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Theorem xnegdi 11431
Description: Extended real version of xnegdi 11431. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegdi  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )

Proof of Theorem xnegdi
StepHypRef Expression
1 elxr 11316 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11316 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 recn 9573 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 9573 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  CC )
5 negdi 9867 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  +  B )  =  (
-u A  +  -u B ) )
63, 4, 5syl2an 477 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u ( A  +  B )  =  (
-u A  +  -u B ) )
7 readdcl 9566 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
8 rexneg 11401 . . . . . . . 8  |-  ( ( A  +  B )  e.  RR  ->  -e
( A  +  B
)  =  -u ( A  +  B )
)
97, 8syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e ( A  +  B )  = 
-u ( A  +  B ) )
10 renegcl 9873 . . . . . . . 8  |-  ( A  e.  RR  ->  -u A  e.  RR )
11 renegcl 9873 . . . . . . . 8  |-  ( B  e.  RR  ->  -u B  e.  RR )
12 rexadd 11422 . . . . . . . 8  |-  ( (
-u A  e.  RR  /\  -u B  e.  RR )  ->  ( -u A +e -u B
)  =  ( -u A  +  -u B ) )
1310, 11, 12syl2an 477 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A +e -u B )  =  ( -u A  +  -u B ) )
146, 9, 133eqtr4d 2513 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e ( A  +  B )  =  ( -u A +e -u B ) )
15 rexadd 11422 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
16 xnegeq 11397 . . . . . . 7  |-  ( ( A +e B )  =  ( A  +  B )  ->  -e ( A +e B )  = 
-e ( A  +  B ) )
1715, 16syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e ( A +e B )  =  -e ( A  +  B ) )
18 rexneg 11401 . . . . . . 7  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
19 rexneg 11401 . . . . . . 7  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
2018, 19oveqan12d 6296 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (  -e A +e  -e
B )  =  (
-u A +e -u B ) )
2114, 17, 203eqtr4d 2513 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e ( A +e B )  =  (  -e
A +e  -e B ) )
22 xnegpnf 11399 . . . . . 6  |-  -e +oo  = -oo
23 oveq2 6285 . . . . . . . 8  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
24 rexr 9630 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  RR* )
25 renemnf 9633 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= -oo )
26 xaddpnf1 11416 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
2724, 25, 26syl2anc 661 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A +e +oo )  = +oo )
2823, 27sylan9eqr 2525 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
29 xnegeq 11397 . . . . . . 7  |-  ( ( A +e B )  = +oo  ->  -e ( A +e B )  = 
-e +oo )
3028, 29syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e ( A +e B )  =  -e +oo )
31 xnegeq 11397 . . . . . . . . 9  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
3231, 22syl6eq 2519 . . . . . . . 8  |-  ( B  = +oo  ->  -e
B  = -oo )
3332oveq2d 6293 . . . . . . 7  |-  ( B  = +oo  ->  (  -e A +e  -e B )  =  (  -e A +e -oo )
)
3418, 10eqeltrd 2550 . . . . . . . 8  |-  ( A  e.  RR  ->  -e
A  e.  RR )
35 rexr 9630 . . . . . . . . 9  |-  (  -e A  e.  RR  -> 
-e A  e. 
RR* )
36 renepnf 9632 . . . . . . . . 9  |-  (  -e A  e.  RR  -> 
-e A  =/= +oo )
37 xaddmnf1 11418 . . . . . . . . 9  |-  ( ( 
-e A  e. 
RR*  /\  -e A  =/= +oo )  -> 
(  -e A +e -oo )  = -oo )
3835, 36, 37syl2anc 661 . . . . . . . 8  |-  (  -e A  e.  RR  ->  (  -e A +e -oo )  = -oo )
3934, 38syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  (  -e A +e -oo )  = -oo )
4033, 39sylan9eqr 2525 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  (  -e A +e  -e
B )  = -oo )
4122, 30, 403eqtr4a 2529 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e ( A +e B )  =  (  -e
A +e  -e B ) )
42 xnegmnf 11400 . . . . . 6  |-  -e -oo  = +oo
43 oveq2 6285 . . . . . . . 8  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
44 renepnf 9632 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= +oo )
45 xaddmnf1 11418 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
4624, 44, 45syl2anc 661 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
4743, 46sylan9eqr 2525 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
48 xnegeq 11397 . . . . . . 7  |-  ( ( A +e B )  = -oo  ->  -e ( A +e B )  = 
-e -oo )
4947, 48syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  -> 
-e ( A +e B )  =  -e -oo )
50 xnegeq 11397 . . . . . . . . 9  |-  ( B  = -oo  ->  -e
B  =  -e -oo )
5150, 42syl6eq 2519 . . . . . . . 8  |-  ( B  = -oo  ->  -e
B  = +oo )
5251oveq2d 6293 . . . . . . 7  |-  ( B  = -oo  ->  (  -e A +e  -e B )  =  (  -e A +e +oo )
)
53 renemnf 9633 . . . . . . . . 9  |-  (  -e A  e.  RR  -> 
-e A  =/= -oo )
54 xaddpnf1 11416 . . . . . . . . 9  |-  ( ( 
-e A  e. 
