set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A1: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A2: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
A3: len (Upper_Seq (C,n)) = (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by Th12;

now

let i1, j1 be set ; :: thesis: ( i1 in dom (Upper_Seq (C,n)) & j1 in dom (Upper_Seq (C,n)) & (Upper_Seq (C,n)) . i1 = (Upper_Seq (C,n)) . j1 implies not i1 <> j1 )
assume that
A4: i1 in dom (Upper_Seq (C,n)) and
A5: j1 in dom (Upper_Seq (C,n)) and
A6: (Upper_Seq (C,n)) . i1 = (Upper_Seq (C,n)) . j1 and
A7: i1 <> j1 ; :: thesis: contradiction
reconsider i2 = i1, j2 = j1 as Element of NAT by A4, A5;
A8: i2 in Seg ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A3, A4, FINSEQ_1:def 3;
then A9: 1 <= i2 by FINSEQ_1:1;
A10: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33;
A11: j2 in Seg ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A3, A5, FINSEQ_1:def 3;
then A12: 1 <= j2 by FINSEQ_1:1;
j2 <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A11, FINSEQ_1:1;
then A13: j2 < len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A10, SPRECT_5:16, XXREAL_0:2;
A14: (Upper_Seq (C,n)) . j1 = (Upper_Seq (C,n)) /. j2 by A5, PARTFUN1:def 6
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. j2 by A2, A11, FINSEQ_5:43 ;
i2 <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A8, FINSEQ_1:1;
then A15: i2 < len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A10, SPRECT_5:16, XXREAL_0:2;
A16: (Upper_Seq (C,n)) . i1 = (Upper_Seq (C,n)) /. i2 by A4, PARTFUN1:def 6
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. i2 by A2, A8, FINSEQ_5:43 ;

per cases ( i2 < j2 or j2 < i2 ) by A7, XXREAL_0:1;

suppose i2 < j2 ; :: thesis: contradiction

hence contradiction by A6, A16, A14, A9, A13, GOBOARD7:36; :: thesis: verum

end;

suppose j2 < i2 ; :: thesis: contradiction

hence contradiction by A6, A16, A14, A12, A15, GOBOARD7:36; :: thesis: verum

end;

end;

end;

hence Upper_Seq (C,n) is one-to-one by FUNCT_1:def 4; :: thesis: ( Upper_Seq (C,n) is special & Upper_Seq (C,n) is unfolded & Upper_Seq (C,n) is s.n.c. )
thus Upper_Seq (C,n) is special ; :: thesis: ( Upper_Seq (C,n) is unfolded & Upper_Seq (C,n) is s.n.c. )
thus Upper_Seq (C,n) is unfolded ; :: thesis: Upper_Seq (C,n) is s.n.c.
let i, j be Nat; :: according to TOPREAL1:def 7 :: thesis: ( j <= i + 1 or LSeg ((Upper_Seq (C,n)),i) misses LSeg ((Upper_Seq (C,n)),j) )
assume A17: i + 1 < j ; :: thesis: LSeg ((Upper_Seq (C,n)),i) misses LSeg ((Upper_Seq (C,n)),j)
len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 0 by NAT_1:3;
then 0 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by NAT_1:13;
then A18: len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) in dom (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_3:25;
A19: ( i in NAT & j in NAT ) by ORDINAL1:def 12;

now

per cases ( j + 1 <= len (Upper_Seq (C,n)) or j + 1 > len (Upper_Seq (C,n)) ) ;

suppose A20: j + 1 <= len (Upper_Seq (C,n)) ; :: thesis: LSeg ((Upper_Seq (C,n)),i) misses LSeg ((Upper_Seq (C,n)),j)

A21: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A2, FINSEQ_4:21;
A22: j + 1 <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A2, A20, FINSEQ_5:42;

A23: now

per cases ( j + 1 < (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) or j + 1 = (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A22, XXREAL_0:1;

suppose j + 1 < (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ; :: thesis: j + 1 < len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))

hence j + 1 < len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A21, XXREAL_0:2; :: thesis: verum

end;

suppose A24: j + 1 = (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ; :: thesis: not j + 1 >= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))

assume j + 1 >= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ; :: thesis: contradiction
then (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A21, A24, XXREAL_0:1;
then E-max (L~ (Cage (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) . (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A2, FINSEQ_4:19
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A18, PARTFUN1:def 6
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def 1
.= W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:92 ;
hence contradiction by TOPREAL5:19; :: thesis: verum

end;

end;

end;

j < len (Upper_Seq (C,n)) by A20, NAT_1:13;
then A25: LSeg ((Upper_Seq (C,n)),i) = LSeg ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),i) by A17, SPPOL_2:9, XXREAL_0:2;
LSeg ((Upper_Seq (C,n)),j) = LSeg ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),j) by A20, SPPOL_2:9;
hence LSeg ((Upper_Seq (C,n)),i) misses LSeg ((Upper_Seq (C,n)),j) by A17, A19, A25, A23, GOBOARD5:def 4; :: thesis: verum

end;

suppose j + 1 > len (Upper_Seq (C,n)) ; :: thesis: LSeg ((Upper_Seq (C,n)),i) misses LSeg ((Upper_Seq (C,n)),j)

then LSeg ((Upper_Seq (C,n)),j) = {} by TOPREAL1:def 3;
then (LSeg ((Upper_Seq (C,n)),i)) /\ (LSeg ((Upper_Seq (C,n)),j)) = {} ;
hence LSeg ((Upper_Seq (C,n)),i) misses LSeg ((Upper_Seq (C,n)),j) by XBOOLE_0:def 7; :: thesis: verum

end;

end;

end;

hence LSeg ((Upper_Seq (C,n)),i) misses LSeg ((Upper_Seq (C,n)),j) ; :: thesis: verum