reserve a,b,c,d,e for Real;

theorem Th01:
  a - b <= c & b <= a implies |. b - a .| <= c
  proof
    assume that
A1: a - b <= c and
A2: b <= a;
    b - b <= a - b by A2,XREAL_1:9; then
A3: 0 <= c by A1;
    (-1) * c <= (-1) * (a - b) by A1,XREAL_1:65; then
A5: -c <= b - a;
    b - a <= a - a by A2,XREAL_1:9;
    hence thesis by ABSVALUE:5,A5,A3;
  end;
