set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A25: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A26: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A25, FINSEQ_6:92;
then (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A25, SPRECT_5:23;
then (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A25, A26, SPRECT_5:22, XXREAL_0:2;
then (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A25, A26, SPRECT_5:24, XXREAL_0:2;
then A27: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A25, A26, SPRECT_5:25, XXREAL_0:2;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A28: 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;
A29: len (Lower_Seq (C,n)) = ((len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) - ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + 1 by Th9;

now :: thesis: for i1, j1 being object st i1 in dom (Lower_Seq (C,n)) & j1 in dom (Lower_Seq (C,n)) & (Lower_Seq (C,n)) . i1 = (Lower_Seq (C,n)) . j1 holds
not i1 <> j1

let i1, j1 be object ; :: thesis: ( i1 in dom (Lower_Seq (C,n)) & j1 in dom (Lower_Seq (C,n)) & (Lower_Seq (C,n)) . i1 = (Lower_Seq (C,n)) . j1 implies not i1 <> j1 )
assume that
A30: i1 in dom (Lower_Seq (C,n)) and
A31: j1 in dom (Lower_Seq (C,n)) and
A32: (Lower_Seq (C,n)) . i1 = (Lower_Seq (C,n)) . j1 and
A33: i1 <> j1 ; :: thesis: contradiction
reconsider i2 = i1, j2 = j1 as Nat by A30, A31;
A34: i2 in Seg (len (Lower_Seq (C,n))) by A30, FINSEQ_1:def 3;
then i2 >= 1 by FINSEQ_1:1;
then A35: i2 = (i2 -' 1) + 1 by XREAL_1:235;
A36: (Lower_Seq (C,n)) . i1 = (Lower_Seq (C,n)) /. i2 by A30, PARTFUN1:def 6
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A28, A30, A35, FINSEQ_5:52 ;
A37: j2 in Seg (len (Lower_Seq (C,n))) by A31, FINSEQ_1:def 3;
then j2 >= 1 by FINSEQ_1:1;
then A38: j2 = (j2 -' 1) + 1 by XREAL_1:235;
A39: (Lower_Seq (C,n)) . j1 = (Lower_Seq (C,n)) /. j2 by A31, PARTFUN1:def 6
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. ((j2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A28, A31, A38, FINSEQ_5:52 ;
0 + 1 <= i2 by A34, FINSEQ_1:1;
then A40: i2 - 1 >= 0 by XREAL_1:19;
i2 <= ((len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) - ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + 1 by A29, A34, FINSEQ_1:1;
then i2 - 1 <= (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) - ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by XREAL_1:20;
then (i2 - 1) + ((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 XREAL_1:19;
then A41: (i2 -' 1) + ((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 A40, XREAL_0:def 2;
0 + 1 <= j2 by A37, FINSEQ_1:1;
then A42: j2 - 1 >= 0 by XREAL_1:19;
j2 <= ((len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) - ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + 1 by A29, A37, FINSEQ_1:1;
then j2 - 1 <= (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) - ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by XREAL_1:20;
then (j2 - 1) + ((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 XREAL_1:19;
then A43: (j2 -' 1) + ((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 A42, XREAL_0:def 2;
(j2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) >= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by NAT_1:11;
then A44: 1 < (j2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A27, XXREAL_0:2;
(i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) >= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by NAT_1:11;
then A45: 1 < (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A27, XXREAL_0:2;

per cases ( i2 < j2 or j2 < i2 ) by A33, XXREAL_0:1;

suppose i2 < j2 ; :: thesis: contradiction

then i2 - 1 < j2 - 1 by XREAL_1:9;
then i2 - 1 < j2 -' 1 by XREAL_0:def 2;
then i2 -' 1 < j2 -' 1 by A40, XREAL_0:def 2;
then (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) < (j2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by XREAL_1:6;
hence contradiction by A32, A36, A39, A43, A45, GOBOARD7:37; :: thesis: verum

end;

suppose j2 < i2 ; :: thesis: contradiction

then j2 - 1 < i2 - 1 by XREAL_1:9;
then j2 - 1 < i2 -' 1 by XREAL_0:def 2;
then j2 -' 1 < i2 -' 1 by A42, XREAL_0:def 2;
then (j2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) < (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by XREAL_1:6;
hence contradiction by A32, A36, A39, A41, A44, GOBOARD7:37; :: thesis: verum

