 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 SegmentAddition:
  between a,b,c & between a9,b9,c9 & a,b equiv a9,b9 & b,c equiv b9,c9
    implies a,c equiv a9,c9
   proof
     assume that
H1:  between a,b,c and
H2:  between a9,b9,c9 and
H3:  a,b equiv a9,b9 and
H4:  b,c equiv b9,c9;
     b,a equiv a,b by A1; then
Z2:  b,a equiv a9,b9 by H3, EquivTransitive;
     per cases;
     suppose a = b;
       hence thesis by H3, H4, A3, EquivSymmetric;
     end;
     suppose
Z1:    a <> b;
       a9,b9 equiv b9,a9 by A1; then
       b,a equiv b9,a9 by Z2, EquivTransitive; then
       a,b,a cong a9,b9,a9 by H3, Baaa;
       hence thesis by Z1, H1, H2, H4, A5;
     end;
   end;
