defpred S1[ object , object ] means $2 = F3($1);
A1: for x being object st x in NAT holds
ex y being object st S1[x,y] ;
consider f being Function such that
A2: dom f = NAT and
A3: for x being object st x in NAT holds
S1[x,f . x] from CLASSES1:sch 1(A1);
now :: thesis: for x being set st x in NAT holds
f . x is PartFunc of F1(),F2()
let x be set ; :: thesis: ( x in NAT implies f . x is PartFunc of F1(),F2() )
assume x in NAT ; :: thesis: f . x is PartFunc of F1(),F2()
then f . x = F3(x) by A3;
hence f . x is PartFunc of F1(),F2() ; :: thesis: verum
end;
then reconsider f = f as Functional_Sequence of F1(),F2() by A2, Lm1;
take f ; :: thesis: for n being Nat holds f . n = F3(n)
let n be Nat; :: thesis: f . n = F3(n)
n in NAT by ORDINAL1:def 12;
hence f . n = F3(n) by A3; :: thesis: verum