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 Th33:
  f is special alternating & 1<=i & i+2<=len f & LSeg(f,i) is
  horizontal implies LSeg(f,i+1) is vertical
proof
  assume that
A1: f is special & f is alternating and
A2: 1<=i and
A3: i+2<=len f and
A4: LSeg(f,i) is horizontal;
  i+1 <= i+2 by XREAL_1:6;
  then i+1 <= len f by A3,XXREAL_0:2;
  then LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A2,TOPREAL1:def 3;
  then (f/.i)`2 = (f/.(i+1))`2 by A4,Th15;
  then (f/.(i+1))`1 = (f/.(i+2))`1 by A1,A2,A3,Th28;
  then
A5: LSeg(f/.(i+1),f/.(i+2)) is vertical by Th16;
  1 <= i+1 & i+1+1 = i+(1+1) by NAT_1:11;
  hence thesis by A3,A5,TOPREAL1:def 3;
end;