RR*  /\  -e A  =/= -oo )  -> 
(  -e A +e +oo )  = +oo )
5535, 53, 54syl2anc 661 . . . . . . . 8  |-  (  -e A  e.  RR  ->  (  -e A +e +oo )  = +oo )
5634, 55syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  (  -e A +e +oo )  = +oo )
5752, 56sylan9eqr 2525 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  (  -e A +e  -e
B )  = +oo )
5842, 49, 573eqtr4a 2529 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  -> 
-e ( A +e B )  =  (  -e
A +e  -e B ) )
5921, 41, 583jaodan 1289 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )
602, 59sylan2b 475 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  -e ( A +e B )  =  (  -e A +e  -e
B ) )
61 xneg0 11402 . . . . . . 7  |-  -e 0  =  0
62 simpr 461 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  B  = -oo )
6362oveq2d 6293 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( +oo +e -oo ) )
64 pnfaddmnf 11420 . . . . . . . . 9  |-  ( +oo +e -oo )  =  0
6563, 64syl6eq 2519 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  0 )
66 xnegeq 11397 . . . . . . . 8  |-  ( ( +oo +e B )  =  0  ->  -e ( +oo +e B )  = 
-e 0 )
6765, 66syl 16 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  -e
( +oo +e B )  =  -e 0 )
6851adantl 466 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  -e
B  = +oo )
6968oveq2d 6293 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( -oo +e  -e
B )  =  ( -oo +e +oo ) )
70 mnfaddpnf 11421 . . . . . . . 8  |-  ( -oo +e +oo )  =  0
7169, 70syl6eq 2519 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( -oo +e  -e
B )  =  0 )
7261, 67, 713eqtr4a 2529 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  -e
( +oo +e B )  =  ( -oo +e  -e B ) )
73 xaddpnf2 11417 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
74 xnegeq 11397 . . . . . . . 8  |-  ( ( +oo +e B )  = +oo  ->  -e ( +oo +e B )  = 
-e +oo )
7573, 74syl 16 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  -e
( +oo +e B )  =  -e +oo )
76 xnegcl 11403 . . . . . . . . 9  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
7776adantr 465 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  -e
B  e.  RR* )
78 xnegeq 11397 . . . . . . . . . . . 12  |-  (  -e B  = +oo  -> 
-e  -e
B  =  -e +oo )
7978, 22syl6eq 2519 . . . . . . . . . . 11  |-  (  -e B  = +oo  -> 
-e  -e
B  = -oo )
80 xnegneg 11404 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  -e  -e B  =  B )
8180eqeq1d 2464 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  (  -e  -e B  = -oo  <->  B  = -oo ) )
8279, 81syl5ib 219 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  (  -e B  = +oo  ->  B  = -oo )
)
8382necon3d 2686 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  =/= -oo  ->  -e
B  =/= +oo )
)
8483imp 429 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  -e
B  =/= +oo )
85 xaddmnf2 11419 . . . . . . . 8  |-  ( ( 
-e B  e. 
