let a, b, c, d be Real; :: thesis: for f being PartFunc of REAL,REAL st a <= b & c <= d & [.a,b.] c= dom f & c in [.a,b.] & d in [.a,b.] & f | ['a,b'] is continuous & ( for t being Real st t in [.c,d.] holds
0 <= f . t ) holds
0 <= integral (f,c,d)
let f be PartFunc of REAL,REAL; :: thesis: ( a <= b & c <= d & [.a,b.] c= dom f & c in [.a,b.] & d in [.a,b.] & f | ['a,b'] is continuous & ( for t being Real st t in [.c,d.] holds
0 <= f . t ) implies 0 <= integral (f,c,d) )
assume A1: ( a <= b & c <= d & [.a,b.] c= dom f & c in [.a,b.] & d in [.a,b.] & f | ['a,b'] is continuous & ( for t being Real st t in [.c,d.] holds
0 <= f . t ) ) ; :: thesis: 0 <= integral (f,c,d)
then A2: ['a,b'] = [.a,b.] by INTEGRA5:def 3;
then ( f | ['a,b'] is bounded & f is_integrable_on ['a,b'] ) by A1, INTEGRA5:10, INTEGRA5:11;
then A3: ( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),c,d) & |.(integral (f,d,c)).| <= integral ((abs f),c,d) ) by A1, A2, INTEGRA6:22;
A5: dom ((abs f) || ['c,d']) = ['c,d'] by A3, RELAT_1:62;
A4: dom (abs f) = dom f by VALUED_1:def 11;
then A6: dom (f || ['c,d']) = ['c,d'] by A3, RELAT_1:62;
A7: for x being object st x in dom (f || ['c,d']) holds
(f || ['c,d']) . x = ((abs f) || ['c,d']) . x

proof

let x be object ; :: thesis: ( x in dom (f || ['c,d']) implies (f || ['c,d']) . x = ((abs f) || ['c,d']) . x )
assume A8: x in dom (f || ['c,d']) ; :: thesis: (f || ['c,d']) . x = ((abs f) || ['c,d']) . x
then A9: (f || ['c,d']) . x = f . x by FUNCT_1:47;
x in dom f by A8, RELAT_1:57;
then A10: x in dom (abs f) by VALUED_1:def 11;
x in [.c,d.] by A1, A6, A8, INTEGRA5:def 3;
then |.(f . x).| = f . x by A1, COMPLEX1:43;
then f . x = (abs f) . x by A10, VALUED_1:def 11;
hence (f || ['c,d']) . x = ((abs f) || ['c,d']) . x by A5, A8, A9, FUNCT_1:47; :: thesis: verum

end;

integral ((abs f),c,d) = integral ((abs f),['c,d']) by A1, INTEGRA5:def 4
.= integral (f,['c,d']) by A3, A4, A5, A7, FUNCT_1:2, RELAT_1:62
.= integral (f,c,d) by A1, INTEGRA5:def 4 ;
hence 0 <= integral (f,c,d) by A3, COMPLEX1:46; :: thesis: verum