let f be non empty FinSequence of (TOP-REAL 2); :: thesis: for p being Point of (TOP-REAL 2) st f is poorly-one-to-one & f is unfolded & f is s.n.c. & p in L~ f & p <> f . (len f) holds
L_Cut (Rev f),p = Rev (R_Cut f,p)
let p be Point of (TOP-REAL 2); :: thesis: ( f is poorly-one-to-one & f is unfolded & f is s.n.c. & p in L~ f & p <> f . (len f) implies L_Cut (Rev f),p = Rev (R_Cut f,p) )
assume that
A1:
( f is poorly-one-to-one & f is unfolded & f is s.n.c. )
and
A2:
p in L~ f
and
A3:
p <> f . (len f)
; :: thesis: L_Cut (Rev f),p = Rev (R_Cut f,p)
A5:
Rev f is s.n.c.
by A1, SPPOL_2:36;
A6:
p in L~ (Rev f)
by A2, SPPOL_2:22;
A7:
Rev (Rev f) = f
by FINSEQ_6:29;
A8:
len f = len (Rev f)
by FINSEQ_5:def 3;
A9:
L~ f = L~ (Rev f)
by SPPOL_2:22;
A10:
dom (Rev f) = dom f
by FINSEQ_5:60;
A11:
1 <= (Index p,(Rev f)) + 1
by NAT_1:11;
Index p,(Rev f) < len (Rev f)
by A2, A9, JORDAN3:41;
then A12:
(Index p,(Rev f)) + 1 <= len f
by A8, NAT_1:13;
then A13:
(Index p,(Rev f)) + 1 in dom f
by A11, FINSEQ_3:27;
A14:
1 <= Index p,f
by A2, JORDAN3:41;
A15:
Index p,f < len f
by A2, JORDAN3:41;
then A16:
1 < len f
by A14, XXREAL_0:2;
per cases
( p = f . 1 or ( p <> f . 1 & p <> f . (len f) & p = f . ((Index p,f) + 1) ) or ( p <> f . 1 & p <> f . ((Index p,f) + 1) ) )
by A3;
suppose A17:
p = f . 1
;
:: thesis: L_Cut (Rev f),p = Rev (R_Cut f,p)then A18:
p = (Rev f) . (len f)
by FINSEQ_5:65;
A19:
len ((Rev f) /^ ((len (Rev f)) -' 1)) = (len (Rev f)) -' ((len (Rev f)) -' 1)
by JORDAN3:19;
A20:
((len (Rev f)) -' 1) + 1
= len (Rev f)
by A8, A14, A15, XREAL_1:237, XXREAL_0:2;
then A21:
((Rev f) /^ ((len (Rev f)) -' 1)) . 1
= (Rev f) . (len (Rev f))
by JORDAN3:23;
1
<= (len (Rev f)) - ((len (Rev f)) -' 1)
by A20;
then A22:
1
<= len ((Rev f) /^ ((len (Rev f)) -' 1))
by A19, NAT_D:39;
then A23:
not
(Rev f) /^ ((len (Rev f)) -' 1) is
empty
;
((len (Rev f)) -' (len (Rev f))) + 1 =
((len (Rev f)) - (len (Rev f))) + 1
by XREAL_1:235
.=
1
;
then A24:
mid (Rev f),
(len (Rev f)),
(len (Rev f)) =
((Rev f) /^ ((len (Rev f)) -' 1)) | 1
by JORDAN3:def 1
.=
<*(((Rev f) /^ ((len (Rev f)) -' 1)) /. 1)*>
by A23, FINSEQ_5:23
.=
<*((Rev f) . (len (Rev f)))*>
by A21, A22, FINSEQ_4:24
;
(Index p,(Rev f)) + 1
= len f
by A1, A5, A8, A16, A18, Th18, SPPOL_2:29;
hence L_Cut (Rev f),
p =
<*p*>
by A8, A18, A24, JORDAN3:def 4
.=
Rev <*p*>
by FINSEQ_5:63
.=
Rev (R_Cut f,p)
by A17, JORDAN3:def 5
;
:: thesis: verum end; suppose that A25:
