let i be Nat; :: thesis: for D being non empty set
for p being Element of D
for f being circular FinSequence of D st p in rng f & len (f :- p) <= i & i <= len f holds
(Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f))
let D be non empty set ; :: thesis: for p being Element of D
for f being circular FinSequence of D st p in rng f & len (f :- p) <= i & i <= len f holds
(Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f))
let p be Element of D; :: thesis: for f being circular FinSequence of D st p in rng f & len (f :- p) <= i & i <= len f holds
(Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f))
let f be circular FinSequence of D; :: thesis: ( p in rng f & len (f :- p) <= i & i <= len f implies (Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f)) )
assume that
A1: p in rng f and
A2: len (f :- p) <= i and
A3: i <= len f ; :: thesis: (Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f))
A4: p .. f <= len f by A1, FINSEQ_4:21;
then A5: (len f) - (p .. f) = (len f) -' (p .. f) by XREAL_1:233;

per cases ( i = len (f :- p) or i > len (f :- p) ) by A2, XXREAL_0:1;

suppose A6: i = len (f :- p) ; :: thesis: (Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f))

then A7: i = ((len f) - (p .. f)) + 1 by A1, FINSEQ_5:50;
then i >= 1 by A5, NAT_1:11;
hence (Rotate (f,p)) /. i = f /. ((i -' 1) + (p .. f)) by A1, A6, Th9
.= f /. (((len f) -' (p .. f)) + (p .. f)) by A5, A7, NAT_D:34
.= f /. (len f) by A4, XREAL_1:235
.= f /. 1 by Def1A
.= f /. (((len f) + 1) -' (len f)) by NAT_D:34
.= f /. ((i + (p .. f)) -' (len f)) by A7 ;
:: thesis: verum

end;

suppose i > len (f :- p) ; :: thesis: (Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f))

hence (Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f)) by A1, A3, Th12; :: thesis: verum

end;

end;