reserve f for non empty FinSequence of TOP-REAL 2,
  i,j,k,k1,k2,n,i1,i2,j1,j2 for Nat,
  r,s,r1,r2 for Real,
  p,q,p1,q1 for Point of TOP-REAL 2,
  G for Go-board;
reserve f for non constant standard special_circular_sequence;

theorem Th29:
  for f being standard non empty FinSequence of TOP-REAL 2 st i
  in dom f & i+1 in dom f holds f/.i <> f/.(i+1)
proof
A1: |.0 .| = 0 by ABSVALUE:2;
  let f be standard non empty FinSequence of TOP-REAL 2 such that
A2: i in dom f and
A3: i+1 in dom f;
A4: f is_sequence_on GoB f by GOBOARD5:def 5;
  then consider i1,j1 such that
A5: [i1,j1] in Indices GoB f & f/.i = (GoB f)*(i1,j1) by A2,GOBOARD1:def 9;
  consider i2,j2 such that
A6: [i2,j2] in Indices GoB f and
A7: f/.(i+1) = (GoB f)*(i2,j2) by A3,A4,GOBOARD1:def 9;
  assume
A8: f/.i = f/.(i+1);
  then j1 = j2 by A5,A6,A7,GOBOARD1:5;
  then
A9: j1-j2 = 0;
  i1 = i2 by A5,A6,A7,A8,GOBOARD1:5;
  then i1-i2 = 0;
  then |.0 .|+|.0 .| = 1 by A2,A3,A4,A5,A7,A9,GOBOARD1:def 9;
  hence contradiction by A1;
end;
