let f2 be PartFunc of REAL ,REAL ; :: thesis: for A being closed-interval Subset of REAL st dom tanh = dom f2 & ( for x being Real st x in REAL holds
f2 . x = 1 / ((cosh . x) ^2 ) ) & f2 | A is continuous holds
integral f2,A = (tanh . (upper_bound A)) - (tanh . (lower_bound A))
let A be closed-interval Subset of REAL ; :: thesis: ( dom tanh = dom f2 & ( for x being Real st x in REAL holds
f2 . x = 1 / ((cosh . x) ^2 ) ) & f2 | A is continuous implies integral f2,A = (tanh . (upper_bound A)) - (tanh . (lower_bound A)) )
assume A1: ( dom tanh = dom f2 & ( for x being Real st x in REAL holds
f2 . x = 1 / ((cosh . x) ^2 ) ) & f2 | A is continuous ) ; :: thesis: integral f2,A = (tanh . (upper_bound A)) - (tanh . (lower_bound A))
dom tanh = REAL by FUNCT_2:def 1;
then A c= dom f2 by A1;
then A2: ( f2 is_integrable_on A & f2 | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A3: dom tanh = REAL by SIN_COS2:30;
A4: for x being Real st x in dom (tanh `| REAL ) holds
(tanh `| REAL ) . x = f2 . x

proof

let x be Real; :: thesis: ( x in dom (tanh `| REAL ) implies (tanh `| REAL ) . x = f2 . x )
assume x in dom (tanh `| REAL ) ; :: thesis: (tanh `| REAL ) . x = f2 . x
(tanh `| REAL ) . x = diff tanh ,x by FDIFF_1:def 8, SIN_COS2:36
.= 1 / ((cosh . x) ^2 ) by SIN_COS2:33
.= f2 . x by A1 ;
hence (tanh `| REAL ) . x = f2 . x ; :: thesis: verum

end;

dom (tanh `| REAL ) = dom f2 by A1, A3, FDIFF_1:def 8, SIN_COS2:36;
then tanh `| REAL = f2 by A4, PARTFUN1:34;
hence integral f2,A = (tanh . (upper_bound A)) - (tanh . (lower_bound A)) by A2, INTEGRA5:13, SIN_COS2:36; :: thesis: verum