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

theorem Th10:
  for A, B being non empty compact Subset of REAL holds dist(A,B) >= 0
proof
  let A, B be non empty compact Subset of REAL;
  set X = {|.r-s.| where r, s is Real : r in
  A & s in B};
  consider r0 being object such that
A1: r0 in A by XBOOLE_0:def 1;
A2: 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;
  consider s0 being object such that
A3: s0 in B by XBOOLE_0:def 1;
  reconsider r0,s0 as Real by A1,A3;
  |.r0-s0.| in X by A1,A3;
  then reconsider X as non empty Subset of REAL by A2;
A4: for t being Real st t in X holds t >= 0
  proof
    let t be Real;
    assume t in X;
    then ex r,s being Real st t = |.r-s.| & r in A & s in B;
    hence thesis by COMPLEX1:46;
  end;
  dist(A,B) = lower_bound X by Def1;
  hence thesis by A4,SEQ_4:43;
end;
