reserve n,i,k,m for Nat;
reserve r,r1,r2,s,s1,s2 for Real;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL 2,
  f,f1,f2 for FinSequence of the carrier of TOP-REAL 2,
  p,p1,p2,p3,q,q3 for Point of TOP-REAL 2;

theorem Th26:
  for f st Q = union{LSeg(f,i): 1<=i & i+1<=len f} holds Q is closed
proof
  let f;
  reconsider F = {LSeg(f,i): 1<=i & i+1<=len f} as Subset-Family of TOP-REAL 2
  by Th25;
  now
    let P;
    assume P in F;
    then ex i st LSeg(f,i)=P & 1<=i & i+1<=len f;
    hence P is closed;
  end;
  then
A1: F is closed by TOPS_2:def 2;
  F is finite by Th23;
  hence thesis by A1,TOPS_2:21;
end;
