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

theorem Th32:
  for m,n being Nat st 1 <= n & n <= num-polytopes(p,k) & 1 <= m &
  m <= num-polytopes(p,k) & n-th-polytope(p,k) = m-th-polytope(p,k) holds m = n
proof
  let m,n be Nat such that
A1: 1 <= n & n <= num-polytopes(p,k) and
A2: 1 <= m & m <= num-polytopes(p,k) and
A3: n-th-polytope(p,k) = m-th-polytope(p,k);
  set s = k-polytope-seq(p);
A4: s is one-to-one by Th31;
  m in Seg (num-polytopes(p,k)) by A2;
  then
A5: m in dom s by Th25;
  n in Seg (num-polytopes(p,k)) by A1;
  then
A6: n in dom s by Th25;
  n-th-polytope(p,k) = s.n & m-th-polytope(p,k) = s.m by A1,A2,Def12;
  hence thesis by A3,A6,A5,A4,FUNCT_1:def 4;
end;
