:: FDIFF_11 semantic presentation

REAL is V1() V46() V47() V48() V52() V62() set
NAT is V46() V47() V48() V49() V50() V51() V52() Element of K6(REAL)
K6(REAL) is set
COMPLEX is V1() V46() V52() V62() set
0 is set
1 is V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() Element of NAT
{0,1} is set
K7(REAL,REAL) is V36() V37() V38() set
K6(K7(REAL,REAL)) is set
K7(NAT,REAL) is V36() V37() V38() set
K6(K7(NAT,REAL)) is set
K7(NAT,COMPLEX) is V36() set
K6(K7(NAT,COMPLEX)) is set
K7(COMPLEX,COMPLEX) is V36() set
K6(K7(COMPLEX,COMPLEX)) is set
NAT is V46() V47() V48() V49() V50() V51() V52() set
K6(NAT) is set
K6(NAT) is set
INT is V1() V46() V47() V48() V49() V50() V52() V62() set
K7(1,1) is V23( RAT ) V23( INT ) V36() V37() V38() V39() set
RAT is V1() V46() V47() V48() V49() V52() V62() set
K6(K7(1,1)) is set
K7(K7(1,1),1) is V23( RAT ) V23( INT ) V36() V37() V38() V39() set
K6(K7(K7(1,1),1)) is set
K7(K7(1,1),REAL) is V36() V37() V38() set
K6(K7(K7(1,1),REAL)) is set
K7(K7(REAL,REAL),REAL) is V36() V37() V38() set
K6(K7(K7(REAL,REAL),REAL)) is set
2 is V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() Element of NAT
K7(2,2) is V23( RAT ) V23( INT ) V36() V37() V38() V39() set
K7(K7(2,2),REAL) is V36() V37() V38() set
K6(K7(K7(2,2),REAL)) is set
K427(2) is V180() L15()
the U1 of K427(2) is set
K485() is L6()
the U1 of K485() is set
K321() is V115() L7()
K490() is V89() L6()
K492() is M13(K490())
K470(K492(),K492()) is V88() V89() L6()
the U1 of K470(K492(),K492()) is set
PFuncs (REAL,REAL) is set
K7(NAT,(PFuncs (REAL,REAL))) is set
K6(K7(NAT,(PFuncs (REAL,REAL)))) is set
0 is V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() Element of NAT
sin is V19() V22( REAL ) V23( REAL ) V24() V33( REAL , REAL ) V36() V37() V38() Element of K6(K7(REAL,REAL))
dom sin is V46() V47() V48() Element of K6(REAL)
cos is V19() V22( REAL ) V23( REAL ) V24() V33( REAL , REAL ) V36() V37() V38() Element of K6(K7(REAL,REAL))
dom cos is V46() V47() V48() Element of K6(REAL)
exp_R is V19() V22( REAL ) V23( REAL ) V24() V33( REAL , REAL ) V36() V37() V38() Element of K6(K7(REAL,REAL))
PI is V1() V11() V12() ext-real positive non negative set
tan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
sin / cos is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
4 is V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() Element of NAT
].0,4.[ is open V46() V47() V48() Element of K6(REAL)
cos / sin is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
{0} is V46() V47() V48() V49() V50() V51() set
[#] REAL is V46() V47() V48() Element of K6(REAL)
dom exp_R is V46() V47() V48() Element of K6(REAL)
rng exp_R is V46() V47() V48() Element of K6(REAL)
K514(0) is V46() V47() V48() Element of K6(REAL)
sec is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
cos ^ is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
cosec is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
sin ^ is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
- 1 is V1() V11() V12() ext-real non positive negative V60() Element of REAL
[.(- 1),1.] is closed V46() V47() V48() Element of K6(REAL)
arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom arctan is V46() V47() V48() Element of K6(REAL)
arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom arccot is V46() V47() V48() Element of K6(REAL)
arccot | [.(- 1),1.] is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
rng (arccot | [.(- 1),1.]) is V46() V47() V48() Element of K6(REAL)
PI is V1() V11() V12() ext-real positive non negative Element of REAL
PI / 4 is V1() V11() V12() ext-real positive non negative Element of REAL
K39(4) is V1() V11() V12() ext-real positive non negative set
K37(PI,K39(4)) is V1() V11() V12() ext-real positive non negative set
3 is V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() Element of NAT
3 / 4 is V1() V11() V12() ext-real positive non negative Element of REAL
K37(3,K39(4)) is V1() V11() V12() ext-real positive non negative set
(3 / 4) * PI is V1() V11() V12() ext-real positive non negative Element of REAL
[.(PI / 4),((3 / 4) * PI).] is closed V46() V47() V48() Element of K6(REAL)
].(- 1),1.[ is open V46() V47() V48() Element of K6(REAL)
dom tan is V46() V47() V48() Element of K6(REAL)
cot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom cot is V46() V47() V48() Element of K6(REAL)
arctan * sin is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arctan * sin) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arctan * sin) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan * sin) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
1 + ((sin . x) ^2) is V11() V12() ext-real Element of REAL
(cos . x) / (1 + ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((sin . x) ^2))) is V11() V12() ext-real set
K37((cos . x),K39((1 + ((sin . x) ^2)))) is V11() V12() ext-real set
diff ((arctan * sin),x) is V11() V12() ext-real Element of REAL
diff (sin,x) is V11() V12() ext-real Element of REAL
(diff (sin,x)) / (1 + ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
K37((diff (sin,x)),K39((1 + ((sin . x) ^2)))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((arctan * sin) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
1 + ((sin . x) ^2) is V11() V12() ext-real Element of REAL
(cos . x) / (1 + ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((sin . x) ^2))) is V11() V12() ext-real set
K37((cos . x),K39((1 + ((sin . x) ^2)))) is V11() V12() ext-real set
arccot * sin is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arccot * sin) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arccot * sin) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * sin) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
1 + ((sin . x) ^2) is V11() V12() ext-real Element of REAL
(cos . x) / (1 + ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((sin . x) ^2))) is V11() V12() ext-real set
K37((cos . x),K39((1 + ((sin . x) ^2)))) is V11() V12() ext-real set
- ((cos . x) / (1 + ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
diff ((arccot * sin),x) is V11() V12() ext-real Element of REAL
diff (sin,x) is V11() V12() ext-real Element of REAL
(diff (sin,x)) / (1 + ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
K37((diff (sin,x)),K39((1 + ((sin . x) ^2)))) is V11() V12() ext-real set
- ((diff (sin,x)) / (1 + ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * sin) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
1 + ((sin . x) ^2) is V11() V12() ext-real Element of REAL
(cos . x) / (1 + ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((sin . x) ^2))) is V11() V12() ext-real set
K37((cos . x),K39((1 + ((sin . x) ^2)))) is V11() V12() ext-real set
- ((cos . x) / (1 + ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
arctan * cos is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arctan * cos) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arctan * cos) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan * cos) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
1 + ((cos . x) ^2) is V11() V12() ext-real Element of REAL
(sin . x) / (1 + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((cos . x) ^2))) is V11() V12() ext-real set
K37((sin . x),K39((1 + ((cos . x) ^2)))) is V11() V12() ext-real set
- ((sin . x) / (1 + ((cos . x) ^2))) is V11() V12() ext-real Element of REAL
diff ((arctan * cos),x) is V11() V12() ext-real Element of REAL
diff (cos,x) is V11() V12() ext-real Element of REAL
(diff (cos,x)) / (1 + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K37((diff (cos,x)),K39((1 + ((cos . x) ^2)))) is V11() V12() ext-real set
- (sin . x) is V11() V12() ext-real Element of REAL
(- (sin . x)) / (1 + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K37((- (sin . x)),K39((1 + ((cos . x) ^2)))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((arctan * cos) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
1 + ((cos . x) ^2) is V11() V12() ext-real Element of REAL
(sin . x) / (1 + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((cos . x) ^2))) is V11() V12() ext-real set
K37((sin . x),K39((1 + ((cos . x) ^2)))) is V11() V12() ext-real set
- ((sin . x) / (1 + ((cos . x) ^2))) is V11() V12() ext-real Element of REAL
arccot * cos is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arccot * cos) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arccot * cos) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * cos) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
1 + ((cos . x) ^2) is V11() V12() ext-real Element of REAL
(sin . x) / (1 + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((cos . x) ^2))) is V11() V12() ext-real set
K37((sin . x),K39((1 + ((cos . x) ^2)))) is V11() V12() ext-real set
diff ((arccot * cos),x) is V11() V12() ext-real Element of REAL
diff (cos,x) is V11() V12() ext-real Element of REAL
(diff (cos,x)) / (1 + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K37((diff (cos,x)),K39((1 + ((cos . x) ^2)))) is V11() V12() ext-real set
- ((diff (cos,x)) / (1 + ((cos . x) ^2))) is V11() V12() ext-real Element of REAL
- (sin . x) is V11() V12() ext-real Element of REAL
(- (sin . x)) / (1 + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K37((- (sin . x)),K39((1 + ((cos . x) ^2)))) is V11() V12() ext-real set
- ((- (sin . x)) / (1 + ((cos . x) ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * cos) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
1 + ((cos . x) ^2) is V11() V12() ext-real Element of REAL
(sin . x) / (1 + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((cos . x) ^2))) is V11() V12() ext-real set
K37((sin . x),K39((1 + ((cos . x) ^2)))) is V11() V12() ext-real set
arctan * tan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arctan * tan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arctan * tan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
tan . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan * tan) `| Z) . x is V11() V12() ext-real Element of REAL
tan . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(sin . x) / (cos . x) is V11() V12() ext-real Element of REAL
K39((cos . x)) is V11() V12() ext-real set
K37((sin . x),K39((cos . x))) is V11() V12() ext-real set
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
diff ((arctan * tan),x) is V11() V12() ext-real Element of REAL
diff (tan,x) is V11() V12() ext-real Element of REAL
(tan . x) ^2 is V11() V12() ext-real Element of REAL
K37((tan . x),(tan . x)) is V11() V12() ext-real set
1 + ((tan . x) ^2) is V11() V12() ext-real Element of REAL
(diff (tan,x)) / (1 + ((tan . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((tan . x) ^2))) is V11() V12() ext-real set
K37((diff (tan,x)),K39((1 + ((tan . x) ^2)))) is V11() V12() ext-real set
1 / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(1,K39(((cos . x) ^2))) is V11() V12() ext-real set
(1 / ((cos . x) ^2)) / (1 + ((tan . x) ^2)) is V11() V12() ext-real Element of REAL
K37((1 / ((cos . x) ^2)),K39((1 + ((tan . x) ^2)))) is V11() V12() ext-real set
((sin . x) / (cos . x)) * ((sin . x) / (cos . x)) is V11() V12() ext-real Element of REAL
1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x))) is V11() V12() ext-real Element of REAL
((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x)))) is V11() V12() ext-real Element of REAL
1 / (((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x))))) is V11() V12() ext-real Element of REAL
K39((((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x)))))) is V11() V12() ext-real set
K37(1,K39((((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x))))))) is V11() V12() ext-real set
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((sin . x) ^2) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37(((sin . x) ^2),K39(((cos . x) ^2))) is V11() V12() ext-real set
1 + (((sin . x) ^2) / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2))) is V11() V12() ext-real Element of REAL
1 / (((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2)))) is V11() V12() ext-real Element of REAL
K39((((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2))))) is V11() V12() ext-real set
K37(1,K39((((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2)))))) is V11() V12() ext-real set
((cos . x) ^2) * ((sin . x) ^2) is V11() V12() ext-real Element of REAL
(((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37((((cos . x) ^2) * ((sin . x) ^2)),K39(((cos . x) ^2))) is V11() V12() ext-real set
((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
1 / (((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2)))) is V11() V12() ext-real set
K37(1,K39((((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2))))) is V11() V12() ext-real set
((cos . x) ^2) + ((sin . x) ^2) is V11() V12() ext-real Element of REAL
1 / (((cos . x) ^2) + ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
K39((((cos . x) ^2) + ((sin . x) ^2))) is V11() V12() ext-real set
K37(1,K39((((cos . x) ^2) + ((sin . x) ^2)))) is V11() V12() ext-real set
1 / 1 is V1() V11() V12() ext-real positive non negative Element of REAL
K39(1) is V1() V11() V12() ext-real positive non negative set
K37(1,K39(1)) is V1() V11() V12() ext-real positive non negative set
x is V11() V12() ext-real Element of REAL
((arctan * tan) `| Z) . x is V11() V12() ext-real Element of REAL
arccot * tan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arccot * tan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arccot * tan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
tan . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * tan) `| Z) . x is V11() V12() ext-real Element of REAL
tan . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(sin . x) / (cos . x) is V11() V12() ext-real Element of REAL
K39((cos . x)) is V11() V12() ext-real set
K37((sin . x),K39((cos . x))) is V11() V12() ext-real set
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
diff ((arccot * tan),x) is V11() V12() ext-real Element of REAL
diff (tan,x) is V11() V12() ext-real Element of REAL
(tan . x) ^2 is V11() V12() ext-real Element of REAL
K37((tan . x),(tan . x)) is V11() V12() ext-real set
1 + ((tan . x) ^2) is V11() V12() ext-real Element of REAL
(diff (tan,x)) / (1 + ((tan . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((tan . x) ^2))) is V11() V12() ext-real set
K37((diff (tan,x)),K39((1 + ((tan . x) ^2)))) is V11() V12() ext-real set
- ((diff (tan,x)) / (1 + ((tan . x) ^2))) is V11() V12() ext-real Element of REAL
1 / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(1,K39(((cos . x) ^2))) is V11() V12() ext-real set
(1 / ((cos . x) ^2)) / (1 + ((tan . x) ^2)) is V11() V12() ext-real Element of REAL
K37((1 / ((cos . x) ^2)),K39((1 + ((tan . x) ^2)))) is V11() V12() ext-real set
- ((1 / ((cos . x) ^2)) / (1 + ((tan . x) ^2))) is V11() V12() ext-real Element of REAL
((sin . x) / (cos . x)) * ((sin . x) / (cos . x)) is V11() V12() ext-real Element of REAL
1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x))) is V11() V12() ext-real Element of REAL
((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x)))) is V11() V12() ext-real Element of REAL
1 / (((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x))))) is V11() V12() ext-real Element of REAL
K39((((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x)))))) is V11() V12() ext-real set
K37(1,K39((((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x))))))) is V11() V12() ext-real set
- (1 / (((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x)))))) is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((sin . x) ^2) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37(((sin . x) ^2),K39(((cos . x) ^2))) is V11() V12() ext-real set
1 + (((sin . x) ^2) / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2))) is V11() V12() ext-real Element of REAL
1 / (((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2)))) is V11() V12() ext-real Element of REAL
K39((((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2))))) is V11() V12() ext-real set
K37(1,K39((((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2)))))) is V11() V12() ext-real set
- (1 / (((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2))))) is V11() V12() ext-real Element of REAL
((cos . x) ^2) * ((sin . x) ^2) is V11() V12() ext-real Element of REAL
(((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37((((cos . x) ^2) * ((sin . x) ^2)),K39(((cos . x) ^2))) is V11() V12() ext-real set
((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
1 / (((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2)))) is V11() V12() ext-real set
K37(1,K39((((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2))))) is V11() V12() ext-real set
- (1 / (((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2)))) is V11() V12() ext-real Element of REAL
((cos . x) ^2) + ((sin . x) ^2) is V11() V12() ext-real Element of REAL
1 / (((cos . x) ^2) + ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
K39((((cos . x) ^2) + ((sin . x) ^2))) is V11() V12() ext-real set
K37(1,K39((((cos . x) ^2) + ((sin . x) ^2)))) is V11() V12() ext-real set
- (1 / (((cos . x) ^2) + ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
1 / 1 is V1() V11() V12() ext-real positive non negative Element of REAL
K39(1) is V1() V11() V12() ext-real positive non negative set
K37(1,K39(1)) is V1() V11() V12() ext-real positive non negative set
- (1 / 1) is V1() V11() V12() ext-real non positive negative Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * tan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan * cot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arctan * cot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arctan * cot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan * cot) `| Z) . x is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(cos . x) / (sin . x) is V11() V12() ext-real Element of REAL
K39((sin . x)) is V11() V12() ext-real set
K37((cos . x),K39((sin . x))) is V11() V12() ext-real set
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
diff ((arctan * cot),x) is V11() V12() ext-real Element of REAL
diff (cot,x) is V11() V12() ext-real Element of REAL
(cot . x) ^2 is V11() V12() ext-real Element of REAL
K37((cot . x),(cot . x)) is V11() V12() ext-real set
1 + ((cot . x) ^2) is V11() V12() ext-real Element of REAL
(diff (cot,x)) / (1 + ((cot . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((cot . x) ^2))) is V11() V12() ext-real set
K37((diff (cot,x)),K39((1 + ((cot . x) ^2)))) is V11() V12() ext-real set
1 / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(1,K39(((sin . x) ^2))) is V11() V12() ext-real set
- (1 / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
(- (1 / ((sin . x) ^2))) / (1 + ((cot . x) ^2)) is V11() V12() ext-real Element of REAL
K37((- (1 / ((sin . x) ^2))),K39((1 + ((cot . x) ^2)))) is V11() V12() ext-real set
(1 / ((sin . x) ^2)) / (1 + ((cot . x) ^2)) is V11() V12() ext-real Element of REAL
K37((1 / ((sin . x) ^2)),K39((1 + ((cot . x) ^2)))) is V11() V12() ext-real set
- ((1 / ((sin . x) ^2)) / (1 + ((cot . x) ^2))) is V11() V12() ext-real Element of REAL
((cos . x) / (sin . x)) * ((cos . x) / (sin . x)) is V11() V12() ext-real Element of REAL
1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x))) is V11() V12() ext-real Element of REAL
((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x)))) is V11() V12() ext-real Element of REAL
1 / (((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x))))) is V11() V12() ext-real Element of REAL
K39((((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x)))))) is V11() V12() ext-real set
K37(1,K39((((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x))))))) is V11() V12() ext-real set
- (1 / (((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x)))))) is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((cos . x) ^2) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37(((cos . x) ^2),K39(((sin . x) ^2))) is V11() V12() ext-real set
1 + (((cos . x) ^2) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
1 / (((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2)))) is V11() V12() ext-real Element of REAL
K39((((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2))))) is V11() V12() ext-real set
K37(1,K39((((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2)))))) is V11() V12() ext-real set
- (1 / (((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2))))) is V11() V12() ext-real Element of REAL
((sin . x) ^2) * ((cos . x) ^2) is V11() V12() ext-real Element of REAL
(((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37((((sin . x) ^2) * ((cos . x) ^2)),K39(((sin . x) ^2))) is V11() V12() ext-real set
((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
1 / (((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2)))) is V11() V12() ext-real set
K37(1,K39((((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2))))) is V11() V12() ext-real set
- (1 / (((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2)))) is V11() V12() ext-real Element of REAL
((sin . x) ^2) + ((cos . x) ^2) is V11() V12() ext-real Element of REAL
1 / (((sin . x) ^2) + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K39((((sin . x) ^2) + ((cos . x) ^2))) is V11() V12() ext-real set
K37(1,K39((((sin . x) ^2) + ((cos . x) ^2)))) is V11() V12() ext-real set
- (1 / (((sin . x) ^2) + ((cos . x) ^2))) is V11() V12() ext-real Element of REAL
1 / 1 is V1() V11() V12() ext-real positive non negative Element of REAL
K39(1) is V1() V11() V12() ext-real positive non negative set
K37(1,K39(1)) is V1() V11() V12() ext-real positive non negative set
- (1 / 1) is V1() V11() V12() ext-real non positive negative Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan * cot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot * cot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arccot * cot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arccot * cot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * cot) `| Z) . x is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(cos . x) / (sin . x) is V11() V12() ext-real Element of REAL
K39((sin . x)) is V11() V12() ext-real set
K37((cos . x),K39((sin . x))) is V11() V12() ext-real set
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
diff ((arccot * cot),x) is V11() V12() ext-real Element of REAL
