 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 ::: SegmentSubtraction:
  between a,b,c & between a9,b9,c9 & a,b equiv a9,b9 & a,c equiv a9,c9
    implies b,c equiv b9,c9
   proof
     assume that
H1:  between a,b,c and
H2:  between a9,b9,c9 and
H3:  a,b equiv a9,b9 and
H4:  a,c equiv a9,c9;
     per cases;
     suppose a = b;
       hence b,c equiv b9,c9 by H4, A3, EquivSymmetric, H3;
     end;
     suppose
Z1:    a <> b;
       consider x such that
Z2:    between a,b,x & b,x equiv b9,c9 by A4;
Z3:    a,x equiv a9,c9 by Z2, H2, H3, SegmentAddition;
       a9,c9 equiv a,c by H4, EquivSymmetric; then
       a,x equiv a,c by Z3, EquivTransitive;
       hence b,c equiv b9,c9 by Z1, Z2, H1, C1prime;
     end;
   end;
