 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem Core1:
  rseq (0,1,1,0) = Reci-Sqf (#) Reci-TSq
  proof
    for n being Nat holds
      (rseq (0,1,1,0)).n = (Reci-Sqf.n) * (Reci-TSq.n)
    proof
      let n be Nat;
A1:   (rseq (0,1,1,0)).n = 1 / (1 * n + 0) by AlgDef;
      per cases;
      suppose
A2:     n = 0; then
        (rseq (0,1,1,0)).n = 1 / (1 * 0 + 0) by AlgDef
          .= (Reci-Sqf.n) * 0
          .= (Reci-Sqf.n) * (Reci-TSq.n) by MySum2Def,A2;
        hence thesis;
      end;
      suppose n <> 0; then
        reconsider nn = n as non zero Nat;
A3:     TSqF nn = (SqF nn) |^2 by Cosik
        .= (SqF nn)^2 by NEWTON:81;
A4:     Reci-TSq.nn = 1 / TSqF nn by MySum2Def;
        1 / nn = 1 / ((SquarefreePart nn) * (TSqF nn)) by Canonical,A3
           .= (1 / (SquarefreePart nn)) * (1 / (TSqF nn)) by XCMPLX_1:102;
        hence thesis by MySumDef,A1,A4;
      end;
    end;
    hence thesis by SEQ_1:8;
  end;