diff (cot,x) is V11() V12() ext-real Element of REAL
(cot . x) ^2 is V11() V12() ext-real Element of REAL
K37((cot . x),(cot . x)) is V11() V12() ext-real set
1 + ((cot . x) ^2) is V11() V12() ext-real Element of REAL
(diff (cot,x)) / (1 + ((cot . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((cot . x) ^2))) is V11() V12() ext-real set
K37((diff (cot,x)),K39((1 + ((cot . x) ^2)))) is V11() V12() ext-real set
- ((diff (cot,x)) / (1 + ((cot . x) ^2))) is V11() V12() ext-real Element of REAL
1 / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(1,K39(((sin . x) ^2))) is V11() V12() ext-real set
- (1 / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
(- (1 / ((sin . x) ^2))) / (1 + ((cot . x) ^2)) is V11() V12() ext-real Element of REAL
K37((- (1 / ((sin . x) ^2))),K39((1 + ((cot . x) ^2)))) is V11() V12() ext-real set
- ((- (1 / ((sin . x) ^2))) / (1 + ((cot . x) ^2))) is V11() V12() ext-real Element of REAL
(1 / ((sin . x) ^2)) / (1 + ((cot . x) ^2)) is V11() V12() ext-real Element of REAL
K37((1 / ((sin . x) ^2)),K39((1 + ((cot . x) ^2)))) is V11() V12() ext-real set
((cos . x) / (sin . x)) * ((cos . x) / (sin . x)) is V11() V12() ext-real Element of REAL
1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x))) is V11() V12() ext-real Element of REAL
((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x)))) is V11() V12() ext-real Element of REAL
1 / (((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x))))) is V11() V12() ext-real Element of REAL
K39((((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x)))))) is V11() V12() ext-real set
K37(1,K39((((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x))))))) is V11() V12() ext-real set
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((cos . x) ^2) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37(((cos . x) ^2),K39(((sin . x) ^2))) is V11() V12() ext-real set
1 + (((cos . x) ^2) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
1 / (((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2)))) is V11() V12() ext-real Element of REAL
K39((((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2))))) is V11() V12() ext-real set
K37(1,K39((((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2)))))) is V11() V12() ext-real set
((sin . x) ^2) * ((cos . x) ^2) is V11() V12() ext-real Element of REAL
(((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37((((sin . x) ^2) * ((cos . x) ^2)),K39(((sin . x) ^2))) is V11() V12() ext-real set
((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
1 / (((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2)))) is V11() V12() ext-real set
K37(1,K39((((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2))))) is V11() V12() ext-real set
((sin . x) ^2) + ((cos . x) ^2) is V11() V12() ext-real Element of REAL
1 / (((sin . x) ^2) + ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
K39((((sin . x) ^2) + ((cos . x) ^2))) is V11() V12() ext-real set
K37(1,K39((((sin . x) ^2) + ((cos . x) ^2)))) is V11() V12() ext-real set
1 / 1 is V1() V11() V12() ext-real positive non negative Element of REAL
K39(1) is V1() V11() V12() ext-real positive non negative set
K37(1,K39(1)) is V1() V11() V12() ext-real positive non negative set
x is V11() V12() ext-real Element of REAL
((arccot * cot) `| Z) . x is V11() V12() ext-real Element of REAL
arctan * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arctan * arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arctan * arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) ^2 is V11() V12() ext-real Element of REAL
K37((arctan . x),(arctan . x)) is V11() V12() ext-real set
1 + ((arctan . x) ^2) is V11() V12() ext-real Element of REAL
(1 + (x ^2)) * (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) is V11() V12() ext-real Element of REAL
K39(((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
K37(1,K39(((1 + (x ^2)) * (1 + ((arctan . x) ^2))))) is V11() V12() ext-real set
diff ((arctan * arctan),x) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(diff (arctan,x)) / (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((arctan . x) ^2))) is V11() V12() ext-real set
K37((diff (arctan,x)),K39((1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
((arctan `| Z) . x) / (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
K37(((arctan `| Z) . x),K39((1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(1 / (1 + (x ^2))) / (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
K37((1 / (1 + (x ^2))),K39((1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((arctan * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) ^2 is V11() V12() ext-real Element of REAL
K37((arctan . x),(arctan . x)) is V11() V12() ext-real set
1 + ((arctan . x) ^2) is V11() V12() ext-real Element of REAL
(1 + (x ^2)) * (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) is V11() V12() ext-real Element of REAL
K39(((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
K37(1,K39(((1 + (x ^2)) * (1 + ((arctan . x) ^2))))) is V11() V12() ext-real set
arccot * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arccot * arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arccot * arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) ^2 is V11() V12() ext-real Element of REAL
K37((arctan . x),(arctan . x)) is V11() V12() ext-real set
1 + ((arctan . x) ^2) is V11() V12() ext-real Element of REAL
(1 + (x ^2)) * (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) is V11() V12() ext-real Element of REAL
K39(((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
K37(1,K39(((1 + (x ^2)) * (1 + ((arctan . x) ^2))))) is V11() V12() ext-real set
- (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) is V11() V12() ext-real Element of REAL
diff ((arccot * arctan),x) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(diff (arctan,x)) / (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((arctan . x) ^2))) is V11() V12() ext-real set
K37((diff (arctan,x)),K39((1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
- ((diff (arctan,x)) / (1 + ((arctan . x) ^2))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
((arctan `| Z) . x) / (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
K37(((arctan `| Z) . x),K39((1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
- (((arctan `| Z) . x) / (1 + ((arctan . x) ^2))) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(1 / (1 + (x ^2))) / (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
K37((1 / (1 + (x ^2))),K39((1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
- ((1 / (1 + (x ^2))) / (1 + ((arctan . x) ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) ^2 is V11() V12() ext-real Element of REAL
K37((arctan . x),(arctan . x)) is V11() V12() ext-real set
1 + ((arctan . x) ^2) is V11() V12() ext-real Element of REAL
(1 + (x ^2)) * (1 + ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) is V11() V12() ext-real Element of REAL
K39(((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) is V11() V12() ext-real set
K37(1,K39(((1 + (x ^2)) * (1 + ((arctan . x) ^2))))) is V11() V12() ext-real set
- (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) is V11() V12() ext-real Element of REAL
arctan * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arctan * arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arctan * arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) ^2 is V11() V12() ext-real Element of REAL
K37((arccot . x),(arccot . x)) is V11() V12() ext-real set
1 + ((arccot . x) ^2) is V11() V12() ext-real Element of REAL
(1 + (x ^2)) * (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) is V11() V12() ext-real Element of REAL
K39(((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
K37(1,K39(((1 + (x ^2)) * (1 + ((arccot . x) ^2))))) is V11() V12() ext-real set
- (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) is V11() V12() ext-real Element of REAL
diff ((arctan * arccot),x) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (arccot,x)) / (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((arccot . x) ^2))) is V11() V12() ext-real set
K37((diff (arccot,x)),K39((1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
((arccot `| Z) . x) / (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
K37(((arccot `| Z) . x),K39((1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- (1 / (1 + (x ^2)))) / (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
K37((- (1 / (1 + (x ^2)))),K39((1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
(1 / (1 + (x ^2))) / (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
K37((1 / (1 + (x ^2))),K39((1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
- ((1 / (1 + (x ^2))) / (1 + ((arccot . x) ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) ^2 is V11() V12() ext-real Element of REAL
K37((arccot . x),(arccot . x)) is V11() V12() ext-real set
1 + ((arccot . x) ^2) is V11() V12() ext-real Element of REAL
(1 + (x ^2)) * (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) is V11() V12() ext-real Element of REAL
K39(((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
K37(1,K39(((1 + (x ^2)) * (1 + ((arccot . x) ^2))))) is V11() V12() ext-real set
- (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) is V11() V12() ext-real Element of REAL
arccot * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arccot * arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(arccot * arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arccot * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) ^2 is V11() V12() ext-real Element of REAL
K37((arccot . x),(arccot . x)) is V11() V12() ext-real set
1 + ((arccot . x) ^2) is V11() V12() ext-real Element of REAL
(1 + (x ^2)) * (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) is V11() V12() ext-real Element of REAL
K39(((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
K37(1,K39(((1 + (x ^2)) * (1 + ((arccot . x) ^2))))) is V11() V12() ext-real set
diff ((arccot * arccot),x) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (arccot,x)) / (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
K39((1 + ((arccot . x) ^2))) is V11() V12() ext-real set
K37((diff (arccot,x)),K39((1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
- ((diff (arccot,x)) / (1 + ((arccot . x) ^2))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
((arccot `| Z) . x) / (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
K37(((arccot `| Z) . x),K39((1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
- (((arccot `| Z) . x) / (1 + ((arccot . x) ^2))) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- (1 / (1 + (x ^2)))) / (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
K37((- (1 / (1 + (x ^2)))),K39((1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
- ((- (1 / (1 + (x ^2)))) / (1 + ((arccot . x) ^2))) is V11() V12() ext-real Element of REAL
(1 / (1 + (x ^2))) / (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
K37((1 / (1 + (x ^2))),K39((1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((arccot * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) ^2 is V11() V12() ext-real Element of REAL
K37((arccot . x),(arccot . x)) is V11() V12() ext-real set
1 + ((arccot . x) ^2) is V11() V12() ext-real Element of REAL
(1 + (x ^2)) * (1 + ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) is V11() V12() ext-real Element of REAL
K39(((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) is V11() V12() ext-real set
K37(1,K39(((1 + (x ^2)) * (1 + ((arccot . x) ^2))))) is V11() V12() ext-real set
sin * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sin * arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sin * arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cos . (arctan . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
diff ((sin * arctan),x) is V11() V12() ext-real Element of REAL
diff (sin,(arctan . x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(diff (sin,(arctan . x))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(cos . (arctan . x)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cos . (arctan . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
sin * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sin * arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sin * arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cos . (arccot . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
- ((cos . (arccot . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
diff ((sin * arccot),x) is V11() V12() ext-real Element of REAL
diff (sin,(arccot . x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (sin,(arccot . x))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cos . (arccot . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
- ((cos . (arccot . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
cos * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cos * arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cos * arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((sin . (arctan . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
- ((sin . (arctan . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
diff ((cos * arctan),x) is V11() V12() ext-real Element of REAL
diff (cos,(arctan . x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(diff (cos,(arctan . x))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
- (sin . (arctan . x)) is V11() V12() ext-real Element of REAL
(- (sin . (arctan . x))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
- ((sin . (arctan . x)) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
- ((sin . (arctan . x)) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(sin . (arctan . x)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
- ((sin . (arctan . x)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((sin . (arctan . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
- ((sin . (arctan . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
cos * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cos * arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cos * arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((sin . (arccot . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
diff ((cos * arccot),x) is V11() V12() ext-real Element of REAL
diff (cos,(arccot . x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (cos,(arccot . x))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
- (sin . (arccot . x)) is V11() V12() ext-real Element of REAL
(- (sin . (arccot . x))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
- ((sin . (arccot . x)) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
- ((sin . (arccot . x)) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
- ((sin . (arccot . x)) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((sin . (arccot . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
tan * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (tan * arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(tan * arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) ^2 is V11() V12() ext-real Element of REAL
K37((cos . (arctan . x)),(cos . (arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((cos . (arctan . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((cos . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
diff ((tan * arctan),x) is V11() V12() ext-real Element of REAL
diff (tan,(arctan . x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(diff (tan,(arctan . x))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
1 / ((cos . (arctan . x)) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . (arctan . x)) ^2)) is V11() V12() ext-real set
K37(1,K39(((cos . (arctan . x)) ^2))) is V11() V12() ext-real set
(1 / ((cos . (arctan . x)) ^2)) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(1 / ((cos . (arctan . x)) ^2)) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(1 / ((cos . (arctan . x)) ^2)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) ^2 is V11() V12() ext-real Element of REAL
K37((cos . (arctan . x)),(cos . (arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((cos . (arctan . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((cos . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
tan * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (tan * arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(tan * arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) ^2 is V11() V12() ext-real Element of REAL
K37((cos . (arccot . x)),(cos . (arccot . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((cos . (arccot . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((cos . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
diff ((tan * arccot),x) is V11() V12() ext-real Element of REAL
diff (tan,(arccot . x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (tan,(arccot . x))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
1 / ((cos . (arccot . x)) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . (arccot . x)) ^2)) is V11() V12() ext-real set
K37(1,K39(((cos . (arccot . x)) ^2))) is V11() V12() ext-real set
(1 / ((cos . (arccot . x)) ^2)) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(1 / ((cos . (arccot . x)) ^2)) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(1 / ((cos . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(1 / ((cos . (arccot . x)) ^2)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
- ((1 / ((cos . (arccot . x)) ^2)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) ^2 is V11() V12() ext-real Element of REAL
K37((cos . (arccot . x)),(cos . (arccot . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((cos . (arccot . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((cos . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
cot * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cot * arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cot * arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) ^2 is V11() V12() ext-real Element of REAL
K37((sin . (arctan . x)),(sin . (arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((sin . (arctan . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((sin . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
diff ((cot * arctan),x) is V11() V12() ext-real Element of REAL
diff (cot,(arctan . x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(diff (cot,(arctan . x))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
1 / ((sin . (arctan . x)) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . (arctan . x)) ^2)) is V11() V12() ext-real set
K37(1,K39(((sin . (arctan . x)) ^2))) is V11() V12() ext-real set
- (1 / ((sin . (arctan . x)) ^2)) is V11() V12() ext-real Element of REAL
(- (1 / ((sin . (arctan . x)) ^2))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
(1 / ((sin . (arctan . x)) ^2)) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
- ((1 / ((sin . (arctan . x)) ^2)) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(1 / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
- ((1 / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(1 / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
- ((1 / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) ^2 is V11() V12() ext-real Element of REAL
K37((sin . (arctan . x)),(sin . (arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((sin . (arctan . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((sin . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
cot * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cot * arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cot * arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) ^2 is V11() V12() ext-real Element of REAL
K37((sin . (arccot . x)),(sin . (arccot . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((sin . (arccot . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((sin . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
diff ((cot * arccot),x) is V11() V12() ext-real Element of REAL
diff (cot,(arccot . x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (cot,(arccot . x))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
1 / ((sin . (arccot . x)) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . (arccot . x)) ^2)) is V11() V12() ext-real set
K37(1,K39(((sin . (arccot . x)) ^2))) is V11() V12() ext-real set
- (1 / ((sin . (arccot . x)) ^2)) is V11() V12() ext-real Element of REAL
(- (1 / ((sin . (arccot . x)) ^2))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
(1 / ((sin . (arccot . x)) ^2)) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
- ((1 / ((sin . (arccot . x)) ^2)) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(1 / ((sin . (arccot . x)) ^2)) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
- ((1 / ((sin . (arccot . x)) ^2)) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(1 / ((sin . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
- ((1 / ((sin . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
(1 / ((sin . (arccot . x)) ^2)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) ^2 is V11() V12() ext-real Element of REAL
K37((sin . (arccot . x)),(sin . (arccot . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((sin . (arccot . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((sin . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
sec * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sec * arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sec * arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
dom sec is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sec * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) ^2 is V11() V12() ext-real Element of REAL
K37((cos . (arctan . x)),(cos . (arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((cos . (arctan . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37((sin . (arctan . x)),K39((((cos . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
diff ((sec * arctan),x) is V11() V12() ext-real Element of REAL
diff (sec,(arctan . x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(diff (sec,(arctan . x))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) / ((cos . (arctan . x)) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . (arctan . x)) ^2)) is V11() V12() ext-real set
K37((sin . (arctan . x)),K39(((cos . (arctan . x)) ^2))) is V11() V12() ext-real set
