let a, b, c, d be real number ; :: thesis: for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
abs (integral (f,d,c)) <= integral ((abs f),c,d)
let f be PartFunc of REAL,REAL; :: thesis: ( a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] implies abs (integral (f,d,c)) <= integral ((abs f),c,d) )
assume that
A1: a <= b and
A2: c <= d and
A3: ( f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] ) and
A4: d in ['a,b'] ; :: thesis: abs (integral (f,d,c)) <= integral ((abs f),c,d)
A5: ( abs (integral (f,c,d)) <= integral ((abs f),c,d) & integral (f,c,d) = integral (f,['c,d']) ) by A1, A2, A3, A4, Lm6, INTEGRA5:def 5;

per cases ( c = d or c <> d ) ;

suppose c = d ; :: thesis: abs (integral (f,d,c)) <= integral ((abs f),c,d)

hence abs (integral (f,d,c)) <= integral ((abs f),c,d) by A1, A3, Lm6; :: thesis: verum

end;

suppose c <> d ; :: thesis: abs (integral (f,d,c)) <= integral ((abs f),c,d)

then c < d by A2, XXREAL_0:1;
then integral (f,d,c) = - (integral (f,['c,d'])) by INTEGRA5:def 5;
hence abs (integral (f,d,c)) <= integral ((abs f),c,d) by A5, COMPLEX1:138; :: thesis: verum

end;

end;