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

theorem Th03:
  a <= b <= c & |.a - d.| <= e & |.c - d.| <= e implies |. b - d .| <= e
  proof
    assume that
A1: a <= b <= c and
A2: |.a - d.| <= e and
A3: |.c - d.| <= e;
    b in [.a,c.] & a <= c by A1,XXREAL_1:1,XXREAL_0:2;
    then consider r be Real such that
A4: 0 <= r & r <= 1 and
A5: b = r * a + (1 - r) * c by SPRECT_1:51;
A6: |. r * (a - d) + (1 - r) * (c - d).| <= |. r * (a - d) .|
         + |.(1 - r) * (c - d).| by COMPLEX1:56;
    |. r * (a - d) .| = |.r.| * |.a - d.| by COMPLEX1:65
                     .= r * |.a - d.| by A4,ABSVALUE:def 1;
    then
A7: |. r * (a - d) .| <= r * e by A4,A2,XREAL_1:64;
A8: r - r <= 1 - r by A4,XREAL_1:9;
    |. (1 - r) * (c - d) .| = |.1 - r.| * |.c - d.| by COMPLEX1:65
                           .= (1 - r) * |.c - d.| by A8,ABSVALUE:def 1;
    then
A9: |. (1 - r) * (c - d) .| <= (1 - r) * e by A8,A3,XREAL_1:64;
    |. b - d .| <= r * e + |.(1 - r) * (c - d).| by A5,A6,A7,Lm01;
    then |. b - d .| <= r * e + (1 - r) * e by A9,Lm01;
    hence thesis;
  end;
