let f2 be PartFunc of REAL ,REAL ; :: thesis: for A being closed-interval Subset of REAL st A c= ].(- 1),1.[ & dom (arcsin `| ].(- 1),1.[) = dom f2 & ( for x being Real holds
( x in ].(- 1),1.[ & f2 . x = 1 / (sqrt (1 - (x ^2 ))) ) ) & f2 | A is continuous holds
integral f2,A = (arcsin . (upper_bound A)) - (arcsin . (lower_bound A))
let A be closed-interval Subset of REAL ; :: thesis: ( A c= ].(- 1),1.[ & dom (arcsin `| ].(- 1),1.[) = dom f2 & ( for x being Real holds
( x in ].(- 1),1.[ & f2 . x = 1 / (sqrt (1 - (x ^2 ))) ) ) & f2 | A is continuous implies integral f2,A = (arcsin . (upper_bound A)) - (arcsin . (lower_bound A)) )
assume that
A1: A c= ].(- 1),1.[ and
A2: dom (arcsin `| ].(- 1),1.[) = dom f2 and
A3: for x being Real holds
( x in ].(- 1),1.[ & f2 . x = 1 / (sqrt (1 - (x ^2 ))) ) and
A4: f2 | A is continuous ; :: thesis: integral f2,A = (arcsin . (upper_bound A)) - (arcsin . (lower_bound A))
for x being Real st x in dom (arcsin `| ].(- 1),1.[) holds
(arcsin `| ].(- 1),1.[) . x = f2 . x

proof

let x be Real; :: thesis: ( x in dom (arcsin `| ].(- 1),1.[) implies (arcsin `| ].(- 1),1.[) . x = f2 . x )
assume A5: x in dom (arcsin `| ].(- 1),1.[) ; :: thesis: (arcsin `| ].(- 1),1.[) . x = f2 . x
then A6: ( - 1 < x & x < 1 ) by Lm18, XXREAL_1:4;
(arcsin `| ].(- 1),1.[) . x = diff arcsin ,x by A5, Lm18, FDIFF_1:def 8, SIN_COS6:84
.= 1 / (sqrt (1 - (x ^2 ))) by A6, SIN_COS6:84
.= f2 . x by A3 ;
hence (arcsin `| ].(- 1),1.[) . x = f2 . x ; :: thesis: verum

end;

then A7: arcsin `| ].(- 1),1.[ = f2 by A2, PARTFUN1:34;
( A c= dom f2 & f2 is_integrable_on A ) by A1, A2, A4, Lm18, FDIFF_1:def 8, INTEGRA5:11, SIN_COS6:84;
hence integral f2,A = (arcsin . (upper_bound A)) - (arcsin . (lower_bound A)) by A1, A4, A7, INTEGRA5:10, INTEGRA5:13, SIN_COS6:84; :: thesis: verum