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 Th19:
  f is special implies LSeg(f,i) is vertical or LSeg(f,i) is horizontal
proof
  assume
A1: for j be Nat st 1 <= j & j+1 <= len f holds (f/.j)`1 = (f/.(j+1))`1
  or (f/.j)`2 = (f/.(j+1))`2;
  set p1=f/.i, p2=f/.(i+1);
  per cases;
  suppose
A2: 1 <= i & i+1 <= len f;
A3: p1`2=p2`2 implies LSeg(p1,p2) is horizontal by Th15;
    p1`1=p2`1 implies LSeg(p1,p2) is vertical by Th16;
    hence thesis by A1,A2,A3,TOPREAL1:def 3;
  end;
  suppose
    not (1<=i & i+1<=len f);
    then for p,q st p in LSeg(f,i) & q in LSeg(f,i) holds p`2=q`2
    by TOPREAL1:def 3;
    hence thesis;
  end;
end;
