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

theorem Th7:
  for A,B being Subset of REAL, r, s st r in A & s in B holds
  |.r-s.| >= dist(A,B)
proof
  let A,B be Subset of REAL;
  set Y = {|.r-s.| where r, s is Real : r in  A & s in B};
  let r,s;
  assume that
A1: r in A and
A2: s in B;
  Y c= REAL
  proof
    let e be object;
    assume e in Y;
    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 Y0 = Y as Subset of REAL;
A3: Y0 is bounded_below
  proof
    take 0;
    let r0 be ExtReal;
    assume r0 in Y0;
    then ex r,s being Real st r0 = |.r-s.| & r in A & s in B;
    hence thesis by COMPLEX1:46;
  end;
A4: dist(A,B) = lower_bound Y0 by Def1;
  |.r-s.| in Y0 by A1,A2;
  hence thesis by A4,A3,SEQ_4:def 2;
end;
