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 Th28:
  f is special alternating & 1 <= i & i+2 <= len f & (f/.i)`2 = (f
  /.(i+1))`2 implies (f/.(i+1))`1 = (f/.(i+2))`1
proof
  assume that
A1: f is special and
A2: f is alternating & 1 <= i and
A3: i+2 <= len f;
  set p2 = f/.(i+1), p3 = f/.(i+2);
  1 <= i+1 & i+1+1 = i+(1+1) by NAT_1:11;
  then p2`1 = p3`1 or p2`2 = p3`2 by A1,A3;
  hence thesis by A2,A3;
end;
