reserve n for Nat,
  k for Integer;
reserve p for polyhedron,
  k for Integer,
  n for Nat;

theorem Th48:
  p is simply-connected iff for n being Integer holds
  n-circuit-space(p) = n-bounding-chain-space(p)
proof
  defpred Q[polyhedron] means for n being Integer holds n-circuit-space($1) =
  n-bounding-chain-space($1);
  thus p is simply-connected implies Q[p]
  proof
    assume
A1: p is simply-connected;
    let n be Integer;
    n-circuits(p) = n-bounding-chains(p) by A1;
    hence thesis by VECTSP_4:29;
  end;
  thus Q[p] implies p is simply-connected;
end;
