reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem Th26:
  for S being Segmentation of C
  for i,j st 1<=i & i< j & j<len S & i,j are_adjacent
  holds Segm(S,i) /\ Segm(S,j)={S/.(i+1)}
proof
  let S be Segmentation of C;
  let i,j such that
A1: 1<=i and
A2: i< j and
A3: j<len S and
A4: i,j are_adjacent;
  i < len S by A2,A3,XXREAL_0:2;
  then
A5: Segm(S,i) = Segment(S/.i,S/.(i+1),C) by A1,Def4;
  1 < j by A1,A2,XXREAL_0:2;
  then
A6: Segm(S,j) = Segment(S/.j,S/.(j+1),C) by A3,Def4;
  j+1 > i by A2,NAT_1:13;
  then j = i+1 by A4,GOBRD10:def 1;
  then j+1 = i+(1+1);
  hence thesis by A1,A3,A5,A6,Def3;
end;
