reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem Th5:
  a <= b implies vol([' a,b '])=b-a
proof
  assume a<=b;
  then
A1: [' a,b '] = [.a,b.] by INTEGRA5:def 3;
  then
A2: [.a,b.] = [.lower_bound [.a,b.], upper_bound [.a,b.].] by INTEGRA1:4;
  then a=lower_bound([.a,b.]) by A1,INTEGRA1:5;
  hence thesis by A1,A2,INTEGRA1:5;
end;
