 reserve n,i for Nat;

theorem RecSub:
  ReciPrime is subsequence of invNAT
  proof
    deffunc F(Nat) = primenumber $1;
    consider f being Real_Sequence such that
A1: for i being Nat holds f.i = F(i) from SEQ_1:sch 1;
    reconsider f as Function of NAT, REAL;
A3: for n being Nat holds f.n is Element of NAT
    proof
      let n be Nat;
      f.n = primenumber n by A1;
      hence thesis;
    end;
    for n,m being Nat st n < m holds f.n < f.m
    proof
      let n,m be Nat;
      assume
C1:   n < m;
      f.n = primenumber n & f.m = primenumber m by A1;
      hence thesis by Cosik1,C1;
    end; then
    reconsider f as increasing sequence of NAT by A3,SEQM_3:13,1;
    take f;
    ReciPrime = invNAT * f
    proof
      for x being Element of NAT holds ReciPrime.x = (invNAT * f).x
      proof
        let x be Element of NAT;
        dom f = NAT by FUNCT_2:def 1; then
        (invNAT * f).x = invNAT.(f.x) by FUNCT_1:13
                .= invNAT.(primenumber x) by A1
                .= 1 / primenumber x by DefRec;
        hence thesis by ReciPr;
      end;
      hence thesis by FUNCT_2:def 8;
    end;
    hence thesis;
  end;