((sin . (arctan . x)) / ((cos . (arctan . x)) ^2)) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
((sin . (arctan . x)) / ((cos . (arctan . x)) ^2)) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
((sin . (arctan . x)) / ((cos . (arctan . x)) ^2)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) * 1 is V11() V12() ext-real Element of REAL
((sin . (arctan . x)) * 1) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K37(((sin . (arctan . x)) * 1),K39((((cos . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((sec * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) ^2 is V11() V12() ext-real Element of REAL
K37((cos . (arctan . x)),(cos . (arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((cos . (arctan . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37((sin . (arctan . x)),K39((((cos . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
sec * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sec * arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sec * arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
dom sec is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sec * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) ^2 is V11() V12() ext-real Element of REAL
K37((cos . (arccot . x)),(cos . (arccot . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((cos . (arccot . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37((sin . (arccot . x)),K39((((cos . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
diff ((sec * arccot),x) is V11() V12() ext-real Element of REAL
diff (sec,(arccot . x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (sec,(arccot . x))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) / ((cos . (arccot . x)) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . (arccot . x)) ^2)) is V11() V12() ext-real set
K37((sin . (arccot . x)),K39(((cos . (arccot . x)) ^2))) is V11() V12() ext-real set
((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
- (((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) * 1 is V11() V12() ext-real Element of REAL
((sin . (arccot . x)) * 1) / (((cos . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K37(((sin . (arccot . x)) * 1),K39((((cos . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- (((sin . (arccot . x)) * 1) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sec * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) ^2 is V11() V12() ext-real Element of REAL
K37((cos . (arccot . x)),(cos . (arccot . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((cos . (arccot . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((cos . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37((sin . (arccot . x)),K39((((cos . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
cosec * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cosec * arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cosec * arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
dom cosec is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cosec * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) ^2 is V11() V12() ext-real Element of REAL
K37((sin . (arctan . x)),(sin . (arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((sin . (arctan . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37((cos . (arctan . x)),K39((((sin . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
diff ((cosec * arctan),x) is V11() V12() ext-real Element of REAL
diff (cosec,(arctan . x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(diff (cosec,(arctan . x))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) / ((sin . (arctan . x)) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . (arctan . x)) ^2)) is V11() V12() ext-real set
K37((cos . (arctan . x)),K39(((sin . (arctan . x)) ^2))) is V11() V12() ext-real set
- ((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) is V11() V12() ext-real Element of REAL
(- ((cos . (arctan . x)) / ((sin . (arctan . x)) ^2))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
- (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
- (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
- (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) * 1 is V11() V12() ext-real Element of REAL
((cos . (arctan . x)) * 1) / (((sin . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K37(((cos . (arctan . x)) * 1),K39((((sin . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- (((cos . (arctan . x)) * 1) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cosec * arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . (arctan . x) is V11() V12() ext-real Element of REAL
sin . (arctan . x) is V11() V12() ext-real Element of REAL
(sin . (arctan . x)) ^2 is V11() V12() ext-real Element of REAL
K37((sin . (arctan . x)),(sin . (arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((sin . (arctan . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
(cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37((cos . (arctan . x)),K39((((sin . (arctan . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
cosec * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cosec * arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cosec * arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
dom cosec is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cosec * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) ^2 is V11() V12() ext-real Element of REAL
K37((sin . (arccot . x)),(sin . (arccot . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((sin . (arccot . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37((cos . (arccot . x)),K39((((sin . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
diff ((cosec * arccot),x) is V11() V12() ext-real Element of REAL
diff (cosec,(arccot . x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (cosec,(arccot . x))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) / ((sin . (arccot . x)) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . (arccot . x)) ^2)) is V11() V12() ext-real set
K37((cos . (arccot . x)),K39(((sin . (arccot . x)) ^2))) is V11() V12() ext-real set
- ((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) is V11() V12() ext-real Element of REAL
(- ((cos . (arccot . x)) / ((sin . (arccot . x)) ^2))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
- (((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
- (((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
- (((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) * 1 is V11() V12() ext-real Element of REAL
((cos . (arccot . x)) * 1) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K37(((cos . (arccot . x)) * 1),K39((((sin . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((cosec * arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . (arccot . x) is V11() V12() ext-real Element of REAL
sin . (arccot . x) is V11() V12() ext-real Element of REAL
(sin . (arccot . x)) ^2 is V11() V12() ext-real Element of REAL
K37((sin . (arccot . x)),(sin . (arccot . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((sin . (arccot . x)) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
(cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((sin . (arccot . x)) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37((cos . (arccot . x)),K39((((sin . (arccot . x)) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
sin (#) arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sin (#) arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sin (#) arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(dom sin) /\ (dom arctan) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((sin (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(cos . x) * (arctan . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((sin . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
diff (sin,x) is V11() V12() ext-real Element of REAL
(arctan . x) * (diff (sin,x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(sin . x) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
((arctan . x) * (diff (sin,x))) + ((sin . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
(arctan . x) * (cos . x) is V11() V12() ext-real Element of REAL
((arctan . x) * (cos . x)) + ((sin . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(sin . x) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
((cos . x) * (arctan . x)) + ((sin . x) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(sin . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((cos . x) * (arctan . x)) + ((sin . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(cos . x) * (arctan . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((sin . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
sin (#) arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sin (#) arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sin (#) arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(dom sin) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((sin (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(cos . x) * (arccot . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((sin . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((sin . x) / (1 + (x ^2)))) is V11() V12() ext-real set
K36(((cos . x) * (arccot . x)),K38(((sin . x) / (1 + (x ^2))))) is V11() V12() ext-real set
diff (sin,x) is V11() V12() ext-real Element of REAL
(arccot . x) * (diff (sin,x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(sin . x) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
((arccot . x) * (diff (sin,x))) + ((sin . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
(arccot . x) * (cos . x) is V11() V12() ext-real Element of REAL
((arccot . x) * (cos . x)) + ((sin . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(sin . x) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
((cos . x) * (arccot . x)) + ((sin . x) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(sin . x) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((cos . x) * (arccot . x)) + ((sin . x) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(cos . x) * (arccot . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((sin . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((sin . x) / (1 + (x ^2)))) is V11() V12() ext-real set
K36(((cos . x) * (arccot . x)),K38(((sin . x) / (1 + (x ^2))))) is V11() V12() ext-real set
cos (#) arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cos (#) arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cos (#) arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(dom cos) /\ (dom arctan) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((cos (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(sin . x) * (arctan . x) is V11() V12() ext-real Element of REAL
- ((sin . x) * (arctan . x)) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cos . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
diff (cos,x) is V11() V12() ext-real Element of REAL
(arctan . x) * (diff (cos,x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(cos . x) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
((arctan . x) * (diff (cos,x))) + ((cos . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
- (sin . x) is V11() V12() ext-real Element of REAL
(arctan . x) * (- (sin . x)) is V11() V12() ext-real Element of REAL
((arctan . x) * (- (sin . x))) + ((cos . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(cos . x) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
(- ((sin . x) * (arctan . x))) + ((cos . x) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(cos . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- ((sin . x) * (arctan . x))) + ((cos . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(sin . x) * (arctan . x) is V11() V12() ext-real Element of REAL
- ((sin . x) * (arctan . x)) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cos . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
cos (#) arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cos (#) arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cos (#) arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(dom cos) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((cos (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(sin . x) * (arccot . x) is V11() V12() ext-real Element of REAL
- ((sin . x) * (arccot . x)) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cos . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((cos . x) / (1 + (x ^2)))) is V11() V12() ext-real set
K36((- ((sin . x) * (arccot . x))),K38(((cos . x) / (1 + (x ^2))))) is V11() V12() ext-real set
diff (cos,x) is V11() V12() ext-real Element of REAL
(arccot . x) * (diff (cos,x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(cos . x) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
((arccot . x) * (diff (cos,x))) + ((cos . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
- (sin . x) is V11() V12() ext-real Element of REAL
(arccot . x) * (- (sin . x)) is V11() V12() ext-real Element of REAL
((arccot . x) * (- (sin . x))) + ((cos . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(cos . x) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
(- ((sin . x) * (arccot . x))) + ((cos . x) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(cos . x) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(- ((sin . x) * (arccot . x))) + ((cos . x) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(sin . x) * (arccot . x) is V11() V12() ext-real Element of REAL
- ((sin . x) * (arccot . x)) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cos . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((cos . x) / (1 + (x ^2)))) is V11() V12() ext-real set
K36((- ((sin . x) * (arccot . x))),K38(((cos . x) / (1 + (x ^2))))) is V11() V12() ext-real set
tan (#) arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (tan (#) arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(tan (#) arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom tan) /\ (dom arctan) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
(arctan . x) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37((arctan . x),K39(((cos . x) ^2))) is V11() V12() ext-real set
tan . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(tan . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((tan . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
diff (tan,x) is V11() V12() ext-real Element of REAL
(arctan . x) * (diff (tan,x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(tan . x) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
((arctan . x) * (diff (tan,x))) + ((tan . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
1 / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37(1,K39(((cos . x) ^2))) is V11() V12() ext-real set
(arctan . x) * (1 / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
((arctan . x) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(tan . x) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
((arctan . x) / ((cos . x) ^2)) + ((tan . x) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(tan . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((arctan . x) / ((cos . x) ^2)) + ((tan . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
(arctan . x) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37((arctan . x),K39(((cos . x) ^2))) is V11() V12() ext-real set
tan . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(tan . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((tan . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
tan (#) arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (tan (#) arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(tan (#) arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom tan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
(arccot . x) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37((arccot . x),K39(((cos . x) ^2))) is V11() V12() ext-real set
tan . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(tan . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((tan . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((tan . x) / (1 + (x ^2)))) is V11() V12() ext-real set
K36(((arccot . x) / ((cos . x) ^2)),K38(((tan . x) / (1 + (x ^2))))) is V11() V12() ext-real set
diff (tan,x) is V11() V12() ext-real Element of REAL
(arccot . x) * (diff (tan,x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(tan . x) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
((arccot . x) * (diff (tan,x))) + ((tan . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
1 / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37(1,K39(((cos . x) ^2))) is V11() V12() ext-real set
(arccot . x) * (1 / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
((arccot . x) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(tan . x) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
((arccot . x) / ((cos . x) ^2)) + ((tan . x) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(tan . x) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((arccot . x) / ((cos . x) ^2)) + ((tan . x) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
(arccot . x) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37((arccot . x),K39(((cos . x) ^2))) is V11() V12() ext-real set
tan . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(tan . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((tan . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((tan . x) / (1 + (x ^2)))) is V11() V12() ext-real set
K36(((arccot . x) / ((cos . x) ^2)),K38(((tan . x) / (1 + (x ^2))))) is V11() V12() ext-real set
cot (#) arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cot (#) arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cot (#) arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom cot) /\ (dom arctan) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
(arctan . x) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37((arctan . x),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((arctan . x) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cot . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cot . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
diff (cot,x) is V11() V12() ext-real Element of REAL
(arctan . x) * (diff (cot,x)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(cot . x) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
((arctan . x) * (diff (cot,x))) + ((cot . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
1 / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37(1,K39(((sin . x) ^2))) is V11() V12() ext-real set
- (1 / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
(arctan . x) * (- (1 / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
((arctan . x) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(cot . x) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
(- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(cot . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
(arctan . x) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37((arctan . x),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((arctan . x) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cot . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cot . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
cot (#) arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cot (#) arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cot (#) arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom cot) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
(arccot . x) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37((arccot . x),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((arccot . x) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cot . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cot . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((cot . x) / (1 + (x ^2)))) is V11() V12() ext-real set
K36((- ((arccot . x) / ((sin . x) ^2))),K38(((cot . x) / (1 + (x ^2))))) is V11() V12() ext-real set
diff (cot,x) is V11() V12() ext-real Element of REAL
(arccot . x) * (diff (cot,x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(cot . x) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
((arccot . x) * (diff (cot,x))) + ((cot . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
1 / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37(1,K39(((sin . x) ^2))) is V11() V12() ext-real set
- (1 / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
(arccot . x) * (- (1 / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
((arccot . x) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(cot . x) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
(- ((arccot . x) / ((sin . x) ^2))) + ((cot . x) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(cot . x) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(- ((arccot . x) / ((sin . x) ^2))) + ((cot . x) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
(arccot . x) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37((arccot . x),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((arccot . x) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cot . x) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((cot . x),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((cot . x) / (1 + (x ^2)))) is V11() V12() ext-real set
K36((- ((arccot . x) / ((sin . x) ^2))),K38(((cot . x) / (1 + (x ^2))))) is V11() V12() ext-real set
sec (#) arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sec (#) arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sec (#) arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom sec is V46() V47() V48() Element of K6(REAL)
(dom sec) /\ (dom arctan) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sec (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(sin . x) * (arctan . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((sin . x) * (arctan . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(((sin . x) * (arctan . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . x) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((cos . x) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((cos . x) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((cos . x) * (1 + (x ^2))))) is V11() V12() ext-real set
(((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
diff (sec,x) is V11() V12() ext-real Element of REAL
