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

theorem Th11:
  for A,B being non empty compact Subset of REAL st A misses B
  holds dist(A,B) > 0
proof
  let A,B being non empty compact Subset of REAL such that
A1: A misses B;
  consider r0,s0 such that
A2: r0 in A and
A3: s0 in B and
A4: dist(A,B) = |.r0-s0.| by Th9;
  reconsider r0,s0 as Real;
  assume dist(A,B) <= 0;
  then |.r0-s0.| = 0 by A4,Th10;
  then r0 = s0 by GOBOARD7:2;
  hence contradiction by A1,A2,A3,XBOOLE_0:3;
end;
