reserve i,j,k,n,n1,n2,m for Nat;
reserve a,r,x,y for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve C for non empty set;
reserve X for set;

theorem Th6:
  for f being PartFunc of A,REAL st vol(A)=0 holds 
    f is integrable & integral(f)=0
proof
  let f be PartFunc of A,REAL;
  assume
A1: vol(A)=0;
  then
A2: upper_integral(f)=0 by Lm2;
A3: lower_integral(f)=0 by A1,Lm3;
A4: f is lower_integrable by A1,Lm3;
  f is upper_integrable by A1,Lm2;
  hence thesis by A2,A4,A3,INTEGRA1:def 16,def 17;
end;