(arctan . x) * (diff (sec,x)) is V11() V12() ext-real Element of REAL
sec . x is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(sec . x) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
((arctan . x) * (diff (sec,x))) + ((sec . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
(sin . x) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37((sin . x),K39(((cos . x) ^2))) is V11() V12() ext-real set
(arctan . x) * ((sin . x) / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
((arctan . x) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(sec . x) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
(((sin . x) * (arctan . x)) / ((cos . x) ^2)) + ((sec . x) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(sec . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((sin . x) * (arctan . x)) / ((cos . x) ^2)) + ((sec . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
1 / (cos . x) is V11() V12() ext-real Element of REAL
K39((cos . x)) is V11() V12() ext-real set
K37(1,K39((cos . x))) is V11() V12() ext-real set
(1 / (cos . x)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((sin . x) * (arctan . x)) / ((cos . x) ^2)) + ((1 / (cos . x)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sec (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(sin . x) * (arctan . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((sin . x) * (arctan . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(((sin . x) * (arctan . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . x) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((cos . x) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((cos . x) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((cos . x) * (1 + (x ^2))))) is V11() V12() ext-real set
(((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
sec (#) arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sec (#) arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sec (#) arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom sec is V46() V47() V48() Element of K6(REAL)
(dom sec) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sec (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(sin . x) * (arccot . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((sin . x) * (arccot . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(((sin . x) * (arccot . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . x) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((cos . x) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((cos . x) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((cos . x) * (1 + (x ^2))))) is V11() V12() ext-real set
(((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K38((1 / ((cos . x) * (1 + (x ^2))))) is V11() V12() ext-real set
K36((((sin . x) * (arccot . x)) / ((cos . x) ^2)),K38((1 / ((cos . x) * (1 + (x ^2)))))) is V11() V12() ext-real set
diff (sec,x) is V11() V12() ext-real Element of REAL
(arccot . x) * (diff (sec,x)) is V11() V12() ext-real Element of REAL
sec . x is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(sec . x) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
((arccot . x) * (diff (sec,x))) + ((sec . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
(sin . x) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37((sin . x),K39(((cos . x) ^2))) is V11() V12() ext-real set
(arccot . x) * ((sin . x) / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
((arccot . x) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(sec . x) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
(((sin . x) * (arccot . x)) / ((cos . x) ^2)) + ((sec . x) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(sec . x) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(((sin . x) * (arccot . x)) / ((cos . x) ^2)) + ((sec . x) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
(sec . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((sin . x) * (arccot . x)) / ((cos . x) ^2)) - ((sec . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K38(((sec . x) * (1 / (1 + (x ^2))))) is V11() V12() ext-real set
K36((((sin . x) * (arccot . x)) / ((cos . x) ^2)),K38(((sec . x) * (1 / (1 + (x ^2)))))) is V11() V12() ext-real set
1 / (cos . x) is V11() V12() ext-real Element of REAL
K39((cos . x)) is V11() V12() ext-real set
K37(1,K39((cos . x))) is V11() V12() ext-real set
(1 / (cos . x)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((sin . x) * (arccot . x)) / ((cos . x) ^2)) - ((1 / (cos . x)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K38(((1 / (cos . x)) * (1 / (1 + (x ^2))))) is V11() V12() ext-real set
K36((((sin . x) * (arccot . x)) / ((cos . x) ^2)),K38(((1 / (cos . x)) * (1 / (1 + (x ^2)))))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((sec (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(sin . x) * (arccot . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((sin . x) * (arccot . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(((sin . x) * (arccot . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(cos . x) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((cos . x) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((cos . x) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((cos . x) * (1 + (x ^2))))) is V11() V12() ext-real set
(((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K38((1 / ((cos . x) * (1 + (x ^2))))) is V11() V12() ext-real set
K36((((sin . x) * (arccot . x)) / ((cos . x) ^2)),K38((1 / ((cos . x) * (1 + (x ^2)))))) is V11() V12() ext-real set
cosec (#) arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cosec (#) arctan) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cosec (#) arctan) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom cosec is V46() V47() V48() Element of K6(REAL)
(dom cosec) /\ (dom arctan) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cosec (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(cos . x) * (arctan . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((cos . x) * (arctan . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(((cos . x) * (arctan . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- (((cos . x) * (arctan . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . x) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((sin . x) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((sin . x) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((sin . x) * (1 + (x ^2))))) is V11() V12() ext-real set
(- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
diff (cosec,x) is V11() V12() ext-real Element of REAL
(arctan . x) * (diff (cosec,x)) is V11() V12() ext-real Element of REAL
cosec . x is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(cosec . x) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
((arctan . x) * (diff (cosec,x))) + ((cosec . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
(cos . x) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37((cos . x),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((cos . x) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
(arctan . x) * (- ((cos . x) / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
((arctan . x) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(cosec . x) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
(- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(cosec . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
1 / (sin . x) is V11() V12() ext-real Element of REAL
K39((sin . x)) is V11() V12() ext-real set
K37(1,K39((sin . x))) is V11() V12() ext-real set
(1 / (sin . x)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((1 / (sin . x)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cosec (#) arctan) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(cos . x) * (arctan . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((cos . x) * (arctan . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(((cos . x) * (arctan . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- (((cos . x) * (arctan . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . x) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((sin . x) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((sin . x) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((sin . x) * (1 + (x ^2))))) is V11() V12() ext-real set
(- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
cosec (#) arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cosec (#) arccot) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cosec (#) arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom cosec is V46() V47() V48() Element of K6(REAL)
(dom cosec) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cosec (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(cos . x) * (arccot . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((cos . x) * (arccot . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(((cos . x) * (arccot . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- (((cos . x) * (arccot . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . x) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((sin . x) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((sin . x) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((sin . x) * (1 + (x ^2))))) is V11() V12() ext-real set
(- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K38((1 / ((sin . x) * (1 + (x ^2))))) is V11() V12() ext-real set
K36((- (((cos . x) * (arccot . x)) / ((sin . x) ^2))),K38((1 / ((sin . x) * (1 + (x ^2)))))) is V11() V12() ext-real set
diff (cosec,x) is V11() V12() ext-real Element of REAL
(arccot . x) * (diff (cosec,x)) is V11() V12() ext-real Element of REAL
cosec . x is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(cosec . x) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
((arccot . x) * (diff (cosec,x))) + ((cosec . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
(cos . x) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37((cos . x),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((cos . x) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
(arccot . x) * (- ((cos . x) / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
((arccot . x) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(cosec . x) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
(- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) + ((cosec . x) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(cosec . x) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) + ((cosec . x) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
(cosec . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - ((cosec . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K38(((cosec . x) * (1 / (1 + (x ^2))))) is V11() V12() ext-real set
K36((- (((cos . x) * (arccot . x)) / ((sin . x) ^2))),K38(((cosec . x) * (1 / (1 + (x ^2)))))) is V11() V12() ext-real set
1 / (sin . x) is V11() V12() ext-real Element of REAL
K39((sin . x)) is V11() V12() ext-real set
K37(1,K39((sin . x))) is V11() V12() ext-real set
(1 / (sin . x)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - ((1 / (sin . x)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K38(((1 / (sin . x)) * (1 / (1 + (x ^2))))) is V11() V12() ext-real set
K36((- (((cos . x) * (arccot . x)) / ((sin . x) ^2))),K38(((1 / (sin . x)) * (1 / (1 + (x ^2)))))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((cosec (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(cos . x) * (arccot . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((cos . x) * (arccot . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(((cos . x) * (arccot . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- (((cos . x) * (arccot . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(sin . x) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((sin . x) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((sin . x) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((sin . x) * (1 + (x ^2))))) is V11() V12() ext-real set
(- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K38((1 / ((sin . x) * (1 + (x ^2))))) is V11() V12() ext-real set
K36((- (((cos . x) * (arccot . x)) / ((sin . x) ^2))),K38((1 / ((sin . x) * (1 + (x ^2)))))) is V11() V12() ext-real set
arctan + arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan + arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (arctan,x)) + (diff (arccot,x)) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
((arctan `| Z) . x) + (diff (arccot,x)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(1 / (1 + (x ^2))) + (diff (arccot,x)) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(1 / (1 + (x ^2))) + ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(1 / (1 + (x ^2))) + (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arctan - arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
diff (arctan,x) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (arctan,x)) - (diff (arccot,x)) is V11() V12() ext-real Element of REAL
K38((diff (arccot,x))) is V11() V12() ext-real set
K36((diff (arctan,x)),K38((diff (arccot,x)))) is V11() V12() ext-real set
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
((arctan `| Z) . x) - (diff (arccot,x)) is V11() V12() ext-real Element of REAL
K36(((arctan `| Z) . x),K38((diff (arccot,x)))) is V11() V12() ext-real set
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(1 / (1 + (x ^2))) - (diff (arccot,x)) is V11() V12() ext-real Element of REAL
K36((1 / (1 + (x ^2))),K38((diff (arccot,x)))) is V11() V12() ext-real set
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(1 / (1 + (x ^2))) - ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
K38(((arccot `| Z) . x)) is V11() V12() ext-real set
K36((1 / (1 + (x ^2))),K38(((arccot `| Z) . x))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(1 / (1 + (x ^2))) - (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K38((- (1 / (1 + (x ^2))))) is V11() V12() ext-real set
K36((1 / (1 + (x ^2))),K38((- (1 / (1 + (x ^2)))))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
sin (#) (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(sin (#) (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan + arccot) is V46() V47() V48() Element of K6(REAL)
(dom sin) /\ (dom (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
dom (sin (#) (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((sin (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
(cos . x) * ((arctan . x) + (arccot . x)) is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff (sin,x) is V11() V12() ext-real Element of REAL
((arctan + arccot) . x) * (diff (sin,x)) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(sin . x) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan + arccot) . x) * (diff (sin,x))) + ((sin . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (diff (sin,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (diff (sin,x))) + ((sin . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (cos . x) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(sin . x) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * (((arctan + arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
(sin . x) * 0 is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * 0) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
(cos . x) * ((arctan . x) + (arccot . x)) is V11() V12() ext-real Element of REAL
sin (#) (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(sin (#) (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
(dom sin) /\ (dom (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
dom (sin (#) (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((sin (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
(cos . x) * ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
2 * (sin . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (sin . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (sin . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff (sin,x) is V11() V12() ext-real Element of REAL
((arctan - arccot) . x) * (diff (sin,x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(sin . x) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan - arccot) . x) * (diff (sin,x))) + ((sin . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (diff (sin,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (diff (sin,x))) + ((sin . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (cos . x) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (cos . x)) + ((sin . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(sin . x) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (cos . x)) + ((sin . x) * (((arctan - arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(sin . x) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (cos . x)) + ((sin . x) * (2 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
(cos . x) * ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
2 * (sin . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (sin . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (sin . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
cos (#) (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(cos (#) (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan + arccot) is V46() V47() V48() Element of K6(REAL)
(dom cos) /\ (dom (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
dom (cos (#) (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((cos (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
(sin . x) * ((arctan . x) + (arccot . x)) is V11() V12() ext-real Element of REAL
- ((sin . x) * ((arctan . x) + (arccot . x))) is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff (cos,x) is V11() V12() ext-real Element of REAL
((arctan + arccot) . x) * (diff (cos,x)) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(cos . x) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan + arccot) . x) * (diff (cos,x))) + ((cos . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (diff (cos,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (diff (cos,x))) + ((cos . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
- (sin . x) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (- (sin . x)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (- (sin . x))) + ((cos . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(cos . x) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (- (sin . x))) + ((cos . x) * (((arctan + arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (sin . x) is V11() V12() ext-real Element of REAL
- (((arctan . x) + (arccot . x)) * (sin . x)) is V11() V12() ext-real Element of REAL
(cos . x) * 0 is V11() V12() ext-real Element of REAL
(- (((arctan . x) + (arccot . x)) * (sin . x))) + ((cos . x) * 0) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
(sin . x) * ((arctan . x) + (arccot . x)) is V11() V12() ext-real Element of REAL
- ((sin . x) * ((arctan . x) + (arccot . x))) is V11() V12() ext-real Element of REAL
cos (#) (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(cos (#) (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
(dom cos) /\ (dom (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
dom (cos (#) (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((cos (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
(sin . x) * ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
- ((sin . x) * ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
2 * (cos . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (cos . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (cos . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff (cos,x) is V11() V12() ext-real Element of REAL
((arctan - arccot) . x) * (diff (cos,x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(cos . x) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan - arccot) . x) * (diff (cos,x))) + ((cos . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (diff (cos,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (diff (cos,x))) + ((cos . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
- (sin . x) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (- (sin . x)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (- (sin . x))) + ((cos . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(cos . x) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (- (sin . x))) + ((cos . x) * (((arctan - arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (sin . x) is V11() V12() ext-real Element of REAL
- (((arctan . x) - (arccot . x)) * (sin . x)) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(cos . x) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- (((arctan . x) - (arccot . x)) * (sin . x))) + ((cos . x) * (2 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
(sin . x) * ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
- ((sin . x) * ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
2 * (cos . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (cos . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (cos . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
tan (#) (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(tan (#) (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan + arccot) is V46() V47() V48() Element of K6(REAL)
(dom tan) /\ (dom (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
dom (tan (#) (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((tan (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((arctan . x) + (arccot . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(((arctan . x) + (arccot . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff (tan,x) is V11() V12() ext-real Element of REAL
((arctan + arccot) . x) * (diff (tan,x)) is V11() V12() ext-real Element of REAL
tan . x is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(tan . x) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan + arccot) . x) * (diff (tan,x))) + ((tan . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (diff (tan,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (diff (tan,x))) + ((tan . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
1 / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37(1,K39(((cos . x) ^2))) is V11() V12() ext-real set
((arctan . x) + (arccot . x)) * (1 / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(tan . x) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * (((arctan + arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
