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

let i be Nat; :: according to TOPREAL1:def 7 :: 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
A2: 1 <= i and
A3: 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 )
A4: i in NAT by ORDINAL1:def 13;
A5: i + 1 >= i by NAT_1:11;
A6: len (Rotate f,p) = len f by Th14;

now

A7: len (f :- p) = ((len f) - (p .. f)) + 1 by A1, FINSEQ_5:53;

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 )

then A9: (Rotate f,p) /. i = f /. ((i -' 1) + (p .. f)) by A1, A2, A4, Th9;
A10: i + 1 <= len (f :- p) by A8, NAT_1:13;
((i + 1) -' 1) + (p .. f) = i + (p .. f) by NAT_D:34
.= ((i -' 1) + 1) + (p .. f) by A2, XREAL_1:237
.= ((i -' 1) + (p .. f)) + 1 ;
then A11: (Rotate f,p) /. (i + 1) = f /. (((i -' 1) + (p .. f)) + 1) by A1, A10, Th9, NAT_1:11;
A12: 0 <= i -' 1 by NAT_1:3;
1 <= p .. f by A1, FINSEQ_4:31;
then A13: 1 + 0 <= (i -' 1) + (p .. f) by A12, XREAL_1:9;
i < ((len f) + 1) - (p .. f) by A7, A8;
then i + (p .. f) < (len f) + 1 by XREAL_1:22;
then i + (p .. f) <= len f by NAT_1:13;
then ((i -' 1) + 1) + (p .. f) <= len f by A2, XREAL_1:237;
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 A9, A11, A13, TOPREAL1:def 7; :: thesis: verum

end;

suppose A14: 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 A3, A5, A6, XXREAL_0:2;
then A15: (Rotate f,p) /. i = f /. ((i + (p .. f)) -' (len f)) by A1, A4, A14, Th17;
A16: i + 1 >= len (f :- p) by A5, A14, XXREAL_0:2;
then i >= (len f) - (p .. f) by A7, XREAL_1:8;
then A17: len f <= i + (p .. f) by XREAL_1:22;
((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, A3, A6, A16, Th17;
i - ((len f) - (p .. f)) >= 1 by A7, A14, XREAL_1:21;
then (i + (p .. f)) - (len f) >= 1 ;
then A19: 1 <= (i + (p .. f)) -' (len f) by NAT_D:39;
p .. f <= len f by A1, FINSEQ_4:31;
then (i + 1) + (p .. f) <= (len f) + (len f) by A3, A6, XREAL_1:9;
then ((i + (p .. f)) + 1) - (len f) <= len f by XREAL_1:22;
then ((i + (p .. f)) - (len f)) + 1 <= len f ;
then ((i + (p .. f)) -' (len f)) + 1 <= len f by A17, XREAL_1:235;
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 A15, A18, A19, TOPREAL1:def 7; :: 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;