let f2 be PartFunc of REAL ,REAL ; :: thesis: for A being closed-interval Subset of REAL st A c= ].(- 1),1.[ & dom (arccos `| ].(- 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 = (arccos . (upper_bound A)) - (arccos . (lower_bound A))
let A be closed-interval Subset of REAL ; :: thesis: ( A c= ].(- 1),1.[ & dom (arccos `| ].(- 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 = (arccos . (upper_bound A)) - (arccos . (lower_bound A)) )
A4: dom (arccos `| ].(- 1),1.[) = ].(- 1),1.[ by FDIFF_1:def 8, SIN_COS6:108;
assume A1: ( A c= ].(- 1),1.[ & dom (arccos `| ].(- 1),1.[) = dom f2 & ( for x being Real holds
( x in ].(- 1),1.[ & f2 . x = - (1 / (sqrt (1 - (x ^2 )))) ) ) & f2 | A is continuous ) ; :: thesis: integral f2,A = (arccos . (upper_bound A)) - (arccos . (lower_bound A))
A c= dom f2 by A1, A4;
then A2: ( f2 is_integrable_on A & f2 | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
for x being Real st x in dom (arccos `| ].(- 1),1.[) holds
(arccos `| ].(- 1),1.[) . x = f2 . x

proof

let x be Real; :: thesis: ( x in dom (arccos `| ].(- 1),1.[) implies (arccos `| ].(- 1),1.[) . x = f2 . x )
assume A3: x in dom (arccos `| ].(- 1),1.[) ; :: thesis: (arccos `| ].(- 1),1.[) . x = f2 . x
A5: ( - 1 < x & x < 1 ) by A3, A4, XXREAL_1:4;
(arccos `| ].(- 1),1.[) . x = diff arccos ,x by A3, A4, FDIFF_1:def 8, SIN_COS6:108
.= - (1 / (sqrt (1 - (x ^2 )))) by A5, SIN_COS6:108
.= f2 . x by A1 ;
hence (arccos `| ].(- 1),1.[) . x = f2 . x ; :: thesis: verum

end;

then arccos `| ].(- 1),1.[ = f2 by A1, PARTFUN1:34;
hence integral f2,A = (arccos . (upper_bound A)) - (arccos . (lower_bound A)) by A1, A2, INTEGRA5:13, SIN_COS6:108; :: thesis: verum