(tan . x) * 0 is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * 0) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((arctan . x) + (arccot . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(((arctan . x) + (arccot . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
tan (#) (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(tan (#) (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
(dom tan) /\ (dom (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
dom (tan (#) (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((tan (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((arctan . x) - (arccot . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(((arctan . x) - (arccot . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
tan . x is V11() V12() ext-real Element of REAL
2 * (tan . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (tan . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (tan . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
(((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff (tan,x) is V11() V12() ext-real Element of REAL
((arctan - arccot) . x) * (diff (tan,x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(tan . x) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan - arccot) . x) * (diff (tan,x))) + ((tan . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (diff (tan,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (diff (tan,x))) + ((tan . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
1 / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37(1,K39(((cos . x) ^2))) is V11() V12() ext-real set
((arctan . x) - (arccot . x)) * (1 / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(tan . x) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * (((arctan - arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(tan . x) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * (2 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((tan (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
((arctan . x) - (arccot . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37(((arctan . x) - (arccot . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
tan . x is V11() V12() ext-real Element of REAL
2 * (tan . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (tan . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (tan . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
(((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
cot (#) (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(cot (#) (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan + arccot) is V46() V47() V48() Element of K6(REAL)
(dom cot) /\ (dom (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
dom (cot (#) (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((cot (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((arctan . x) + (arccot . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(((arctan . x) + (arccot . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff (cot,x) is V11() V12() ext-real Element of REAL
((arctan + arccot) . x) * (diff (cot,x)) is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(cot . x) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan + arccot) . x) * (diff (cot,x))) + ((cot . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (diff (cot,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (diff (cot,x))) + ((cot . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
1 / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37(1,K39(((sin . x) ^2))) is V11() V12() ext-real set
- (1 / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (- (1 / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(cot . x) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(- (((arctan . x) + (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (((arctan + arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
(cot . x) * 0 is V11() V12() ext-real Element of REAL
(- (((arctan . x) + (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * 0) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((arctan . x) + (arccot . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(((arctan . x) + (arccot . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cot (#) (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(cot (#) (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
(dom cot) /\ (dom (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
dom (cot (#) (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((cot (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((arctan . x) - (arccot . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(((arctan . x) - (arccot . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- (((arctan . x) - (arccot . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
2 * (cot . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (cot . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (cot . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff (cot,x) is V11() V12() ext-real Element of REAL
((arctan - arccot) . x) * (diff (cot,x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(cot . x) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan - arccot) . x) * (diff (cot,x))) + ((cot . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (diff (cot,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (diff (cot,x))) + ((cot . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
1 / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37(1,K39(((sin . x) ^2))) is V11() V12() ext-real set
- (1 / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (- (1 / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(cot . x) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (((arctan - arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(cot . x) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (2 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cot (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
((arctan . x) - (arccot . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37(((arctan . x) - (arccot . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- (((arctan . x) - (arccot . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cot . x is V11() V12() ext-real Element of REAL
2 * (cot . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (cot . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (cot . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
dom sec is V46() V47() V48() Element of K6(REAL)
sec (#) (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(sec (#) (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan + arccot) is V46() V47() V48() Element of K6(REAL)
(dom sec) /\ (dom (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
dom (sec (#) (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((sec (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (sin . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
(((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37((((arctan . x) + (arccot . x)) * (sin . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff (sec,x) is V11() V12() ext-real Element of REAL
((arctan + arccot) . x) * (diff (sec,x)) is V11() V12() ext-real Element of REAL
sec . x is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(sec . x) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan + arccot) . x) * (diff (sec,x))) + ((sec . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (diff (sec,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (diff (sec,x))) + ((sec . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
(sin . x) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37((sin . x),K39(((cos . x) ^2))) is V11() V12() ext-real set
((arctan . x) + (arccot . x)) * ((sin . x) / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(sec . x) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
((((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * (((arctan + arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
(sec . x) * 0 is V11() V12() ext-real Element of REAL
((((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * 0) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sec (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (sin . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
(((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37((((arctan . x) + (arccot . x)) * (sin . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
sec (#) (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(sec (#) (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
(dom sec) /\ (dom (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
dom (sec (#) (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((sec (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
sin . x is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (sin . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
(((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37((((arctan . x) - (arccot . x)) * (sin . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
sec . x is V11() V12() ext-real Element of REAL
2 * (sec . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (sec . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (sec . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff (sec,x) is V11() V12() ext-real Element of REAL
((arctan - arccot) . x) * (diff (sec,x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(sec . x) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan - arccot) . x) * (diff (sec,x))) + ((sec . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (diff (sec,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (diff (sec,x))) + ((sec . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
(sin . x) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K37((sin . x),K39(((cos . x) ^2))) is V11() V12() ext-real set
((arctan . x) - (arccot . x)) * ((sin . x) / ((cos . x) ^2)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(sec . x) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * (((arctan - arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(sec . x) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * (2 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sec (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
sin . x is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (sin . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
(cos . x) ^2 is V11() V12() ext-real Element of REAL
K37((cos . x),(cos . x)) is V11() V12() ext-real set
(((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2) is V11() V12() ext-real Element of REAL
K39(((cos . x) ^2)) is V11() V12() ext-real set
K37((((arctan . x) - (arccot . x)) * (sin . x)),K39(((cos . x) ^2))) is V11() V12() ext-real set
sec . x is V11() V12() ext-real Element of REAL
2 * (sec . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (sec . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (sec . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
dom cosec is V46() V47() V48() Element of K6(REAL)
cosec (#) (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(cosec (#) (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan + arccot) is V46() V47() V48() Element of K6(REAL)
(dom cosec) /\ (dom (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
dom (cosec (#) (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((cosec (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (cos . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
(((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37((((arctan . x) + (arccot . x)) * (cos . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff (cosec,x) is V11() V12() ext-real Element of REAL
((arctan + arccot) . x) * (diff (cosec,x)) is V11() V12() ext-real Element of REAL
cosec . x is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(cosec . x) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan + arccot) . x) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (diff (cosec,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
(cos . x) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37((cos . x),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((cos . x) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (- ((cos . x) / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(cosec . x) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (((arctan + arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
(cosec . x) * 0 is V11() V12() ext-real Element of REAL
(- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * 0) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cosec (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
cos . x is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (cos . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
(((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37((((arctan . x) + (arccot . x)) * (cos . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cosec (#) (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(cosec (#) (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
(dom cosec) /\ (dom (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
dom (cosec (#) (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((cosec (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
cos . x is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (cos . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
(((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37((((arctan . x) - (arccot . x)) * (cos . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cosec . x is V11() V12() ext-real Element of REAL
2 * (cosec . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (cosec . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (cosec . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff (cosec,x) is V11() V12() ext-real Element of REAL
((arctan - arccot) . x) * (diff (cosec,x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(cosec . x) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan - arccot) . x) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (diff (cosec,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
(cos . x) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K37((cos . x),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((cos . x) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (- ((cos . x) / ((sin . x) ^2))) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(cosec . x) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (((arctan - arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(cosec . x) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (2 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cosec (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
cos . x is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (cos . x) is V11() V12() ext-real Element of REAL
sin . x is V11() V12() ext-real Element of REAL
(sin . x) ^2 is V11() V12() ext-real Element of REAL
K37((sin . x),(sin . x)) is V11() V12() ext-real set
(((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2) is V11() V12() ext-real Element of REAL
K39(((sin . x) ^2)) is V11() V12() ext-real set
K37((((arctan . x) - (arccot . x)) * (cos . x)),K39(((sin . x) ^2))) is V11() V12() ext-real set
- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2)) is V11() V12() ext-real Element of REAL
cosec . x is V11() V12() ext-real Element of REAL
2 * (cosec . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (cosec . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (cosec . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
(- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
exp_R (#) (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(exp_R (#) (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan + arccot) is V46() V47() V48() Element of K6(REAL)
(dom exp_R) /\ (dom (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
dom (exp_R (#) (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((exp_R (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
(exp_R . x) * ((arctan . x) + (arccot . x)) is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff (exp_R,x) is V11() V12() ext-real Element of REAL
((arctan + arccot) . x) * (diff (exp_R,x)) is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(exp_R . x) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan + arccot) . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (diff (exp_R,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (exp_R . x) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (diff ((arctan + arccot),x))) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(exp_R . x) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (((arctan + arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
(exp_R . x) * 0 is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (exp_R . x)) + ((exp_R . x) * 0) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((exp_R (#) (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
(exp_R . x) * ((arctan . x) + (arccot . x)) is V11() V12() ext-real Element of REAL
exp_R (#) (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(exp_R (#) (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
(dom exp_R) /\ (dom (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
dom (exp_R (#) (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((exp_R (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
(exp_R . x) * ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
2 * (exp_R . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (exp_R . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (exp_R . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff (exp_R,x) is V11() V12() ext-real Element of REAL
((arctan - arccot) . x) * (diff (exp_R,x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(exp_R . x) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(((arctan - arccot) . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (diff (exp_R,x)) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
((arctan . x) - (arccot . x)) * (exp_R . x) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (diff ((arctan - arccot),x))) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(exp_R . x) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (((arctan - arccot) `| Z) . x)) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(exp_R . x) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (2 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((exp_R (#) (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
(exp_R . x) * ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
2 * (exp_R . x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (exp_R . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (exp_R . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan + arccot) / exp_R is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
((arctan + arccot) / exp_R) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan + arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
(((arctan + arccot) / exp_R) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) / (exp_R . x) is V11() V12() ext-real Element of REAL
K39((exp_R . x)) is V11() V12() ext-real set
K37(((arctan . x) + (arccot . x)),K39((exp_R . x))) is V11() V12() ext-real set
- (((arctan . x) + (arccot . x)) / (exp_R . x)) is V11() V12() ext-real Element of REAL
diff (((arctan + arccot) / exp_R),x) is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(diff ((arctan + arccot),x)) * (exp_R . x) is V11() V12() ext-real Element of REAL
diff (exp_R,x) is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
(diff (exp_R,x)) * ((arctan + arccot) . x) is V11() V12() ext-real Element of REAL
((diff ((arctan + arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x)) is V11() V12() ext-real Element of REAL
K38(((diff (exp_R,x)) * ((arctan + arccot) . x))) is V11() V12() ext-real set
K36(((diff ((arctan + arccot),x)) * (exp_R . x)),K38(((diff (exp_R,x)) * ((arctan + arccot) . x)))) is V11() V12() ext-real set
(exp_R . x) ^2 is V11() V12() ext-real Element of REAL
K37((exp_R . x),(exp_R . x)) is V11() V12() ext-real set
(((diff ((arctan + arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K39(((exp_R . x) ^2)) is V11() V12() ext-real set
K37((((diff ((arctan + arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(((arctan + arccot) `| Z) . x) * (exp_R . x) is V11() V12() ext-real Element of REAL
((((arctan + arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x)) is V11() V12() ext-real Element of REAL
K36(((((arctan + arccot) `| Z) . x) * (exp_R . x)),K38(((diff (exp_R,x)) * ((arctan + arccot) . x)))) is V11() V12() ext-real set
(((((arctan + arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K37((((((arctan + arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
0 * (exp_R . x) is V11() V12() ext-real Element of REAL
(0 * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x)) is V11() V12() ext-real Element of REAL
K36((0 * (exp_R . x)),K38(((diff (exp_R,x)) * ((arctan + arccot) . x)))) is V11() V12() ext-real set
((0 * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K37(((0 * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
((diff (exp_R,x)) * ((arctan + arccot) . x)) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K37(((diff (exp_R,x)) * ((arctan + arccot) . x)),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
- (((diff (exp_R,x)) * ((arctan + arccot) . x)) / ((exp_R . x) ^2)) is V11() V12() ext-real Element of REAL
(exp_R . x) * ((arctan + arccot) . x) is V11() V12() ext-real Element of REAL
((exp_R . x) * ((arctan + arccot) . x)) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K37(((exp_R . x) * ((arctan + arccot) . x)),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
- (((exp_R . x) * ((arctan + arccot) . x)) / ((exp_R . x) ^2)) is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) * (exp_R . x) is V11() V12() ext-real Element of REAL
(exp_R . x) * (exp_R . x) is V11() V12() ext-real Element of REAL
(((arctan . x) + (arccot . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x)) is V11() V12() ext-real Element of REAL
K39(((exp_R . x) * (exp_R . x))) is V11() V12() ext-real set
K37((((arctan . x) + (arccot . x)) * (exp_R . x)),K39(((exp_R . x) * (exp_R . x)))) is V11() V12() ext-real set
