reserve A for non empty closed_interval Subset of REAL;

theorem
for f be Function of REAL,REAL st
(for x be Real st x in A holds f.x = 0) holds
integral(f,A) = 0
proof
 let f be Function of REAL,REAL;
 assume A1: (for x be Real st x in A holds f.x = 0);
 reconsider f as PartFunc of REAL,REAL;
B1x: dom f = REAL by FUNCT_2:def 1;
 A = [.(lower_bound A),(upper_bound A).] by INTEGRA1:4
  .= ['(lower_bound A),(upper_bound A)'] by INTEGRA5:def 3,SEQ_4:11;
 then integral(f,A)
 =integral (f,(lower_bound A),(upper_bound A)) by INTEGRA5:def 4,SEQ_4:11
 .= 0 * ((upper_bound A) - (lower_bound A)) by Lm5,B1x,A1
 .=0;
 hence thesis;
end;
