let n be Element of NAT ; :: thesis: for a, b, c, d being Real
for E being Point of (REAL-NS n)
for f being PartFunc of REAL,(REAL-NS n) st a <= b & ['a,b'] c= dom f & ( for x being Real st x in ['a,b'] holds
f . x = E ) & c in ['a,b'] & d in ['a,b'] holds
integral (f,c,d) = (d - c) * E
let a, b, c, d be Real; :: thesis: for E being Point of (REAL-NS n)
for f being PartFunc of REAL,(REAL-NS n) st a <= b & ['a,b'] c= dom f & ( for x being Real st x in ['a,b'] holds
f . x = E ) & c in ['a,b'] & d in ['a,b'] holds
integral (f,c,d) = (d - c) * E
let e be Point of (REAL-NS n); :: thesis: for f being PartFunc of REAL,(REAL-NS n) st a <= b & ['a,b'] c= dom f & ( for x being Real st x in ['a,b'] holds
f . x = e ) & c in ['a,b'] & d in ['a,b'] holds
integral (f,c,d) = (d - c) * e
let f be PartFunc of REAL,(REAL-NS n); :: thesis: ( a <= b & ['a,b'] c= dom f & ( for x being Real st x in ['a,b'] holds
f . x = e ) & c in ['a,b'] & d in ['a,b'] implies integral (f,c,d) = (d - c) * e )
assume A1: ( a <= b & ['a,b'] c= dom f & ( for x being Real st x in ['a,b'] holds
f . x = e ) & c in ['a,b'] & d in ['a,b'] ) ; :: thesis: integral (f,c,d) = (d - c) * e
reconsider f1 = f as PartFunc of REAL,(REAL n) by REAL_NS1:def 4;
reconsider e1 = e as Element of REAL n by REAL_NS1:def 4;
A2: for x being Real st x in ['a,b'] holds
f1 . x = e1 by A1;
A3: ( f1 is_integrable_on ['a,b'] & f1 | ['a,b'] is bounded & integral (f1,a,b) = (b - a) * e1 ) by Th29, A1, A2;
A4: integral (f1,c,d) = (d - c) * e1 by Th30, A1;
['a,b'] = [.a,b.] by A1, INTEGRA5:def 3;
then A5: ( a <= c & d <= b & a <= d & c <= b ) by A1, XXREAL_1:1;

per cases ( c <= d or not c <= d ) ;

suppose A6: c <= d ; :: thesis: integral (f,c,d) = (d - c) * e

A7: ( ['c,d'] c= dom f1 & f1 | ['c,d'] is bounded & f1 is_integrable_on ['c,d'] ) by A3, A1, Th9, A5, A6, Th2;
integral (f1,c,d) = integral (f,c,d) by A7, A6, Th45;
hence integral (f,c,d) = (d - c) * e by A4, REAL_NS1:3; :: thesis: verum

end;

suppose A8: not c <= d ; :: thesis: integral (f,c,d) = (d - c) * e

A9: ( ['d,c'] c= dom f1 & f1 | ['d,c'] is bounded & f1 is_integrable_on ['d,c'] ) by A3, A1, Th9, A5, A8, Th2;
A10: integral (f1,d,c) = integral (f,d,c) by A9, A8, Th45;
integral (f1,c,d) = - (integral (f1,d,c)) by Th33
.= - (integral (f,d,c)) by A10, REAL_NS1:4
.= integral (f,c,d) by A8, A9, Th47 ;
hence integral (f,c,d) = (d - c) * e by A4, REAL_NS1:3; :: thesis: verum

end;

end;