reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve p for Prime;

theorem Th15:
  p < primenumber(k) implies p in rng primesFinS k
  proof
    set f = primesFinS k;
    set i = primeindex p;
    assume
A1: p < primenumber(k);
    then
A2: 1+i in dom f by Th13,Th12;
    f.(1+i) = p by A1,Th12,Th14;
    hence thesis by A2,FUNCT_1:def 3;
  end;
