reserve r for Real,
  X for set,
  f, g, h for real-valued Function;

theorem Th1:
  for a,b,c being Real st |.a-b.| <= c holds b-c <= a & a <= b+c
proof
  let a,b,c be Real;
  assume
A1: |.a-b.| <= c;
A2: |.a-b.| >= 0 by COMPLEX1:46;
  then
A3: b <= b+c by A1,XREAL_1:31;
A4: b >= b-c by A1,A2,XREAL_1:43;
  per cases;
  suppose
    a-b >= 0;
    then |.a-b.| = a-b & a >= 0 qua Nat + b by ABSVALUE:def 1,XREAL_1:19;
    hence thesis by A1,A4,XREAL_1:20,XXREAL_0:2;
  end;
  suppose
    a-b < 0;
    then
A5: |.a-b.| = -(a-b) by ABSVALUE:def 1
      .= b-a;
    then 0 qua Nat+a <= b by A2,XREAL_1:19;
    hence thesis by A1,A3,A5,XREAL_1:12,XXREAL_0:2;
  end;
end;
