reserve
  r,s,r0,s0,t for Real;

theorem Th8:
  for A,B being Subset of REAL, C,D being non empty Subset of REAL
  st C c= A & D c= B holds dist(A,B) <= dist(C,D)
proof
  let A,B be Subset of REAL, C,D be non empty Subset of REAL such that
A1: C c= A and
A2: D c= B;
  consider s0 being object such that
A3: s0 in D by XBOOLE_0:def 1;
  set Y = {|.r-s.| where r, s is Real : r in
  C & s in D};
  consider r0 being object such that
A4: r0 in C by XBOOLE_0:def 1;
A5: Y c= REAL
  proof
    let e be object;
    assume e in Y;
    then ex r,s being Real st e = |.r-s.| & r in C & s in D;
    hence thesis by XREAL_0:def 1;
  end;
  reconsider r0,s0 as Real by A4,A3;
  |.r0-s0.| in Y by A4,A3;
  then reconsider Y as non empty Subset of REAL by A5;
  set X = {|.r-s.| where r, s is Real : r in
  A & s in B};
  X c= REAL
  proof
    let e be object;
    assume e in X;
    then ex r,s being Real st e = |.r-s.| & r in A & s in B;
    hence thesis by XREAL_0:def 1;
  end;
  then reconsider X as Subset of REAL;
A6: Y c= X
  proof
    let x be object;
    assume x in Y;
    then ex r,s being Real st x = |.r-s.| & r in C & s in D;
    hence thesis by A1,A2;
  end;
A7: X is bounded_below
  proof
    take 0;
    let r0 be ExtReal;
    assume r0 in X;
    then ex r,s being Real st r0 = |.r-s.| & r in A & s in B;
    hence thesis by COMPLEX1:46;
  end;
A8: dist(C,D) = lower_bound Y by Def1;
  dist(A,B) = lower_bound X by Def1;
  hence thesis by A7,A8,A6,SEQ_4:47;
end;
