reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;
reserve X for non empty set;

theorem Th27:
  for a,b,r1,r2 being Real st a<=r1 & r1<=b & a<=r2 & r2<=b holds
  |.r1-r2.|<=b-a
proof
  let a,b,r1,r2 be Real;
  assume that
A1: a<=r1 and
A2: r1<=b & a<=r2 and
A3: r2<=b;
  per cases;
  suppose
A4: r1-r2>=0;
A5: r1-r2<=b-r2 & b-r2<=b-a by A2,XREAL_1:9,10;
    |.r1-r2.|=r1-r2 by A4,ABSVALUE:def 1;
    hence thesis by A5,XXREAL_0:2;
  end;
  suppose
    r1-r2<0;
    then
A6: |.r1-r2.|=-(r1-r2) by ABSVALUE:def 1
      .=r2-r1;
    r2-r1<=b-r1 & b-r1<=b-a by A1,A3,XREAL_1:9,10;
    hence thesis by A6,XXREAL_0:2;
  end;
end;
