reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th1:
  s - r < t & s + r > t iff |.t-s.| < r
proof
  thus s - r < t & s + r > t implies |.t-s.| < r
  proof
    assume that
A1: s - r < t and
A2: s + r > t;
    -r + s < t by A1;
    then
A3: -r < t - s by XREAL_1:20;
    t - s < r by A2,XREAL_1:19;
    hence thesis by A3,SEQ_2:1;
  end;
  assume
A4: |.t-s.| < r;
  then -r < t - s by SEQ_2:1;
  then
A5: s + -r < t by XREAL_1:20;
  t - s < r by A4,SEQ_2:1;
  hence thesis by A5,XREAL_1:19;
end;
