f,n)) . a

assume f orbit a c= A ; :: thesis: i . a = g . a
then A22: a in Z by A9, A21;
then a nin (dom f) \ Z by XBOOLE_0:def 5;
then A23: i . a = ((Z --> 0) +* (g | Z)) . a by A19, FUNCT_4:11;

per cases ( a in dom (g | Z) or a nin dom (g | Z) ) ;

suppose a in dom (g | Z) ; :: thesis: i . a = g . a

then i . a = (g | Z) . a by A23, FUNCT_4:13;
hence i . a = g . a by A22, FUNCT_1:49; :: thesis: verum

end;

suppose A24: a nin dom (g | Z) ; :: thesis: i . a = g . a

then A25: i . a = (Z --> 0) . a by A23, FUNCT_4:11;
A26: (Z --> 0) . a = 0 by A22, FUNCOP_1:7;
a nin dom g by A22, A24, RELAT_1:57;
hence i . a = g . a by A25, A26, FUNCT_1:def 2; :: thesis: verum

end;

end;

end;

let n be Nat; :: thesis: ( (iter (f,n)) . a nin A & ( for i being Nat st i < n holds
(iter (f,i)) . a in A ) implies i . a = (iter (f,n)) . a )
reconsider n9 = n as Element of NAT by ORDINAL1:def 12;
assume that
A27: (iter (f,n)) . a nin A and
A28: for i being Nat st i < n holds
(iter (f,i)) . a in A ; :: thesis: i . a = (iter (f,n)) . a
a in dom (iter (f,n9)) by A8, A21;
then (iter (f,n)) . a in f orbit a ;
then not f orbit a c= A by A27;
then a nin Z by A9;
then A29: a in (dom f) \ Z by A21, XBOOLE_0:def 5;
then consider n2 being Nat such that
A30: h . a = (iter (f,n2)) . a and
A31: h . a nin A and
A32: for i being Nat st i < n2 holds
(iter (f,i)) . a in A by A20;
A33: n <= n2 by A28, A30, A31;
n2 <= n by A27, A32;
then n = n2 by A33, XXREAL_0:1;
hence i . a = (iter (f,n)) . a by A19, A29, A30, FUNCT_4:13; :: thesis: verum