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 Th29:
  f is special alternating & 1 <= i & i+2 <= len f & p1 = f/.i &
p2 = f/.(i+1) & p3 = f/.(i+2) implies p1`1 = p2`1 & p3`1 <> p2`1 or p1`2 = p2`2
  & p3`2 <> p2`2
proof
  assume that
A1: f is special and
A2: f is alternating and
A3: 1 <= i and
A4: i+2 <= len f and
A5: p1 = f/.i and
A6: p2 = f/.(i+1) and
A7: p3 = f/.(i+2);
  i+1 <= i+2 by XREAL_1:6;
  then i+1 <= len f by A4,XXREAL_0:2;
  then p1`1 = p2`1 or p1`2 = p2`2 by A1,A3,A5,A6;
  hence thesis by A2,A3,A4,A5,A7;
end;
