reserve i,i1,i2,i9,i19,j,j1,j2,j9,j19,k,k1,k2,l,m,n for Nat;
reserve r,s,r9,s9 for Real;
reserve D for non empty set, f for FinSequence of D;
reserve f for FinSequence of TOP-REAL 2, G for Go-board;

theorem Th13:
  for f being standard special_circular_sequence st f
  is_sequence_on G holds Values GoB f c= Values G
proof
  let f be standard special_circular_sequence such that
A1: f is_sequence_on G;
  let p be object;
  set F = GoB f;
  assume p in Values F;
  then p in { F*(i,j): [i,j] in Indices F } by MATRIX_0:39;
  then consider i,j such that
A2: p = F*(i,j) and
A3: [i,j] in Indices F;
  reconsider p as Point of TOP-REAL 2 by A2;
A4: 1 <= j & j <= width F by A3,MATRIX_0:32;
A5: 1 <= i & i <= len F by A3,MATRIX_0:32;
  then consider k1 such that
A6: k1 in dom f and
A7: p`1 = (f/.k1)`1 by A2,A4,Lm1;
  consider k2 such that
A8: k2 in dom f and
A9: p`2 = (f/.k2)`2 by A2,A5,A4,Lm2;
  consider i2,j2 such that
A10: [i2,j2] in Indices G and
A11: f/.k2 = G*(i2,j2) by A1,A8,GOBOARD1:def 9;
A12: 1 <= i2 & i2 <= len G by A10,MATRIX_0:32;
  consider i1,j1 such that
A13: [i1,j1] in Indices G and
A14: f/.k1 = G*(i1,j1) by A1,A6,GOBOARD1:def 9;
A15: 1 <= j1 & j1 <= width G by A13,MATRIX_0:32;
A16: p = |[p`1, p`2]| by EUCLID:53;
A17: 1 <= j2 & j2 <= width G by A10,MATRIX_0:32;
A18: 1 <= i1 & i1 <= len G by A13,MATRIX_0:32;
  then
A19: [i1,j2] in Indices G by A17,MATRIX_0:30;
A20: G*(i1,j2)`2 = G*(1,j2)`2 by A18,A17,GOBOARD5:1
    .= G*(i2,j2)`2 by A12,A17,GOBOARD5:1;
  G*(i1,j2)`1 = G*(i1,1)`1 by A18,A17,GOBOARD5:2
    .= G*(i1,j1)`1 by A18,A15,GOBOARD5:2;
  then p = G*(i1,j2) by A7,A9,A14,A11,A20,A16,EUCLID:53;
  then p in { G*(k,l): [k,l] in Indices G } by A19;
  hence thesis by MATRIX_0:39;
end;
