 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 Inner5Segments:
  a,b,c cong a9,b9,c9 &
    between a,x,c & between a9,x9,c9 & c,x equiv c9,x9
      implies b,x equiv b9,x9
   proof
     assume that
H1:  a,b,c cong a9,b9,c9 and
H2:  between a,x,c and
H3:  between a9,x9,c9 and
H4:  c,x equiv c9,x9;
X1:  a,b equiv a9,b9 & a,c equiv a9,c9 & b,c equiv b9,c9 by H1;
     per cases;
     suppose
       x = c;
       hence thesis by H1, A3, H4, EquivSymmetric;
     end;
     suppose x <> c; then
X2:    a <> c by H2, A6;
       consider y such that
X3:    between a,c,y & c,y equiv a,c by A4;
       consider y9 such that
X4:    between a9,c9,y9 & c9,y9 equiv a,c by A4;
       a,c equiv c9,y9 by X4, EquivSymmetric; then
X5:    c,y equiv c9,y9 by X3, EquivTransitive;
X6:    c,b equiv c9,b9 by H1, CongruenceDoubleSymmetry; then
       a,c,b cong a9,c9,b9 by X1; then
X7:    b,y equiv b9,y9 by X2, X3, X4, X5, A5;
X8:    y <> c by X3, EquivSymmetric, A3, X2;
       between y,c,a & between c,x,a by X3, H2, Bsymmetry; then
X9:    between y,c,x by B124and234then123;
       between y9,c9,a9 & between c9,x9,a9 by X4, H3, Bsymmetry;
       then
X10:   between y9,c9,x9 by B124and234then123;
       y,c,b cong y9,c9,b9 by X6, X5, X7, CongruenceDoubleSymmetry;
       hence thesis by X8, X9, X10, H4, A5;
     end;
   end;