- ((((arctan . x) + (arccot . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x))) is V11() V12() ext-real Element of REAL
(exp_R . x) / ((exp_R . x) * (exp_R . x)) is V11() V12() ext-real Element of REAL
K37((exp_R . x),K39(((exp_R . x) * (exp_R . x)))) is V11() V12() ext-real set
((arctan . x) + (arccot . x)) * ((exp_R . x) / ((exp_R . x) * (exp_R . x))) is V11() V12() ext-real Element of REAL
- (((arctan . x) + (arccot . x)) * ((exp_R . x) / ((exp_R . x) * (exp_R . x)))) is V11() V12() ext-real Element of REAL
(exp_R . x) / (exp_R . x) is V11() V12() ext-real Element of REAL
K37((exp_R . x),K39((exp_R . x))) is V11() V12() ext-real set
((exp_R . x) / (exp_R . x)) / (exp_R . x) is V11() V12() ext-real Element of REAL
K37(((exp_R . x) / (exp_R . x)),K39((exp_R . x))) is V11() V12() ext-real set
((arctan . x) + (arccot . x)) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) is V11() V12() ext-real Element of REAL
- (((arctan . x) + (arccot . x)) * (((exp_R . x) / (exp_R . x)) / (exp_R . x))) is V11() V12() ext-real Element of REAL
1 / (exp_R . x) is V11() V12() ext-real Element of REAL
K37(1,K39((exp_R . x))) is V11() V12() ext-real set
((arctan . x) + (arccot . x)) * (1 / (exp_R . x)) is V11() V12() ext-real Element of REAL
- (((arctan . x) + (arccot . x)) * (1 / (exp_R . x))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
(((arctan + arccot) / exp_R) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) + (arccot . x) is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
((arctan . x) + (arccot . x)) / (exp_R . x) is V11() V12() ext-real Element of REAL
K39((exp_R . x)) is V11() V12() ext-real set
K37(((arctan . x) + (arccot . x)),K39((exp_R . x))) is V11() V12() ext-real set
- (((arctan . x) + (arccot . x)) / (exp_R . x)) is V11() V12() ext-real Element of REAL
(arctan - arccot) / exp_R is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
((arctan - arccot) / exp_R) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
(((arctan - arccot) / exp_R) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
arctan . x is V11() V12() ext-real Element of REAL
(2 / (1 + (x ^2))) - (arctan . x) is V11() V12() ext-real Element of REAL
K38((arctan . x)) is V11() V12() ext-real set
K36((2 / (1 + (x ^2))),K38((arctan . x))) is V11() V12() ext-real set
arccot . x is V11() V12() ext-real Element of REAL
((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x) is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
(((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) is V11() V12() ext-real Element of REAL
K39((exp_R . x)) is V11() V12() ext-real set
K37((((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)),K39((exp_R . x))) is V11() V12() ext-real set
diff (((arctan - arccot) / exp_R),x) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(diff ((arctan - arccot),x)) * (exp_R . x) is V11() V12() ext-real Element of REAL
diff (exp_R,x) is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
(diff (exp_R,x)) * ((arctan - arccot) . x) is V11() V12() ext-real Element of REAL
((diff ((arctan - arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x)) is V11() V12() ext-real Element of REAL
K38(((diff (exp_R,x)) * ((arctan - arccot) . x))) is V11() V12() ext-real set
K36(((diff ((arctan - arccot),x)) * (exp_R . x)),K38(((diff (exp_R,x)) * ((arctan - arccot) . x)))) is V11() V12() ext-real set
(exp_R . x) ^2 is V11() V12() ext-real Element of REAL
K37((exp_R . x),(exp_R . x)) is V11() V12() ext-real set
(((diff ((arctan - arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K39(((exp_R . x) ^2)) is V11() V12() ext-real set
K37((((diff ((arctan - arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(((arctan - arccot) `| Z) . x) * (exp_R . x) is V11() V12() ext-real Element of REAL
((((arctan - arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x)) is V11() V12() ext-real Element of REAL
K36(((((arctan - arccot) `| Z) . x) * (exp_R . x)),K38(((diff (exp_R,x)) * ((arctan - arccot) . x)))) is V11() V12() ext-real set
(((((arctan - arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K37((((((arctan - arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
(2 / (1 + (x ^2))) * (exp_R . x) is V11() V12() ext-real Element of REAL
((2 / (1 + (x ^2))) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x)) is V11() V12() ext-real Element of REAL
K36(((2 / (1 + (x ^2))) * (exp_R . x)),K38(((diff (exp_R,x)) * ((arctan - arccot) . x)))) is V11() V12() ext-real set
(((2 / (1 + (x ^2))) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K37((((2 / (1 + (x ^2))) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
(exp_R . x) * ((arctan - arccot) . x) is V11() V12() ext-real Element of REAL
((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan - arccot) . x)) is V11() V12() ext-real Element of REAL
K38(((exp_R . x) * ((arctan - arccot) . x))) is V11() V12() ext-real set
K36(((2 / (1 + (x ^2))) * (exp_R . x)),K38(((exp_R . x) * ((arctan - arccot) . x)))) is V11() V12() ext-real set
(((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K37((((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan - arccot) . x))),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
(exp_R . x) * ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
K38(((exp_R . x) * ((arctan . x) - (arccot . x)))) is V11() V12() ext-real set
K36(((2 / (1 + (x ^2))) * (exp_R . x)),K38(((exp_R . x) * ((arctan . x) - (arccot . x))))) is V11() V12() ext-real set
(((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan . x) - (arccot . x)))) / ((exp_R . x) ^2) is V11() V12() ext-real Element of REAL
K37((((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan . x) - (arccot . x)))),K39(((exp_R . x) ^2))) is V11() V12() ext-real set
(2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
K38(((arctan . x) - (arccot . x))) is V11() V12() ext-real set
K36((2 / (1 + (x ^2))),K38(((arctan . x) - (arccot . x)))) is V11() V12() ext-real set
(exp_R . x) * (exp_R . x) is V11() V12() ext-real Element of REAL
(exp_R . x) / ((exp_R . x) * (exp_R . x)) is V11() V12() ext-real Element of REAL
K39(((exp_R . x) * (exp_R . x))) is V11() V12() ext-real set
K37((exp_R . x),K39(((exp_R . x) * (exp_R . x)))) is V11() V12() ext-real set
((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * ((exp_R . x) / ((exp_R . x) * (exp_R . x))) is V11() V12() ext-real Element of REAL
(exp_R . x) / (exp_R . x) is V11() V12() ext-real Element of REAL
K37((exp_R . x),K39((exp_R . x))) is V11() V12() ext-real set
((exp_R . x) / (exp_R . x)) / (exp_R . x) is V11() V12() ext-real Element of REAL
K37(((exp_R . x) / (exp_R . x)),K39((exp_R . x))) is V11() V12() ext-real set
((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) is V11() V12() ext-real Element of REAL
1 / (exp_R . x) is V11() V12() ext-real Element of REAL
K37(1,K39((exp_R . x))) is V11() V12() ext-real set
((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * (1 / (exp_R . x)) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
(((arctan - arccot) / exp_R) `| Z) . x is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
arctan . x is V11() V12() ext-real Element of REAL
(2 / (1 + (x ^2))) - (arctan . x) is V11() V12() ext-real Element of REAL
K38((arctan . x)) is V11() V12() ext-real set
K36((2 / (1 + (x ^2))),K38((arctan . x))) is V11() V12() ext-real set
arccot . x is V11() V12() ext-real Element of REAL
((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x) is V11() V12() ext-real Element of REAL
exp_R . x is V11() V12() ext-real Element of REAL
(((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) is V11() V12() ext-real Element of REAL
K39((exp_R . x)) is V11() V12() ext-real set
K37((((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)),K39((exp_R . x))) is V11() V12() ext-real set
exp_R * (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (exp_R * (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(exp_R * (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((exp_R * (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff ((exp_R * (arctan + arccot)),x) is V11() V12() ext-real Element of REAL
diff (exp_R,((arctan + arccot) . x)) is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(diff (exp_R,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
exp_R . ((arctan + arccot) . x) is V11() V12() ext-real Element of REAL
(exp_R . ((arctan + arccot) . x)) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(exp_R . ((arctan + arccot) . x)) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(exp_R . ((arctan + arccot) . x)) * 0 is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((exp_R * (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
exp_R * (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (exp_R * (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(exp_R * (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((exp_R * (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
exp_R . ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
2 * (exp_R . ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (exp_R . ((arctan . x) - (arccot . x)))),K39((1 + (x ^2)))) is V11() V12() ext-real set
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff ((exp_R * (arctan - arccot)),x) is V11() V12() ext-real Element of REAL
diff (exp_R,((arctan - arccot) . x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(diff (exp_R,((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
exp_R . ((arctan - arccot) . x) is V11() V12() ext-real Element of REAL
(exp_R . ((arctan - arccot) . x)) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(exp_R . ((arctan - arccot) . x)) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(exp_R . ((arctan - arccot) . x)) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(exp_R . ((arctan . x) - (arccot . x))) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((exp_R * (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
exp_R . ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
2 * (exp_R . ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (exp_R . ((arctan . x) - (arccot . x)))),K39((1 + (x ^2)))) is V11() V12() ext-real set
sin * (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sin * (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sin * (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin * (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff ((sin * (arctan + arccot)),x) is V11() V12() ext-real Element of REAL
diff (sin,((arctan + arccot) . x)) is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(diff (sin,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
cos . ((arctan + arccot) . x) is V11() V12() ext-real Element of REAL
(cos . ((arctan + arccot) . x)) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(cos . ((arctan + arccot) . x)) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(cos . ((arctan + arccot) . x)) * 0 is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin * (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
sin * (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (sin * (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(sin * (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin * (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
cos . ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
2 * (cos . ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (cos . ((arctan . x) - (arccot . x)))),K39((1 + (x ^2)))) is V11() V12() ext-real set
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff ((sin * (arctan - arccot)),x) is V11() V12() ext-real Element of REAL
diff (sin,((arctan - arccot) . x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(diff (sin,((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
cos . ((arctan - arccot) . x) is V11() V12() ext-real Element of REAL
(cos . ((arctan - arccot) . x)) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(cos . ((arctan - arccot) . x)) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(cos . ((arctan - arccot) . x)) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(cos . ((arctan . x) - (arccot . x))) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((sin * (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
cos . ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
2 * (cos . ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (cos . ((arctan . x) - (arccot . x)))),K39((1 + (x ^2)))) is V11() V12() ext-real set
cos * (arctan + arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cos * (arctan + arccot)) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cos * (arctan + arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos * (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
(arctan + arccot) . x is V11() V12() ext-real Element of REAL
diff ((cos * (arctan + arccot)),x) is V11() V12() ext-real Element of REAL
diff (cos,((arctan + arccot) . x)) is V11() V12() ext-real Element of REAL
diff ((arctan + arccot),x) is V11() V12() ext-real Element of REAL
(diff (cos,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
sin . ((arctan + arccot) . x) is V11() V12() ext-real Element of REAL
- (sin . ((arctan + arccot) . x)) is V11() V12() ext-real Element of REAL
(- (sin . ((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) is V11() V12() ext-real Element of REAL
(arctan + arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan + arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(- (sin . ((arctan + arccot) . x))) * (((arctan + arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
(- (sin . ((arctan + arccot) . x))) * 0 is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos * (arctan + arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
cos * (arctan - arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cos * (arctan - arccot)) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(cos * (arctan - arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan - arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos * (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
sin . ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
2 * (sin . ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (sin . ((arctan . x) - (arccot . x)))),K39((1 + (x ^2)))) is V11() V12() ext-real set
- ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan - arccot) . x is V11() V12() ext-real Element of REAL
diff ((cos * (arctan - arccot)),x) is V11() V12() ext-real Element of REAL
diff (cos,((arctan - arccot) . x)) is V11() V12() ext-real Element of REAL
diff ((arctan - arccot),x) is V11() V12() ext-real Element of REAL
(diff (cos,((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
sin . ((arctan - arccot) . x) is V11() V12() ext-real Element of REAL
- (sin . ((arctan - arccot) . x)) is V11() V12() ext-real Element of REAL
(- (sin . ((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) is V11() V12() ext-real Element of REAL
(arctan - arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan - arccot) `| Z) . x is V11() V12() ext-real Element of REAL
(- (sin . ((arctan - arccot) . x))) * (((arctan - arccot) `| Z) . x) is V11() V12() ext-real Element of REAL
2 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(2,K39((1 + (x ^2)))) is V11() V12() ext-real set
(- (sin . ((arctan - arccot) . x))) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
- (sin . ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
(- (sin . ((arctan . x) - (arccot . x)))) * (2 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((cos * (arctan - arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arctan . x) - (arccot . x) is V11() V12() ext-real Element of REAL
K38((arccot . x)) is V11() V12() ext-real set
K36((arctan . x),K38((arccot . x))) is V11() V12() ext-real set
sin . ((arctan . x) - (arccot . x)) is V11() V12() ext-real Element of REAL
2 * (sin . ((arctan . x) - (arccot . x))) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((2 * (sin . ((arctan . x) - (arccot . x)))),K39((1 + (x ^2)))) is V11() V12() ext-real set
- ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
arctan (#) arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(arctan (#) arccot) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom arctan) /\ (dom arccot) is V46() V47() V48() Element of K6(REAL)
dom (arctan (#) arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((arctan (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arccot . x) - (arctan . x) is V11() V12() ext-real Element of REAL
K38((arctan . x)) is V11() V12() ext-real set
K36((arccot . x),K38((arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) - (arctan . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arccot . x) - (arctan . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
diff (arctan,x) is V11() V12() ext-real Element of REAL
(arccot . x) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(arctan . x) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
((arccot . x) * (diff (arctan,x))) + ((arctan . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
(arccot . x) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
((arccot . x) * ((arctan `| Z) . x)) + ((arctan . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(arccot . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
(arctan . x) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arctan . x) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan (#) arccot) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arccot . x) - (arctan . x) is V11() V12() ext-real Element of REAL
K38((arctan . x)) is V11() V12() ext-real set
K36((arccot . x),K38((arctan . x))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) - (arctan . x)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arccot . x) - (arctan . x)),K39((1 + (x ^2)))) is V11() V12() ext-real set
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) ^ is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
arctan * ((id Z) ^) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
arccot * ((id Z) ^) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (arctan * ((id Z) ^)) is V46() V47() V48() Element of K6(REAL)
dom (arccot * ((id Z) ^)) is V46() V47() V48() Element of K6(REAL)
(dom (arctan * ((id Z) ^))) /\ (dom (arccot * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
dom ((id Z) ^) is V46() V47() V48() Element of K6(REAL)
x is set
x is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arctan . (1 / x) is V11() V12() ext-real Element of REAL
arccot . (1 / x) is V11() V12() ext-real Element of REAL
(arctan . (1 / x)) - (arccot . (1 / x)) is V11() V12() ext-real Element of REAL
K38((arccot . (1 / x))) is V11() V12() ext-real set
K36((arctan . (1 / x)),K38((arccot . (1 / x)))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arctan . (1 / x)) - (arccot . (1 / x))),K39((1 + (x ^2)))) is V11() V12() ext-real set
(arccot * ((id Z) ^)) . x is V11() V12() ext-real Element of REAL
diff ((arctan * ((id Z) ^)),x) is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) * (diff ((arctan * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(arctan * ((id Z) ^)) . x is V11() V12() ext-real Element of REAL
diff ((arccot * ((id Z) ^)),x) is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (diff ((arctan * ((id Z) ^)),x))) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(arctan * ((id Z) ^)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan * ((id Z) ^)) `| Z) . x is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) * (((arctan * ((id Z) ^)) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (((arctan * ((id Z) ^)) `| Z) . x)) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(arccot * ((id Z) ^)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arccot * ((id Z) ^)) `| Z) . x is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) * (((arccot * ((id Z) ^)) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (((arccot * ((id Z) ^)) `| Z) . x)) is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((id Z) ^) . x is V11() V12() ext-real Element of REAL
arccot . (((id Z) ^) . x) is V11() V12() ext-real Element of REAL
(arccot . (((id Z) ^) . x)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((arccot . (((id Z) ^) . x)) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(id Z) . x is V11() V12() ext-real Element of REAL
((id Z) . x) " is V11() V12() ext-real Element of REAL
arccot . (((id Z) . x) ") is V11() V12() ext-real Element of REAL
(arccot . (((id Z) . x) ")) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((arccot . (((id Z) . x) ")) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(arccot . (1 / x)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
arctan . (((id Z) ^) . x) is V11() V12() ext-real Element of REAL
(arctan . (((id Z) ^) . x)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + ((arctan . (((id Z) ^) . x)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
arctan . (((id Z) . x) ") is V11() V12() ext-real Element of REAL
(arctan . (((id Z) . x) ")) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + ((arctan . (((id Z) . x) ")) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(arccot . (1 / x)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
- ((arccot . (1 / x)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(arctan . (1 / x)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- ((arccot . (1 / x)) * (1 / (1 + (x ^2))))) + ((arctan . (1 / x)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arctan . (1 / x) is V11() V12() ext-real Element of REAL
arccot . (1 / x) is V11() V12() ext-real Element of REAL
(arctan . (1 / x)) - (arccot . (1 / x)) is V11() V12() ext-real Element of REAL
K38((arccot . (1 / x))) is V11() V12() ext-real set
K36((arctan . (1 / x)),K38((arccot . (1 / x)))) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arctan . (1 / x)) - (arccot . (1 / x))),K39((1 + (x ^2)))) is V11() V12() ext-real set
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) ^ is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
arctan * ((id Z) ^) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) (#) (arctan * ((id Z) ^)) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id Z) (#) (arctan * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
((id Z) (#) (arctan * ((id Z) ^))) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (id Z) is V46() V47() V48() Element of K6(REAL)
