 reserve S for satisfying_Tarski-model TarskiGeometryStruct;
 reserve a, b, c, d, e, f, o, p, q, r, s,
    v, w, u, x, y, z, a9, b9, c9, d9, x9, y9, z for POINT of S;

theorem
  for a, b, c being Element of RealSpace st
    b in [.a,c.] holds b is_Between a,c
  proof
    let a, b, c be Element of RealSpace;
    assume b in [.a,c.]; then
B1: b >= a + 0 & c >= b + 0 by XXREAL_1:1; then
b2: c >= a + 0 by XXREAL_0:2;
A0: |.a-c.| = |.c - a.| by COMPLEX1:60
           .= (c - b) + (b - a) by b2,COMPLEX1:43,XREAL_1:19
           .= |.c-b.| + (b - a) by B1,XREAL_1:19,COMPLEX1:43
           .= |.c-b.| + |.b-a.| by B1,XREAL_1:19,COMPLEX1:43
           .= |.b-c.| + |.b-a.| by COMPLEX1:60
           .= |.a-b.| + |.b-c.| by COMPLEX1:60;
A1: dist (a,c) = |.a-c.| by TOPMETR:11;
    dist (a,b) = |.a-b.| by TOPMETR:11;
    hence thesis by A1,A0,TOPMETR:11;
  end;
