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

suppose not p in rng f ; :: thesis: Rotate f,p is unfolded

hence Rotate f,p is unfolded by FINSEQ_6:def 2; :: thesis: verum

end;

suppose A1: p in rng f ; :: thesis: Rotate f,p is unfolded

let i be Nat; :: according to TOPREAL1:def 8 :: thesis: ( not 1 <= i or not i + 2 <= len (Rotate f,p) or (LSeg (Rotate f,p),i) /\ (LSeg (Rotate f,p),(i + 1)) = {((Rotate f,p) /. (i + 1))} )
assume that
A2: 1 <= i and
A3: i + 2 <= len (Rotate f,p) ; :: thesis: (LSeg (Rotate f,p),i) /\ (LSeg (Rotate f,p),(i + 1)) = {((Rotate f,p) /. (i + 1))}
A4: i in NAT by ORDINAL1:def 13;
A5: len f > 4 by GOBOARD7:36;
thus (LSeg (Rotate f,p),i) /\ (LSeg (Rotate f,p),(i + 1)) = {((Rotate f,p) /. (i + 1))} :: thesis: verum

proof

A6: ((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 ;
A7: len f = len (Rotate f,p) by Th14;
i + 1 < i + 2 by XREAL_1:8;
then A8: i + 1 < len f by A3, A7, XXREAL_0:2;
A9: len (f :- p) = ((len f) - (p .. f)) + 1 by A1, FINSEQ_5:53;

per cases ( i + 1 = len (f :- p) or i + 1 < len (f :- p) or len (f :- p) < i + 1 ) by XXREAL_0:1;

suppose A10: i + 1 = len (f :- p) ; :: thesis: (LSeg (Rotate f,p),i) /\ (LSeg (Rotate f,p),(i + 1)) = {((Rotate f,p) /. (i + 1))}

((len (f :- p)) + (p .. f)) -' (len f) = ((len f) + 1) - (len f) by A9, NAT_1:11, XREAL_1:235
.= 1 ;
then A11: LSeg (Rotate f,p),(len (f :- p)) = LSeg f,1 by A1, A8, A10, Th31;
A12: i < len (f :- p) by A10, XREAL_1:31;
(i -' 1) + (p .. f) = (((((len f) - (p .. f)) + 1) - 1) - 1) + (p .. f) by A2, A9, A10, XREAL_1:235
.= (((len f) - (p .. f)) + (p .. f)) - 1
.= (len f) -' 1 by A5, XREAL_1:235, XXREAL_0:2 ;
then LSeg (Rotate f,p),i = LSeg f,((len f) -' 1) by A1, A2, A4, A12, Th24;
hence (LSeg (Rotate f,p),i) /\ (LSeg (Rotate f,p),(i + 1)) = {(f /. 1)} by A10, A11, Th30, GOBOARD7:36
.= {(f /. (len f))} by FINSEQ_6:def 1
.= {((Rotate f,p) /. (i + 1))} by A1, A10, Th11 ;
:: thesis: verum

end;

suppose A13: i + 1 < len (f :- p) ; :: thesis: (LSeg (Rotate f,p),i) /\ (LSeg (Rotate f,p),(i + 1)) = {((Rotate f,p) /. (i + 1))}

then i + 1 < ((len f) - (p .. f)) + 1 by A1, FINSEQ_5:53;
then i < (len f) - (p .. f) by XREAL_1:8;
then i + (p .. f) < len f by XREAL_1:22;
then ((i -' 1) + 1) + (p .. f) < len f by A2, XREAL_1:237;
then (((i -' 1) + (p .. f)) + 1) + 1 <= len f by NAT_1:13;
then A14: ((i -' 1) + (p .. f)) + (1 + 1) <= len f ;
( (i -' 1) + (p .. f) >= p .. f & p .. f >= 1 ) by A1, FINSEQ_4:31, NAT_1:11;
then A15: 1 <= (i -' 1) + (p .. f) by XXREAL_0:2;
i + 0 < i + 1 by XREAL_1:8;
then i < len (f :- p) by A13, XXREAL_0:2;
then A16: LSeg (Rotate f,p),i = LSeg f,((i -' 1) + (p .. f)) by A1, A2, A4, Th24;
LSeg (Rotate f,p),(i + 1) = LSeg f,(((i + 1) -' 1) + (p .. f)) by A1, A13, Th24, NAT_1:11;
hence (LSeg (Rotate f,p),i) /\ (LSeg (Rotate f,p),(i + 1)) = {(f /. (((i + 1) -' 1) + (p .. f)))} by A6, A16, A15, A14, TOPREAL1:def 8
.= {((Rotate f,p) /. (i + 1))} by A1, A13, Th9, NAT_1:11 ;
:: thesis: verum

end;

suppose A17: len (f :- p) < i + 1 ; :: thesis: (LSeg (Rotate f,p),i) /\ (LSeg (Rotate f,p),(i + 1)) = {((Rotate f,p) /. (i + 1))}

p .. f <= len f by A1, FINSEQ_4:31;
then (i + 2) + (p .. f) <= (len f) + (len f) by A3, A7, XREAL_1:9;
then ((i + (p .. f)) + 2) - (len f) <= len f by XREAL_1:22;
then A18: ((i + (p .. f)) - (len f)) + 2 <= len f ;
(i + 1) + 1 <= len f by A3, Th14;
then A19: i + 1 < len f by NAT_1:13;
i + 1 > ((len f) - (p .. f)) + 1 by A1, A17, FINSEQ_5:53;
then (len f) - (p .. f) < i by XREAL_1:8;
then A20: len f < i + (p .. f) by XREAL_1:21;
then A21: (i + (p .. f)) -' (len f) = (i + (p .. f)) - (len f) by XREAL_1:235;
0 < (i + (p .. f)) - (len f) by A20, XREAL_1:52;
then A22: 0 + 1 <= (i + (p .. f)) -' (len f) by A21, NAT_1:13;
A23: ((i + 1) + (p .. f)) -' (len f) = ((i + (p .. f)) + 1) -' (len f)
.= ((i + (p .. f)) -' (len f)) + 1 by A20, NAT_D:38 ;
A24: len (f :- p) <= i by A17, NAT_1:13;
i + 0 < i + 1 by XREAL_1:8;
then i < len f by A19, XXREAL_0:2;
then A25: LSeg (Rotate f,p),i = LSeg f,((i + (p .. f)) -' (len f)) by A1, A4, A24, Th31;
LSeg (Rotate f,p),(i + 1) = LSeg f,(((i + 1) + (p .. f)) -' (len f)) by A1, A17, A19, Th31;
hence (LSeg (Rotate f,p),i) /\ (LSeg (Rotate f,p),(i + 1)) = {(f /. (((i + 1) + (p .. f)) -' (len f)))} by A25, A21, A22, A23, A18, TOPREAL1:def 8
.= {((Rotate f,p) /. (i + 1))} by A1, A8, A17, Th17 ;
:: thesis: verum

end;

end;

end;

end;

end;