reserve A for non empty closed_interval Subset of REAL;

theorem Lm5:
for e being Real
for f being PartFunc of REAL,REAL st
A c= dom f & ( for x being Real st x in A holds f . x = e ) holds
( f is_integrable_on A & f | A is bounded &
integral (f,(lower_bound A),(upper_bound A))
        = e * ((upper_bound A) - (lower_bound A)) )
proof
 let e be Real;
 let f be PartFunc of REAL,REAL;
 assume that
 A1: A c= dom f and
 A2: for x being Real st x in A holds f . x = e;
 B2: (lower_bound A)<=(upper_bound A) by SEQ_4:11;
 A = [.(lower_bound A),(upper_bound A).] by INTEGRA1:4
  .= ['(lower_bound A),(upper_bound A)'] by INTEGRA5:def 3,SEQ_4:11;
 hence thesis by INTEGRA6:26,B2,A1,A2;
end;