end;

end;

end;

hence Lower_Seq (C,n) is one-to-one by FUNCT_1:def 4; :: thesis: ( Lower_Seq (C,n) is special & Lower_Seq (C,n) is unfolded & Lower_Seq (C,n) is s.n.c. )
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A46: 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;
hence Lower_Seq (C,n) is special by SPPOL_2:39; :: thesis: ( Lower_Seq (C,n) is unfolded & Lower_Seq (C,n) is s.n.c. )
thus Lower_Seq (C,n) is unfolded by A46, SPPOL_2:27; :: thesis: Lower_Seq (C,n) is s.n.c.
let i, j be Nat; :: according to TOPREAL1:def 7 :: thesis: ( j <= i + 1 or LSeg ((Lower_Seq (C,n)),i) misses LSeg ((Lower_Seq (C,n)),j) )
assume A47: i + 1 < j ; :: thesis: LSeg ((Lower_Seq (C,n)),i) misses LSeg ((Lower_Seq (C,n)),j)
i + 1 >= 1 by NAT_1:11;
then j = (j -' 1) + 1 by A47, XREAL_1:235, XXREAL_0:2;
then A48: LSeg ((Lower_Seq (C,n)),j) = LSeg ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),((j -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))))) by A46, SPPOL_2:10;

now :: thesis: (LSeg ((Lower_Seq (C,n)),i)) /\ (LSeg ((Lower_Seq (C,n)),j)) = {}

per cases ( i > 0 or i = 0 ) ;

suppose i > 0 ; :: thesis: (LSeg ((Lower_Seq (C,n)),i)) /\ (LSeg ((Lower_Seq (C,n)),j)) = {}

then A49: i >= 0 + 1 by NAT_1:13;
then i = (i -' 1) + 1 by XREAL_1:235;
then A50: LSeg ((Lower_Seq (C,n)),i) = LSeg ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),((i -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))))) by A28, SPPOL_2:10;
(i + 1) - 1 < j - 1 by A47, XREAL_1:9;
then A51: (i - 1) + 1 < j -' 1 by XREAL_0:def 2;
i - 1 >= 0 by A49, XREAL_1:19;
then (i -' 1) + 1 < j -' 1 by A51, XREAL_0:def 2;
then ((i -' 1) + 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) < (j -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by XREAL_1:6;
then A52: ((i -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + 1 < (j -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) ;
A53: (i -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) > 0 + 1 by A27, XREAL_1:8;

now :: thesis: (LSeg ((Lower_Seq (C,n)),i)) /\ (LSeg ((Lower_Seq (C,n)),j)) = {}

per cases ( ((j -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + 1 <= len (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)))))))) + 1 > len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) ;

suppose ((j -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ; :: thesis: (LSeg ((Lower_Seq (C,n)),i)) /\ (LSeg ((Lower_Seq (C,n)),j)) = {}

then (j -' 1) + ((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 NAT_1:13;
then LSeg ((Lower_Seq (C,n)),i) misses LSeg ((Lower_Seq (C,n)),j) by A48, A50, A52, A53, GOBOARD5:def 4;
hence (LSeg ((Lower_Seq (C,n)),i)) /\ (LSeg ((Lower_Seq (C,n)),j)) = {} ; :: thesis: verum

end;

suppose ((j -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + 1 > len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ; :: thesis: (LSeg ((Lower_Seq (C,n)),i)) /\ (LSeg ((Lower_Seq (C,n)),j)) = {}

then LSeg ((Lower_Seq (C,n)),j) = {} by A48, TOPREAL1:def 3;
hence (LSeg ((Lower_Seq (C,n)),i)) /\ (LSeg ((Lower_Seq (C,n)),j)) = {} ; :: thesis: verum

end;

end;

end;

hence (LSeg ((Lower_Seq (C,n)),i)) /\ (LSeg ((Lower_Seq (C,n)),j)) = {} ; :: thesis: verum

end;

suppose i = 0 ; :: thesis: (LSeg ((Lower_Seq (C,n)),i)) /\ (LSeg ((Lower_Seq (C,n)),j)) = {}

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

end;

end;

end;

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