p <> f . 1
and A26:
p <> f . (len f)
and A27:
p = f . ((Index p,f) + 1)
;
:: thesis: L_Cut (Rev f),p = Rev (R_Cut f,p)A28:
p <> (Rev f) . (len f)
by A25, FINSEQ_5:65;
len f =
((Index p,(Rev f)) + (Index p,f)) + 1
by A1, A2, A26, A27, Th36
.=
(Index p,(Rev f)) + ((Index p,f) + 1)
;
then A29:
p =
f . (((len f) - ((Index p,(Rev f)) + 1)) + 1)
by A27
.=
(Rev f) . ((Index p,(Rev f)) + 1)
by A13, FINSEQ_5:61
;
A30:
len f =
((Index p,(Rev f)) + (Index p,f)) + 1
by A1, A2, A26, A27, Th36
.=
(Index p,f) + ((Index p,(Rev f)) + 1)
;
A31:
(len f) -' (Index p,f) =
(len f) - (Index p,f)
by A15, XREAL_1:235
.=
(Index p,(Rev f)) + 1
by A30
;
A32:
(Index p,(Rev f)) + 1
< len f
by A12, A28, A29, XXREAL_0:1;
thus L_Cut (Rev f),
p =
mid (Rev f),
((Index p,(Rev f)) + 1),
(len f)
by A8, A29, JORDAN3:def 4
.=
<*p*> ^ (mid (Rev f),(((len f) -' (Index p,f)) + 1),(len f))
by A10, A13, A29, A31, A32, JORDAN3:56
.=
<*p*> ^ (mid (Rev f),(((len f) -' (Index p,f)) + 1),(((len f) -' 1) + 1))
by A14, A15, XREAL_1:237, XXREAL_0:2
.=
<*p*> ^ (Rev (mid f,1,(Index p,f)))
by A14, A15, A16, JORDAN3:22
.=
Rev ((mid f,1,(Index p,f)) ^ <*p*>)
by FINSEQ_5:66
.=
Rev (R_Cut f,p)
by A25, JORDAN3:def 5
;
:: thesis: verum end; suppose that A33:
p <> f . 1
and A34:
p <> f . ((Index p,f) + 1)
;
:: thesis: L_Cut (Rev f),p = Rev (R_Cut f,p)A35:
p <> (Rev f) . (len f)
by A33, FINSEQ_5:65;
A36:
now assume A37:
p = (Rev f) . ((Index p,(Rev f)) + 1)
;
:: thesis: contradictionthen A38:
len (Rev f) =
((Index p,(Rev (Rev f))) + (Index p,(Rev f))) + 1
by A1, A5, A6, A8, A35, Th36, SPPOL_2:29
.=
((Index p,f) + 1) + (Index p,(Rev f))
by A7
;
p =
f . (((len f) - ((Index p,(Rev f)) + 1)) + 1)
by A13, A37, FINSEQ_5:61
.=
f . ((Index p,f) + 1)
by A8, A38
;
hence
contradiction
by A34;
:: thesis: verum end; A39:
len f = (Index p,(Rev f)) + (Index p,f)
by A1, A2, A34, JORDAN3:54;
A40:
Index p,
f < len f
by A2, JORDAN3:41;
Index p,
(Rev f) =
(len f) - (Index p,f)
by A39
.=
(len f) -' (Index p,f)
by A40, XREAL_1:235
;
hence L_Cut (Rev f),
p =
<*p*> ^ (mid (Rev f),(((len f) -' (Index p,f)) + 1),(len f))
by A8, A36, JORDAN3:def 4
.=
<*p*> ^ (mid (Rev f),(((len f) -' (Index p,f)) + 1),(((len f) -' 1) + 1))
by A14, A15, XREAL_1:237, XXREAL_0:2
.=
<*p*> ^ (Rev (mid f,1,(Index p,f)))
by A14, A15, A16, JORDAN3:22
.=
Rev ((mid f,1,(Index p,f)) ^ <*p*>)
by FINSEQ_5:66
.=
Rev (R_Cut f,p)
by A33, JORDAN3:def 5
;
:: thesis: verum end;