
theorem Th10:
  for X being set, R being Order of X, B being finite Subset of X,
    x being object st B = {x} holds
      SgmX(R,B) = <*x*>
proof
  let X be set, R be Order of X, B be finite Subset of X, x be object;
  assume A1: B = {x};
  set fin = <*x*>;
  A2: rng fin = B by A1,FINSEQ_1:38;
  then reconsider fin as FinSequence of X by FINSEQ_1:def 4;
  for n,m be Nat st n in dom fin & m in dom fin & n < m holds
    fin/.n <> fin/.m & [fin/.n, fin/.m] in R
  proof
    let n, m be Nat;
    assume that
      A3: n in dom fin and
      A4: m in dom fin and
      A5: n < m;
    assume not (fin/.n <> fin/.m & [fin/.n, fin/.m] in R);
    n in Seg 1 & m in Seg 1 by A3, A4, FINSEQ_1:38;
    then n = 1 & m = 1 by FINSEQ_1:2, TARSKI:def 1;
    hence contradiction by A5;
  end;
  hence SgmX(R,B) = <*x*> by A2, PRE_POLY:9;
end;