dom (arctan * ((id Z) ^)) is V46() V47() V48() Element of K6(REAL)
(dom (id Z)) /\ (dom (arctan * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
(id Z) . x is V11() V12() ext-real Element of REAL
1 * x is V11() V12() ext-real Element of REAL
(1 * x) + 0 is V11() V12() ext-real Element of REAL
dom ((id Z) ^) is V46() V47() V48() Element of K6(REAL)
x is set
x is V11() V12() ext-real Element of REAL
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arctan . (1 / x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
x / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(x,K39((1 + (x ^2)))) is V11() V12() ext-real set
(arctan . (1 / x)) - (x / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38((x / (1 + (x ^2)))) is V11() V12() ext-real set
K36((arctan . (1 / x)),K38((x / (1 + (x ^2))))) is V11() V12() ext-real set
(arctan * ((id Z) ^)) . x is V11() V12() ext-real Element of REAL
diff ((id Z),x) is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) * (diff ((id Z),x)) is V11() V12() ext-real Element of REAL
(id Z) . x is V11() V12() ext-real Element of REAL
diff ((arctan * ((id Z) ^)),x) is V11() V12() ext-real Element of REAL
((id Z) . x) * (diff ((arctan * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(id Z) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) `| Z) . x is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) * (((id Z) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) * 1 is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * 1) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
x * (diff ((arctan * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * 1) + (x * (diff ((arctan * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(arctan * ((id Z) ^)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan * ((id Z) ^)) `| Z) . x is V11() V12() ext-real Element of REAL
x * (((arctan * ((id Z) ^)) `| Z) . x) is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) + (x * (((arctan * ((id Z) ^)) `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
x * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) + (x * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
((id Z) ^) . x is V11() V12() ext-real Element of REAL
arctan . (((id Z) ^) . x) is V11() V12() ext-real Element of REAL
(arctan . (((id Z) ^) . x)) - (x / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K36((arctan . (((id Z) ^) . x)),K38((x / (1 + (x ^2))))) is V11() V12() ext-real set
((id Z) . x) " is V11() V12() ext-real Element of REAL
arctan . (((id Z) . x) ") is V11() V12() ext-real Element of REAL
(arctan . (((id Z) . x) ")) - (x / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K36((arctan . (((id Z) . x) ")),K38((x / (1 + (x ^2))))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arctan . (1 / x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
x / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(x,K39((1 + (x ^2)))) is V11() V12() ext-real set
(arctan . (1 / x)) - (x / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38((x / (1 + (x ^2)))) is V11() V12() ext-real set
K36((arctan . (1 / x)),K38((x / (1 + (x ^2))))) is V11() V12() ext-real set
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) ^ is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
arccot * ((id Z) ^) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) (#) (arccot * ((id Z) ^)) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id Z) (#) (arccot * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
((id Z) (#) (arccot * ((id Z) ^))) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (id Z) is V46() V47() V48() Element of K6(REAL)
dom (arccot * ((id Z) ^)) is V46() V47() V48() Element of K6(REAL)
(dom (id Z)) /\ (dom (arccot * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
(id Z) . x is V11() V12() ext-real Element of REAL
1 * x is V11() V12() ext-real Element of REAL
(1 * x) + 0 is V11() V12() ext-real Element of REAL
dom ((id Z) ^) is V46() V47() V48() Element of K6(REAL)
x is set
x is V11() V12() ext-real Element of REAL
(((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arccot . (1 / x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
x / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(x,K39((1 + (x ^2)))) is V11() V12() ext-real set
(arccot . (1 / x)) + (x / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arccot * ((id Z) ^)) . x is V11() V12() ext-real Element of REAL
diff ((id Z),x) is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) * (diff ((id Z),x)) is V11() V12() ext-real Element of REAL
(id Z) . x is V11() V12() ext-real Element of REAL
diff ((arccot * ((id Z) ^)),x) is V11() V12() ext-real Element of REAL
((id Z) . x) * (diff ((arccot * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(id Z) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) `| Z) . x is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) * (((id Z) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) * 1 is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * 1) + (((id Z) . x) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
x * (diff ((arccot * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * 1) + (x * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(arccot * ((id Z) ^)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arccot * ((id Z) ^)) `| Z) . x is V11() V12() ext-real Element of REAL
x * (((arccot * ((id Z) ^)) `| Z) . x) is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) + (x * (((arccot * ((id Z) ^)) `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
x * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) + (x * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((id Z) ^) . x is V11() V12() ext-real Element of REAL
arccot . (((id Z) ^) . x) is V11() V12() ext-real Element of REAL
(arccot . (((id Z) ^) . x)) + (x / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((id Z) . x) " is V11() V12() ext-real Element of REAL
arccot . (((id Z) . x) ") is V11() V12() ext-real Element of REAL
(arccot . (((id Z) . x) ")) + (x / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
(((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arccot . (1 / x) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
x / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(x,K39((1 + (x ^2)))) is V11() V12() ext-real set
(arccot . (1 / x)) + (x / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
#Z 2 is V19() V22( REAL ) V23( REAL ) V24() V33( REAL , REAL ) V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) ^ is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
arctan * ((id Z) ^) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x (#) (arctan * ((id Z) ^)) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (x (#) (arctan * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
(x (#) (arctan * ((id Z) ^))) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
dom x is V46() V47() V48() Element of K6(REAL)
dom (arctan * ((id Z) ^)) is V46() V47() V48() Element of K6(REAL)
(dom x) /\ (dom (arctan * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
x `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(x `| Z) . x is V11() V12() ext-real Element of REAL
2 * x is V11() V12() ext-real Element of REAL
diff (x,x) is V11() V12() ext-real Element of REAL
2 - 1 is V11() V12() ext-real V60() Element of REAL
K38(1) is V1() V11() V12() ext-real non positive negative V60() set
K36(2,K38(1)) is V11() V12() ext-real V60() set
x #Z (2 - 1) is V11() V12() ext-real Element of REAL
2 * (x #Z (2 - 1)) is V11() V12() ext-real Element of REAL
dom ((id Z) ^) is V46() V47() V48() Element of K6(REAL)
x is set
x is V11() V12() ext-real Element of REAL
((x (#) (arctan * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
2 * x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arctan . (1 / x) is V11() V12() ext-real Element of REAL
(2 * x) * (arctan . (1 / x)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(x ^2) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((x ^2),K39((1 + (x ^2)))) is V11() V12() ext-real set
((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((x ^2) / (1 + (x ^2)))) is V11() V12() ext-real set
K36(((2 * x) * (arctan . (1 / x))),K38(((x ^2) / (1 + (x ^2))))) is V11() V12() ext-real set
(arctan * ((id Z) ^)) . x is V11() V12() ext-real Element of REAL
diff (x,x) is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) * (diff (x,x)) is V11() V12() ext-real Element of REAL
x . x is V11() V12() ext-real Element of REAL
diff ((arctan * ((id Z) ^)),x) is V11() V12() ext-real Element of REAL
(x . x) * (diff ((arctan * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (diff (x,x))) + ((x . x) * (diff ((arctan * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(x `| Z) . x is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) * ((x `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * ((x `| Z) . x)) + ((x . x) * (diff ((arctan * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
((arctan * ((id Z) ^)) . x) * (2 * x) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x . x) * (diff ((arctan * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
x #Z 2 is V11() V12() ext-real Element of REAL
(x #Z 2) * (diff ((arctan * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (diff ((arctan * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(arctan * ((id Z) ^)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan * ((id Z) ^)) `| Z) . x is V11() V12() ext-real Element of REAL
(x #Z 2) * (((arctan * ((id Z) ^)) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (((arctan * ((id Z) ^)) `| Z) . x)) is V11() V12() ext-real Element of REAL
1 + 1 is V1() V11() V12() ext-real positive non negative V60() Element of REAL
x #Z (1 + 1) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(x #Z (1 + 1)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x #Z (1 + 1)) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x #Z 1 is V11() V12() ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V11() V12() ext-real Element of REAL
((x #Z 1) * (x #Z 1)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (2 * x)) + (((x #Z 1) * (x #Z 1)) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x * (x #Z 1) is V11() V12() ext-real Element of REAL
(x * (x #Z 1)) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x * (x #Z 1)) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
(x ^2) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x ^2) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
((id Z) ^) . x is V11() V12() ext-real Element of REAL
arctan . (((id Z) ^) . x) is V11() V12() ext-real Element of REAL
(arctan . (((id Z) ^) . x)) * (2 * x) is V11() V12() ext-real Element of REAL
((arctan . (((id Z) ^) . x)) * (2 * x)) - ((x ^2) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K36(((arctan . (((id Z) ^) . x)) * (2 * x)),K38(((x ^2) / (1 + (x ^2))))) is V11() V12() ext-real set
(id Z) . x is V11() V12() ext-real Element of REAL
((id Z) . x) " is V11() V12() ext-real Element of REAL
arctan . (((id Z) . x) ") is V11() V12() ext-real Element of REAL
(arctan . (((id Z) . x) ")) * (2 * x) is V11() V12() ext-real Element of REAL
((arctan . (((id Z) . x) ")) * (2 * x)) - ((x ^2) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K36(((arctan . (((id Z) . x) ")) * (2 * x)),K38(((x ^2) / (1 + (x ^2))))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((x (#) (arctan * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
2 * x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arctan . (1 / x) is V11() V12() ext-real Element of REAL
(2 * x) * (arctan . (1 / x)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(x ^2) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((x ^2),K39((1 + (x ^2)))) is V11() V12() ext-real set
((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K38(((x ^2) / (1 + (x ^2)))) is V11() V12() ext-real set
K36(((2 * x) * (arctan . (1 / x))),K38(((x ^2) / (1 + (x ^2))))) is V11() V12() ext-real set
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) ^ is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
arccot * ((id Z) ^) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x (#) (arccot * ((id Z) ^)) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (x (#) (arccot * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
(x (#) (arccot * ((id Z) ^))) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
dom x is V46() V47() V48() Element of K6(REAL)
dom (arccot * ((id Z) ^)) is V46() V47() V48() Element of K6(REAL)
(dom x) /\ (dom (arccot * ((id Z) ^))) is V46() V47() V48() Element of K6(REAL)
x `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
(x `| Z) . x is V11() V12() ext-real Element of REAL
2 * x is V11() V12() ext-real Element of REAL
diff (x,x) is V11() V12() ext-real Element of REAL
2 - 1 is V11() V12() ext-real V60() Element of REAL
K38(1) is V1() V11() V12() ext-real non positive negative V60() set
K36(2,K38(1)) is V11() V12() ext-real V60() set
x #Z (2 - 1) is V11() V12() ext-real Element of REAL
2 * (x #Z (2 - 1)) is V11() V12() ext-real Element of REAL
dom ((id Z) ^) is V46() V47() V48() Element of K6(REAL)
x is set
x is V11() V12() ext-real Element of REAL
((x (#) (arccot * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
2 * x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arccot . (1 / x) is V11() V12() ext-real Element of REAL
(2 * x) * (arccot . (1 / x)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(x ^2) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((x ^2),K39((1 + (x ^2)))) is V11() V12() ext-real set
((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(arccot * ((id Z) ^)) . x is V11() V12() ext-real Element of REAL
diff (x,x) is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) * (diff (x,x)) is V11() V12() ext-real Element of REAL
x . x is V11() V12() ext-real Element of REAL
diff ((arccot * ((id Z) ^)),x) is V11() V12() ext-real Element of REAL
(x . x) * (diff ((arccot * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (diff (x,x))) + ((x . x) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(x `| Z) . x is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) * ((x `| Z) . x) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * ((x `| Z) . x)) + ((x . x) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
((arccot * ((id Z) ^)) . x) * (2 * x) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x . x) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
x #Z 2 is V11() V12() ext-real Element of REAL
(x #Z 2) * (diff ((arccot * ((id Z) ^)),x)) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (diff ((arccot * ((id Z) ^)),x))) is V11() V12() ext-real Element of REAL
(arccot * ((id Z) ^)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arccot * ((id Z) ^)) `| Z) . x is V11() V12() ext-real Element of REAL
(x #Z 2) * (((arccot * ((id Z) ^)) `| Z) . x) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (((arccot * ((id Z) ^)) `| Z) . x)) is V11() V12() ext-real Element of REAL
1 + 1 is V1() V11() V12() ext-real positive non negative V60() Element of REAL
x #Z (1 + 1) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(x #Z (1 + 1)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x #Z (1 + 1)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x #Z 1 is V11() V12() ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V11() V12() ext-real Element of REAL
((x #Z 1) * (x #Z 1)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (2 * x)) + (((x #Z 1) * (x #Z 1)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x * (x #Z 1) is V11() V12() ext-real Element of REAL
(x * (x #Z 1)) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x * (x #Z 1)) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x ^2) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((id Z) ^) . x is V11() V12() ext-real Element of REAL
arccot . (((id Z) ^) . x) is V11() V12() ext-real Element of REAL
(arccot . (((id Z) ^) . x)) * (2 * x) is V11() V12() ext-real Element of REAL
((arccot . (((id Z) ^) . x)) * (2 * x)) + ((x ^2) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(id Z) . x is V11() V12() ext-real Element of REAL
((id Z) . x) " is V11() V12() ext-real Element of REAL
arccot . (((id Z) . x) ") is V11() V12() ext-real Element of REAL
(arccot . (((id Z) . x) ")) * (2 * x) is V11() V12() ext-real Element of REAL
((arccot . (((id Z) . x) ")) * (2 * x)) + ((x ^2) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((x (#) (arccot * ((id Z) ^))) `| Z) . x is V11() V12() ext-real Element of REAL
2 * x is V11() V12() ext-real Element of REAL
1 / x is V11() V12() ext-real Element of REAL
K39(x) is V11() V12() ext-real set
K37(1,K39(x)) is V11() V12() ext-real set
arccot . (1 / x) is V11() V12() ext-real Element of REAL
(2 * x) * (arccot . (1 / x)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
(x ^2) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37((x ^2),K39((1 + (x ^2)))) is V11() V12() ext-real set
((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
arctan ^ is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(arctan ^) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
((arctan ^) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) ^2 is V11() V12() ext-real Element of REAL
K37((arctan . x),(arctan . x)) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . x) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arctan . x) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arctan . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arctan . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((arctan . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
diff ((arctan ^),x) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
(diff (arctan,x)) / ((arctan . x) ^2) is V11() V12() ext-real Element of REAL
K39(((arctan . x) ^2)) is V11() V12() ext-real set
K37((diff (arctan,x)),K39(((arctan . x) ^2))) is V11() V12() ext-real set
- ((diff (arctan,x)) / ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
((arctan `| Z) . x) / ((arctan . x) ^2) is V11() V12() ext-real Element of REAL
K37(((arctan `| Z) . x),K39(((arctan . x) ^2))) is V11() V12() ext-real set
- (((arctan `| Z) . x) / ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
(1 / (1 + (x ^2))) / ((arctan . x) ^2) is V11() V12() ext-real Element of REAL
K37((1 / (1 + (x ^2))),K39(((arctan . x) ^2))) is V11() V12() ext-real set
- ((1 / (1 + (x ^2))) / ((arctan . x) ^2)) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((arctan ^) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) ^2 is V11() V12() ext-real Element of REAL
K37((arctan . x),(arctan . x)) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . x) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arctan . x) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arctan . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arctan . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((arctan . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
arccot ^ is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(arccot ^) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
0 / 4 is V11() V12() ext-real Element of REAL
K37(0,K39(4)) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
arccot .: [.(- 1),1.] is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((arccot ^) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) ^2 is V11() V12() ext-real Element of REAL
K37((arccot . x),(arccot . x)) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arccot . x) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arccot . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arccot . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
diff ((arccot ^),x) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
(diff (arccot,x)) / ((arccot . x) ^2) is V11() V12() ext-real Element of REAL
K39(((arccot . x) ^2)) is V11() V12() ext-real set
K37((diff (arccot,x)),K39(((arccot . x) ^2))) is V11() V12() ext-real set
- ((diff (arccot,x)) / ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
((arccot `| Z) . x) / ((arccot . x) ^2) is V11() V12() ext-real Element of REAL
K37(((arccot `| Z) . x),K39(((arccot . x) ^2))) is V11() V12() ext-real set
- (((arccot `| Z) . x) / ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(- (1 / (1 + (x ^2)))) / ((arccot . x) ^2) is V11() V12() ext-real Element of REAL
K37((- (1 / (1 + (x ^2)))),K39(((arccot . x) ^2))) is V11() V12() ext-real set
- ((- (1 / (1 + (x ^2)))) / ((arccot . x) ^2)) is V11() V12() ext-real Element of REAL
(1 / (1 + (x ^2))) / ((arccot . x) ^2) is V11() V12() ext-real Element of REAL
K37((1 / (1 + (x ^2))),K39(((arccot . x) ^2))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
((arccot ^) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) ^2 is V11() V12() ext-real Element of REAL
K37((arccot . x),(arccot . x)) is V11() V12() ext-real set
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arccot . x) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arccot . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arccot . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
Z is V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() Element of NAT
#Z Z is V19() V22( REAL ) V23( REAL ) V24() V33( REAL , REAL ) V36() V37() V38() Element of K6(K7(REAL,REAL))
(#Z Z) * (arctan ^) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
1 / Z is V11() V12() ext-real Element of REAL
K39(Z) is V11() V12() ext-real set
K37(1,K39(Z)) is V11() V12() ext-real set
(1 / Z) (#) ((#Z Z) * (arctan ^)) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((1 / Z) (#) ((#Z Z) * (arctan ^))) is V46() V47() V48() Element of K6(REAL)
Z + 1 is V11() V12() ext-real V60() Element of REAL
x is open V46() V47() V48() Element of K6(REAL)
((1 / Z) (#) ((#Z Z) * (arctan ^))) `| x is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((#Z Z) * (arctan ^)) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
dom (arctan ^) is V46() V47() V48() Element of K6(REAL)
x is set
x is V11() V12() ext-real Element of REAL
