reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem
  for a,b being Real st a<=b holds integral(Cst(1),a,b)=b-a
proof
  let a,b be Real;
  assume a<=b;
  then [.a,b.]=[' a,b '] by INTEGRA5:def 3;
  then reconsider A=[.a,b.] as non empty closed_interval Subset of REAL;
  upper_bound A=b & lower_bound A=a by Th37;
  then
A1: vol(A)=b-a by INTEGRA1:def 5;
  integral(Cst(1),a,b)=integral(Cst(1),A) & (Cst(1))||A = chi(A,A) by Th36,
INTEGRA5:19;
  hence thesis by A1,INTEGRA4:2;
end;