RR*  /\  -e B  =/= +oo )  -> 
( -oo +e  -e B )  = -oo )
8677, 84, 85syl2anc 661 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( -oo +e  -e
B )  = -oo )
8722, 75, 863eqtr4a 2529 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  -e
( +oo +e B )  =  ( -oo +e  -e B ) )
8872, 87pm2.61dane 2780 . . . . 5  |-  ( B  e.  RR*  ->  -e
( +oo +e B )  =  ( -oo +e  -e B ) )
8988adantl 466 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e ( +oo +e B )  =  ( -oo +e  -e B ) )
90 simpl 457 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
9190oveq1d 6292 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( +oo +e B ) )
92 xnegeq 11397 . . . . 5  |-  ( ( A +e B )  =  ( +oo +e B )  ->  -e ( A +e B )  =  -e ( +oo +e B ) )
9391, 92syl 16 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e ( A +e B )  = 
-e ( +oo +e B ) )
94 xnegeq 11397 . . . . . . 7  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
9594adantr 465 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e A  =  -e +oo )
9695, 22syl6eq 2519 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e A  = -oo )
9796oveq1d 6292 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
(  -e A +e  -e B )  =  ( -oo +e  -e B ) )
9889, 93, 973eqtr4d 2513 . . 3  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e ( A +e B )  =  (  -e A +e  -e
B ) )
99 simpr 461 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  B  = +oo )
10099oveq2d 6293 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( -oo +e +oo ) )
101100, 70syl6eq 2519 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  0 )
102 xnegeq 11397 . . . . . . . 8  |-  ( ( -oo +e B )  =  0  ->  -e ( -oo +e B )  = 
-e 0 )
103101, 102syl 16 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  -e
( -oo +e B )  =  -e 0 )
10432adantl 466 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  -e
B  = -oo )
105104oveq2d 6293 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( +oo +e  -e
B )  =  ( +oo +e -oo ) )
106105, 64syl6eq 2519 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( +oo +e  -e
B )  =  0 )
10761, 103, 1063eqtr4a 2529 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  -e
( -oo +e B )  =  ( +oo +e  -e B ) )
108 xaddmnf2 11419 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
109 xnegeq 11397 . . . . . . . 8  |-  ( ( -oo +e B )  = -oo  ->  -e ( -oo +e B )  = 
-e -oo )
110108, 109syl 16 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  -e
( -oo +e B )  =  -e -oo )
11176adantr 465 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  -e
B  e.  RR* )
112 xnegeq 11397 . . . . . . . . . . . 12  |-  (  -e B  = -oo  -> 
-e  -e
B  =  -e -oo )
113112, 42syl6eq 2519 . . . . . . . . . . 11  |-  (  -e B  = -oo  -> 
-e  -e
B  = +oo )
11480eqeq1d 2464 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  (  -e  -e B  = +oo  <->  B  = +oo ) )
115113, 114syl5ib 219 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  (  -e B  = -oo  ->  B  = +oo )
)
116115necon3d 2686 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  =/= +oo  ->  -e
B  =/= -oo )
)
117116imp 429 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  -e
B  =/= -oo )
118 xaddpnf2 11417 . . . . . . . 8  |-  ( ( 
-e B  e. 
RR*  /\  -e B  =/= -oo )  -> 
( +oo +e  -e B )  = +oo )
119111, 117, 118syl2anc 661 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( +oo +e  -e
B )  = +oo )
12042, 110, 1193eqtr4a 2529 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  -e
( -oo +e B )  =  ( +oo +e  -e B ) )
121107, 120pm2.61dane 2780 . . . . 5  |-  ( B  e.  RR*  ->  -e
( -oo +e B )  =  ( +oo +e  -e B ) )
122121adantl 466 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e ( -oo +e B )  =  ( +oo +e  -e B ) )
123 simpl 457 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  A  = -oo )
124123oveq1d 6292 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( -oo +e B ) )
125 xnegeq 11397 . . . . 5  |-  ( ( A +e B )  =  ( -oo +e B )  ->  -e ( A +e B )  =  -e ( -oo +e B ) )
126124, 125syl 16 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e ( A +e B )  = 
-e ( -oo +e B ) )
127 xnegeq 11397 . . . . . . 7  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
128127adantr 465 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e A  =  -e -oo )
129128, 42syl6eq 2519 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e A  = +oo )
130129oveq1d 6292 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
(  -e A +e  -e B )  =  ( +oo +e  -e B ) )
131122, 126, 1303eqtr4d 2513 . . 3  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e ( A +e B )  =  (  -e A +e  -e
B ) )
13260, 98, 1313jaoian 1288 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  -e ( A +e B )  =  (  -e
A +e  -e B ) )
1331, 132sylanb 472 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    \/ w3o 967    = wceq 1374    e. wcel 1762    =/= wne 2657  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483    + caddc 9486   +oocpnf 9616   -oocmnf 9617   RR*cxr 9618   -ucneg 9797    -ecxne 11306   +ecxad 11307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-sub 9798  df-neg 9799  df-xneg 11309  df-xadd 11310
This theorem is referenced by:  xaddass2  11433  xposdif  11445  xadddi  11478  xrsxmet  21044
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