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 Th6:
  vol([' min(a,b),max(a,b) ']) = |.b-a.|
proof
  per cases;
  suppose
A1: a<= b;
    then min(a,b) = a & max(a,b) = b by XXREAL_0:def 9,def 10;
    then
A2: vol([' min(a,b),max(a,b) '])=b-a by Th5,XXREAL_0:25;
    0 <= b-a by A1,XREAL_1:48;
    hence thesis by A2,ABSVALUE:def 1;
  end;
  suppose
A3: not a<= b;
    then 0 <= a-b by XREAL_1:48;
    then
A4: a-b = |.a-b.| by ABSVALUE:def 1
      .=|.b-a.| by COMPLEX1:60;
    min(a,b) = b & max(a,b) = a by A3,XXREAL_0:def 9,def 10;
    hence thesis by A4,Th5,XXREAL_0:17;
  end;
end;
