per cases ( not p in rng f or p in rng f ) ;

suppose not p in rng f ; :: thesis: Rotate (f,p) is special

hence Rotate (f,p) is special by FINSEQ_6:def 2; :: thesis: verum

end;

suppose A1: p in rng f ; :: thesis: Rotate (f,p) is special

A2: len (Rotate (f,p)) = len f by Th14;
let i be Nat; :: according to TOPREAL1:def 5 :: thesis: ( not 1 <= i or not i + 1 <= len (Rotate (f,p)) or ((Rotate (f,p)) /. i) `1 = ((Rotate (f,p)) /. (i + 1)) `1 or ((Rotate (f,p)) /. i) `2 = ((Rotate (f,p)) /. (i + 1)) `2 )
assume that
A3: 1 <= i and
A4: i + 1 <= len (Rotate (f,p)) ; :: thesis: ( ((Rotate (f,p)) /. i) `1 = ((Rotate (f,p)) /. (i + 1)) `1 or ((Rotate (f,p)) /. i) `2 = ((Rotate (f,p)) /. (i + 1)) `2 )
A5: i + 1 >= i by NAT_1:11;
A6: i in NAT by ORDINAL1:def 12;

now

A7: len (f :- p) = ((len f) - (p .. f)) + 1 by A1, FINSEQ_5:50;

per cases ( i < len (f :- p) or i >= len (f :- p) ) ;

suppose A8: i < len (f :- p) ; :: thesis: ( ((Rotate (f,p)) /. i) `1 = ((Rotate (f,p)) /. (i + 1)) `1 or ((Rotate (f,p)) /. i) `2 = ((Rotate (f,p)) /. (i + 1)) `2 )

A9: ((i + 1) -' 1) + (p .. f) = i + (p .. f) by NAT_D:34
.= ((i -' 1) + 1) + (p .. f) by A3, XREAL_1:235
.= ((i -' 1) + (p .. f)) + 1 ;
i + 1 <= len (f :- p) by A8, NAT_1:13;
then A10: (Rotate (f,p)) /. (i + 1) = f /. (((i -' 1) + (p .. f)) + 1) by A1, A9, Th9, NAT_1:11;
( 0 <= i -' 1 & 1 <= p .. f ) by A1, FINSEQ_4:21, NAT_1:3;
then A11: 1 + 0 <= (i -' 1) + (p .. f) by XREAL_1:7;
i < ((len f) + 1) - (p .. f) by A7, A8;
then i + (p .. f) < (len f) + 1 by XREAL_1:20;
then i + (p .. f) <= len f by NAT_1:13;
then A12: ((i -' 1) + 1) + (p .. f) <= len f by A3, XREAL_1:235;
(Rotate (f,p)) /. i = f /. ((i -' 1) + (p .. f)) by A1, A3, A6, A8, Th9;
hence ( ((Rotate (f,p)) /. i) `1 = ((Rotate (f,p)) /. (i + 1)) `1 or ((Rotate (f,p)) /. i) `2 = ((Rotate (f,p)) /. (i + 1)) `2 ) by A10, A11, A12, TOPREAL1:def 5; :: thesis: verum

end;

suppose A13: i >= len (f :- p) ; :: thesis: ( ((Rotate (f,p)) /. i) `1 = ((Rotate (f,p)) /. (i + 1)) `1 or ((Rotate (f,p)) /. i) `2 = ((Rotate (f,p)) /. (i + 1)) `2 )

i <= len f by A4, A5, A2, XXREAL_0:2;
then A14: (Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f)) by A1, A6, A13, Th17;
i - ((len f) - (p .. f)) >= 1 by A7, A13, XREAL_1:19;
then (i + (p .. f)) - (len f) >= 1 ;
then A15: 1 <= (i + (p .. f)) -' (len f) by NAT_D:39;
A16: i + 1 >= len (f :- p) by A5, A13, XXREAL_0:2;
then i >= (len f) - (p .. f) by A7, XREAL_1:6;
then A17: len f <= i + (p .. f) by XREAL_1:20;
((i + 1) + (p .. f)) -' (len f) = ((i + (p .. f)) + 1) -' (len f)
.= ((i + (p .. f)) -' (len f)) + 1 by A17, NAT_D:38 ;
then A18: (Rotate (f,p)) /. (i + 1) = f /. (((i + (p .. f)) -' (len f)) + 1) by A1, A4, A2, A16, Th17;
p .. f <= len f by A1, FINSEQ_4:21;
then (i + 1) + (p .. f) <= (len f) + (len f) by A4, A2, XREAL_1:7;
then ((i + (p .. f)) + 1) - (len f) <= len f by XREAL_1:20;
then ((i + (p .. f)) - (len f)) + 1 <= len f ;
then ((i + (p .. f)) -' (len f)) + 1 <= len f by A17, XREAL_1:233;
hence ( ((Rotate (f,p)) /. i) `1 = ((Rotate (f,p)) /. (i + 1)) `1 or ((Rotate (f,p)) /. i) `2 = ((Rotate (f,p)) /. (i + 1)) `2 ) by A14, A18, A15, TOPREAL1:def 5; :: thesis: verum

end;

end;

end;

hence ( ((Rotate (f,p)) /. i) `1 = ((Rotate (f,p)) /. (i + 1)) `1 or ((Rotate (f,p)) /. i) `2 = ((Rotate (f,p)) /. (i + 1)) `2 ) ; :: thesis: verum

end;

end;