defpred S₁[ Element of NAT , Element of ExtREAL ] means ex Y being non empty Subset of ExtREAL st
( Y = { (seq . k) where k is Element of NAT : $1 <= k } & $2 = sup Y );
A1: for n being Element of NAT ex y being Element of ExtREAL st S₁[n,y]

proof

let n be Element of NAT ; :: thesis: ex y being Element of ExtREAL st S₁[n,y]
reconsider Y = { (seq . k) where k is Element of NAT : n <= k } as non empty Subset of ExtREAL by Th5;
reconsider y = sup Y as Element of ExtREAL ;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

thus ex f being Function of NAT,ExtREAL st
for n being Element of NAT holds S₁[n,f . n] from FUNCT_2:sch 3(A1); :: thesis: verum