
theorem
  for f be non constant standard special_circular_sequence,
  j be Nat, P be Subset of TOP-REAL 2 st
  1 <= j & j <= width GoB f &
  P = LSeg ((GoB f)*(1,j), (GoB f)*(len (GoB f),j)) holds
  P is_S-P_arc_joining (GoB f)*(1,j), (GoB f)*(len (GoB f),j)
proof
  let f be non constant standard special_circular_sequence,
  j be Nat, P be Subset of TOP-REAL 2;
  assume that
A1: 1 <= j and
A2: j <= width GoB f and
A3: P = LSeg ((GoB f)*(1,j), (GoB f)*(len (GoB f),j));
  set p = (GoB f)*(1,j), q = (GoB f)*(len GoB f, j);
  1 <= len GoB f by GOBOARD7:32;
  then
A4: p`2 = q`2 by A1,A2,GOBOARD5:1;
A5: p`1 <> q`1
  proof
    assume
A6: p`1 = q`1;
A7: GoB f = GoB(Incr(X_axis(f)),Incr(Y_axis(f))) by GOBOARD2:def 2;
A8: 1 <= len GoB f by GOBOARD7:32;
    then
A9: [1,j] in Indices GoB(Incr(X_axis f),Incr(Y_axis(f))) by A1,A2,A7,
MATRIX_0:30;
A10: [len GoB f,j] in Indices GoB(Incr(X_axis f),Incr(Y_axis(f))) by A1,A2,A7
,A8,MATRIX_0:30;
    (GoB f)*(1,j)= (GoB(Incr(X_axis(f)),Incr(Y_axis(f))))*(1,j)
    by GOBOARD2:def 2
      .= |[Incr(X_axis(f)).1,Incr(Y_axis(f)).j]|
    by A9,GOBOARD2:def 1;
    then
A11: p`1 = Incr(X_axis(f)).1 by EUCLID:52;
A12: (GoB f)*(len GoB f, j) =
    (GoB(Incr(X_axis(f)),Incr(Y_axis(f))))*(len GoB f, j) by GOBOARD2:def 2
      .= |[Incr(X_axis(f)).(len GoB f), Incr(Y_axis(f)).j]|
    by A10,GOBOARD2:def 1;
A13: len Incr (X_axis(f)) = len GoB f by A7,GOBOARD2:def 1;
A14: 1 <= len GoB f by GOBOARD7:32;
A15: 1 <= len Incr (X_axis(f)) by A13,GOBOARD7:32;
A16: len GoB f in dom Incr (X_axis(f)) by A13,A14,FINSEQ_3:25;
    1 in dom Incr (X_axis(f)) by A15,FINSEQ_3:25;
    then len GoB f = 1 by A6,A11,A12,A16,EUCLID:52,SEQ_4:138;
    hence thesis by GOBOARD7:32;
  end;
  reconsider gg = <*p,q*> as FinSequence of the carrier of TOP-REAL 2;
A17: len gg = 2 by FINSEQ_1:44;
  take gg;
  thus gg is being_S-Seq by A4,A5,SPPOL_2:43;
  thus P = L~gg by A3,SPPOL_2:21;
  thus thesis by A17,FINSEQ_4:17;
end;
