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

theorem Th13:
  primeindex p < k implies 1+primeindex p in dom primesFinS k
  proof
    set i = primeindex p;
    assume i < k;
    then
A1: i+1 <= k by NAT_1:13;
A2: len primesFinS k = k by Def1;
    0+1 <= i+1 by XREAL_1:6;
    hence thesis by A1,A2,FINSEQ_3:25;
  end;