(((1 / Z) (#) ((#Z Z) * (arctan ^))) `| x) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) #Z (Z + 1) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . x) #Z (Z + 1)) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arctan . x) #Z (Z + 1)) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arctan . x) #Z (Z + 1)) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arctan . x) #Z (Z + 1)) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((arctan . x) #Z (Z + 1)) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(arctan ^) . x is V11() V12() ext-real Element of REAL
1 / (arctan . x) is V11() V12() ext-real Element of REAL
K39((arctan . x)) is V11() V12() ext-real set
K37(1,K39((arctan . x))) is V11() V12() ext-real set
diff (((#Z Z) * (arctan ^)),x) is V11() V12() ext-real Element of REAL
(1 / Z) * (diff (((#Z Z) * (arctan ^)),x)) is V11() V12() ext-real Element of REAL
Z - 1 is V11() V12() ext-real V60() Element of REAL
K38(1) is V1() V11() V12() ext-real non positive negative V60() set
K36(Z,K38(1)) is V11() V12() ext-real V60() set
((arctan ^) . x) #Z (Z - 1) is V11() V12() ext-real Element of REAL
Z * (((arctan ^) . x) #Z (Z - 1)) is V11() V12() ext-real Element of REAL
diff ((arctan ^),x) is V11() V12() ext-real Element of REAL
(Z * (((arctan ^) . x) #Z (Z - 1))) * (diff ((arctan ^),x)) is V11() V12() ext-real Element of REAL
(1 / Z) * ((Z * (((arctan ^) . x) #Z (Z - 1))) * (diff ((arctan ^),x))) is V11() V12() ext-real Element of REAL
(arctan ^) `| x is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arctan ^) `| x) . x is V11() V12() ext-real Element of REAL
(Z * (((arctan ^) . x) #Z (Z - 1))) * (((arctan ^) `| x) . x) is V11() V12() ext-real Element of REAL
(1 / Z) * ((Z * (((arctan ^) . x) #Z (Z - 1))) * (((arctan ^) `| x) . x)) is V11() V12() ext-real Element of REAL
(arctan . x) ^2 is V11() V12() ext-real Element of REAL
K37((arctan . x),(arctan . x)) is V11() V12() ext-real set
((arctan . x) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arctan . x) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arctan . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arctan . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((arctan . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(Z * (((arctan ^) . x) #Z (Z - 1))) * (- (1 / (((arctan . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
(1 / Z) * ((Z * (((arctan ^) . x) #Z (Z - 1))) * (- (1 / (((arctan . x) ^2) * (1 + (x ^2)))))) is V11() V12() ext-real Element of REAL
(1 / Z) * Z is V11() V12() ext-real Element of REAL
((1 / Z) * Z) * (((arctan ^) . x) #Z (Z - 1)) is V11() V12() ext-real Element of REAL
(((1 / Z) * Z) * (((arctan ^) . x) #Z (Z - 1))) * (1 / (((arctan . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
- ((((1 / Z) * Z) * (((arctan ^) . x) #Z (Z - 1))) * (1 / (((arctan . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
1 * (((arctan ^) . x) #Z (Z - 1)) is V11() V12() ext-real Element of REAL
(1 * (((arctan ^) . x) #Z (Z - 1))) * (1 / (((arctan . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
- ((1 * (((arctan ^) . x) #Z (Z - 1))) * (1 / (((arctan . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
(1 / (arctan . x)) #Z (Z - 1) is V11() V12() ext-real Element of REAL
(arctan . x) #Z 2 is V11() V12() ext-real Element of REAL
((arctan . x) #Z 2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arctan . x) #Z 2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arctan . x) #Z 2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arctan . x) #Z 2) * (1 + (x ^2))))) is V11() V12() ext-real set
((1 / (arctan . x)) #Z (Z - 1)) * (1 / (((arctan . x) #Z 2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
- (((1 / (arctan . x)) #Z (Z - 1)) * (1 / (((arctan . x) #Z 2) * (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
(arctan . x) #Z (Z - 1) is V11() V12() ext-real Element of REAL
1 / ((arctan . x) #Z (Z - 1)) is V11() V12() ext-real Element of REAL
K39(((arctan . x) #Z (Z - 1))) is V11() V12() ext-real set
K37(1,K39(((arctan . x) #Z (Z - 1)))) is V11() V12() ext-real set
(1 / ((arctan . x) #Z (Z - 1))) / (((arctan . x) #Z 2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K37((1 / ((arctan . x) #Z (Z - 1))),K39((((arctan . x) #Z 2) * (1 + (x ^2))))) is V11() V12() ext-real set
- ((1 / ((arctan . x) #Z (Z - 1))) / (((arctan . x) #Z 2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
((arctan . x) #Z (Z - 1)) * (((arctan . x) #Z 2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
1 / (((arctan . x) #Z (Z - 1)) * (((arctan . x) #Z 2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K39((((arctan . x) #Z (Z - 1)) * (((arctan . x) #Z 2) * (1 + (x ^2))))) is V11() V12() ext-real set
K37(1,K39((((arctan . x) #Z (Z - 1)) * (((arctan . x) #Z 2) * (1 + (x ^2)))))) is V11() V12() ext-real set
- (1 / (((arctan . x) #Z (Z - 1)) * (((arctan . x) #Z 2) * (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
((arctan . x) #Z (Z - 1)) * ((arctan . x) #Z 2) is V11() V12() ext-real Element of REAL
(((arctan . x) #Z (Z - 1)) * ((arctan . x) #Z 2)) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((((arctan . x) #Z (Z - 1)) * ((arctan . x) #Z 2)) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((((arctan . x) #Z (Z - 1)) * ((arctan . x) #Z 2)) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((((arctan . x) #Z (Z - 1)) * ((arctan . x) #Z 2)) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / ((((arctan . x) #Z (Z - 1)) * ((arctan . x) #Z 2)) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(Z - 1) + 2 is V11() V12() ext-real V60() Element of REAL
(arctan . x) #Z ((Z - 1) + 2) is V11() V12() ext-real Element of REAL
((arctan . x) #Z ((Z - 1) + 2)) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arctan . x) #Z ((Z - 1) + 2)) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arctan . x) #Z ((Z - 1) + 2)) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arctan . x) #Z ((Z - 1) + 2)) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((arctan . x) #Z ((Z - 1) + 2)) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
(((1 / Z) (#) ((#Z Z) * (arctan ^))) `| x) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) #Z (Z + 1) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . x) #Z (Z + 1)) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arctan . x) #Z (Z + 1)) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arctan . x) #Z (Z + 1)) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arctan . x) #Z (Z + 1)) * (1 + (x ^2))))) is V11() V12() ext-real set
- (1 / (((arctan . x) #Z (Z + 1)) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
Z is V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() Element of NAT
#Z Z is V19() V22( REAL ) V23( REAL ) V24() V33( REAL , REAL ) V36() V37() V38() Element of K6(K7(REAL,REAL))
(#Z Z) * (arccot ^) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
1 / Z is V11() V12() ext-real Element of REAL
K39(Z) is V11() V12() ext-real set
K37(1,K39(Z)) is V11() V12() ext-real set
(1 / Z) (#) ((#Z Z) * (arccot ^)) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((1 / Z) (#) ((#Z Z) * (arccot ^))) is V46() V47() V48() Element of K6(REAL)
Z + 1 is V11() V12() ext-real V60() Element of REAL
x is open V46() V47() V48() Element of K6(REAL)
((1 / Z) (#) ((#Z Z) * (arccot ^))) `| x is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((#Z Z) * (arccot ^)) is V46() V47() V48() Element of K6(REAL)
0 / 4 is V11() V12() ext-real Element of REAL
K37(0,K39(4)) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
arccot .: [.(- 1),1.] is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
dom (arccot ^) is V46() V47() V48() Element of K6(REAL)
x is set
x is V11() V12() ext-real Element of REAL
(((1 / Z) (#) ((#Z Z) * (arccot ^))) `| x) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) #Z (Z + 1) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) #Z (Z + 1)) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arccot . x) #Z (Z + 1)) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arccot . x) #Z (Z + 1)) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arccot . x) #Z (Z + 1)) * (1 + (x ^2))))) is V11() V12() ext-real set
(arccot ^) . x is V11() V12() ext-real Element of REAL
1 / (arccot . x) is V11() V12() ext-real Element of REAL
K39((arccot . x)) is V11() V12() ext-real set
K37(1,K39((arccot . x))) is V11() V12() ext-real set
diff (((#Z Z) * (arccot ^)),x) is V11() V12() ext-real Element of REAL
(1 / Z) * (diff (((#Z Z) * (arccot ^)),x)) is V11() V12() ext-real Element of REAL
Z - 1 is V11() V12() ext-real V60() Element of REAL
K38(1) is V1() V11() V12() ext-real non positive negative V60() set
K36(Z,K38(1)) is V11() V12() ext-real V60() set
((arccot ^) . x) #Z (Z - 1) is V11() V12() ext-real Element of REAL
Z * (((arccot ^) . x) #Z (Z - 1)) is V11() V12() ext-real Element of REAL
diff ((arccot ^),x) is V11() V12() ext-real Element of REAL
(Z * (((arccot ^) . x) #Z (Z - 1))) * (diff ((arccot ^),x)) is V11() V12() ext-real Element of REAL
(1 / Z) * ((Z * (((arccot ^) . x) #Z (Z - 1))) * (diff ((arccot ^),x))) is V11() V12() ext-real Element of REAL
(arccot ^) `| x is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
((arccot ^) `| x) . x is V11() V12() ext-real Element of REAL
(Z * (((arccot ^) . x) #Z (Z - 1))) * (((arccot ^) `| x) . x) is V11() V12() ext-real Element of REAL
(1 / Z) * ((Z * (((arccot ^) . x) #Z (Z - 1))) * (((arccot ^) `| x) . x)) is V11() V12() ext-real Element of REAL
(arccot . x) ^2 is V11() V12() ext-real Element of REAL
K37((arccot . x),(arccot . x)) is V11() V12() ext-real set
((arccot . x) ^2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arccot . x) ^2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arccot . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arccot . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real set
(Z * (((arccot ^) . x) #Z (Z - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(1 / Z) * ((Z * (((arccot ^) . x) #Z (Z - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
(1 / Z) * Z is V11() V12() ext-real Element of REAL
((1 / Z) * Z) * (((arccot ^) . x) #Z (Z - 1)) is V11() V12() ext-real Element of REAL
(((1 / Z) * Z) * (((arccot ^) . x) #Z (Z - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
1 * (((arccot ^) . x) #Z (Z - 1)) is V11() V12() ext-real Element of REAL
(1 * (((arccot ^) . x) #Z (Z - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(1 / (arccot . x)) #Z (Z - 1) is V11() V12() ext-real Element of REAL
(arccot . x) #Z 2 is V11() V12() ext-real Element of REAL
((arccot . x) #Z 2) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arccot . x) #Z 2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arccot . x) #Z 2) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arccot . x) #Z 2) * (1 + (x ^2))))) is V11() V12() ext-real set
((1 / (arccot . x)) #Z (Z - 1)) * (1 / (((arccot . x) #Z 2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(arccot . x) #Z (Z - 1) is V11() V12() ext-real Element of REAL
1 / ((arccot . x) #Z (Z - 1)) is V11() V12() ext-real Element of REAL
K39(((arccot . x) #Z (Z - 1))) is V11() V12() ext-real set
K37(1,K39(((arccot . x) #Z (Z - 1)))) is V11() V12() ext-real set
(1 / ((arccot . x) #Z (Z - 1))) / (((arccot . x) #Z 2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K37((1 / ((arccot . x) #Z (Z - 1))),K39((((arccot . x) #Z 2) * (1 + (x ^2))))) is V11() V12() ext-real set
((arccot . x) #Z (Z - 1)) * (((arccot . x) #Z 2) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
1 / (((arccot . x) #Z (Z - 1)) * (((arccot . x) #Z 2) * (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
K39((((arccot . x) #Z (Z - 1)) * (((arccot . x) #Z 2) * (1 + (x ^2))))) is V11() V12() ext-real set
K37(1,K39((((arccot . x) #Z (Z - 1)) * (((arccot . x) #Z 2) * (1 + (x ^2)))))) is V11() V12() ext-real set
((arccot . x) #Z (Z - 1)) * ((arccot . x) #Z 2) is V11() V12() ext-real Element of REAL
(((arccot . x) #Z (Z - 1)) * ((arccot . x) #Z 2)) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / ((((arccot . x) #Z (Z - 1)) * ((arccot . x) #Z 2)) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39(((((arccot . x) #Z (Z - 1)) * ((arccot . x) #Z 2)) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39(((((arccot . x) #Z (Z - 1)) * ((arccot . x) #Z 2)) * (1 + (x ^2))))) is V11() V12() ext-real set
(Z - 1) + 2 is V11() V12() ext-real V60() Element of REAL
(arccot . x) #Z ((Z - 1) + 2) is V11() V12() ext-real Element of REAL
((arccot . x) #Z ((Z - 1) + 2)) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arccot . x) #Z ((Z - 1) + 2)) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arccot . x) #Z ((Z - 1) + 2)) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arccot . x) #Z ((Z - 1) + 2)) * (1 + (x ^2))))) is V11() V12() ext-real set
x is V11() V12() ext-real Element of REAL
(((1 / Z) (#) ((#Z Z) * (arccot ^))) `| x) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) #Z (Z + 1) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) #Z (Z + 1)) * (1 + (x ^2)) is V11() V12() ext-real Element of REAL
1 / (((arccot . x) #Z (Z + 1)) * (1 + (x ^2))) is V11() V12() ext-real Element of REAL
K39((((arccot . x) #Z (Z + 1)) * (1 + (x ^2)))) is V11() V12() ext-real set
K37(1,K39((((arccot . x) #Z (Z + 1)) * (1 + (x ^2))))) is V11() V12() ext-real set
1 / 2 is V1() V11() V12() ext-real positive non negative Element of REAL
K39(2) is V1() V11() V12() ext-real positive non negative set
K37(1,K39(2)) is V1() V11() V12() ext-real positive non negative set
#R (1 / 2) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(#R (1 / 2)) * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
2 (#) ((#R (1 / 2)) * arctan) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (2 (#) ((#R (1 / 2)) * arctan)) is V46() V47() V48() Element of K6(REAL)
- (1 / 2) is V1() V11() V12() ext-real non positive negative Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
(2 (#) ((#R (1 / 2)) * arctan)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
dom ((#R (1 / 2)) * arctan) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) #R (- (1 / 2)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arctan . x) #R (- (1 / 2))),K39((1 + (x ^2)))) is V11() V12() ext-real set
diff (((#R (1 / 2)) * arctan),x) is V11() V12() ext-real Element of REAL
2 * (diff (((#R (1 / 2)) * arctan),x)) is V11() V12() ext-real Element of REAL
(1 / 2) - 1 is V11() V12() ext-real Element of REAL
K38(1) is V1() V11() V12() ext-real non positive negative V60() set
K36((1 / 2),K38(1)) is V11() V12() ext-real set
(arctan . x) #R ((1 / 2) - 1) is V11() V12() ext-real Element of REAL
(1 / 2) * ((arctan . x) #R ((1 / 2) - 1)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) #R (- (1 / 2)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arctan . x) #R (- (1 / 2))),K39((1 + (x ^2)))) is V11() V12() ext-real set
(#R (1 / 2)) * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
2 (#) ((#R (1 / 2)) * arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (2 (#) ((#R (1 / 2)) * arccot)) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(2 (#) ((#R (1 / 2)) * arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
arccot .: [.(- 1),1.] is V46() V47() V48() Element of K6(REAL)
(arccot . x) + 0 is V11() V12() ext-real Element of REAL
- (PI / 4) is V1() V11() V12() ext-real non positive negative Element of REAL
(PI / 4) + (- (PI / 4)) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
dom ((#R (1 / 2)) * arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) #R (- (1 / 2)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) #R (- (1 / 2))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arccot . x) #R (- (1 / 2))),K39((1 + (x ^2)))) is V11() V12() ext-real set
- (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
diff (((#R (1 / 2)) * arccot),x) is V11() V12() ext-real Element of REAL
2 * (diff (((#R (1 / 2)) * arccot),x)) is V11() V12() ext-real Element of REAL
(1 / 2) - 1 is V11() V12() ext-real Element of REAL
K38(1) is V1() V11() V12() ext-real non positive negative V60() set
K36((1 / 2),K38(1)) is V11() V12() ext-real set
(arccot . x) #R ((1 / 2) - 1) is V11() V12() ext-real Element of REAL
(1 / 2) * ((arccot . x) #R ((1 / 2) - 1)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
2 * (((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
2 * (((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
2 * (((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) #R (- (1 / 2)) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) #R (- (1 / 2))) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arccot . x) #R (- (1 / 2))),K39((1 + (x ^2)))) is V11() V12() ext-real set
- (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
3 / 2 is V1() V11() V12() ext-real positive non negative Element of REAL
K37(3,K39(2)) is V1() V11() V12() ext-real positive non negative set
#R (3 / 2) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(#R (3 / 2)) * arctan is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
2 / 3 is V1() V11() V12() ext-real positive non negative Element of REAL
K39(3) is V1() V11() V12() ext-real positive non negative set
K37(2,K39(3)) is V1() V11() V12() ext-real positive non negative set
(2 / 3) (#) ((#R (3 / 2)) * arctan) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((2 / 3) (#) ((#R (3 / 2)) * arctan)) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
dom ((#R (3 / 2)) * arctan) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
(((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) #R (1 / 2) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . x) #R (1 / 2)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arctan . x) #R (1 / 2)),K39((1 + (x ^2)))) is V11() V12() ext-real set
diff (((#R (3 / 2)) * arctan),x) is V11() V12() ext-real Element of REAL
(2 / 3) * (diff (((#R (3 / 2)) * arctan),x)) is V11() V12() ext-real Element of REAL
(3 / 2) - 1 is V11() V12() ext-real Element of REAL
K38(1) is V1() V11() V12() ext-real non positive negative V60() set
K36((3 / 2),K38(1)) is V11() V12() ext-real set
(arctan . x) #R ((3 / 2) - 1) is V11() V12() ext-real Element of REAL
(3 / 2) * ((arctan . x) #R ((3 / 2) - 1)) is V11() V12() ext-real Element of REAL
diff (arctan,x) is V11() V12() ext-real Element of REAL
((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * (diff (arctan,x)) is V11() V12() ext-real Element of REAL
(2 / 3) * (((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * (diff (arctan,x))) is V11() V12() ext-real Element of REAL
arctan `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arctan `| Z) . x is V11() V12() ext-real Element of REAL
((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * ((arctan `| Z) . x) is V11() V12() ext-real Element of REAL
(2 / 3) * (((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * ((arctan `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
(2 / 3) * (((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
(((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x is V11() V12() ext-real Element of REAL
arctan . x is V11() V12() ext-real Element of REAL
(arctan . x) #R (1 / 2) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arctan . x) #R (1 / 2)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arctan . x) #R (1 / 2)),K39((1 + (x ^2)))) is V11() V12() ext-real set
(#R (3 / 2)) * arccot is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(2 / 3) (#) ((#R (3 / 2)) * arccot) is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((2 / 3) (#) ((#R (3 / 2)) * arccot)) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
arccot .: [.(- 1),1.] is V46() V47() V48() Element of K6(REAL)
(arccot . x) + 0 is V11() V12() ext-real Element of REAL
- (PI / 4) is V1() V11() V12() ext-real non positive negative Element of REAL
(PI / 4) + (- (PI / 4)) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
dom ((#R (3 / 2)) * arccot) is V46() V47() V48() Element of K6(REAL)
x is V11() V12() ext-real Element of REAL
(((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) #R (1 / 2) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) #R (1 / 2)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arccot . x) #R (1 / 2)),K39((1 + (x ^2)))) is V11() V12() ext-real set
- (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
diff (((#R (3 / 2)) * arccot),x) is V11() V12() ext-real Element of REAL
(2 / 3) * (diff (((#R (3 / 2)) * arccot),x)) is V11() V12() ext-real Element of REAL
(3 / 2) - 1 is V11() V12() ext-real Element of REAL
K38(1) is V1() V11() V12() ext-real non positive negative V60() set
K36((3 / 2),K38(1)) is V11() V12() ext-real set
(arccot . x) #R ((3 / 2) - 1) is V11() V12() ext-real Element of REAL
(3 / 2) * ((arccot . x) #R ((3 / 2) - 1)) is V11() V12() ext-real Element of REAL
diff (arccot,x) is V11() V12() ext-real Element of REAL
((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * (diff (arccot,x)) is V11() V12() ext-real Element of REAL
(2 / 3) * (((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * (diff (arccot,x))) is V11() V12() ext-real Element of REAL
arccot `| Z is V19() V22( REAL ) V23( REAL ) V24() V36() V37() V38() Element of K6(K7(REAL,REAL))
(arccot `| Z) . x is V11() V12() ext-real Element of REAL
((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * ((arccot `| Z) . x) is V11() V12() ext-real Element of REAL
(2 / 3) * (((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * ((arccot `| Z) . x)) is V11() V12() ext-real Element of REAL
1 / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K37(1,K39((1 + (x ^2)))) is V11() V12() ext-real set
- (1 / (1 + (x ^2))) is V11() V12() ext-real Element of REAL
((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * (- (1 / (1 + (x ^2)))) is V11() V12() ext-real Element of REAL
(2 / 3) * (((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * (- (1 / (1 + (x ^2))))) is V11() V12() ext-real Element of REAL
x is V11() V12() ext-real Element of REAL
(((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x is V11() V12() ext-real Element of REAL
arccot . x is V11() V12() ext-real Element of REAL
(arccot . x) #R (1 / 2) is V11() V12() ext-real Element of REAL
x ^2 is V11() V12() ext-real Element of REAL
K37(x,x) is V11() V12() ext-real set
1 + (x ^2) is V11() V12() ext-real Element of REAL
((arccot . x) #R (1 / 2)) / (1 + (x ^2)) is V11() V12() ext-real Element of REAL
K39((1 + (x ^2))) is V11() V12() ext-real set
K37(((arccot . x) #R (1 / 2)),K39((1 + (x ^2)))) is V11() V12() ext-real set
- (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) is V11() V12() ext-real Element of REAL