reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct;
reserve a,b for POINT of S;
reserve A for Subset of S;
reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct;
reserve a,b,c,m,r,s for POINT of S;
reserve A for Subset of S;

theorem Th6:
  between a,A,c & m in A & Middle a,m,c & r in A implies
  (for b st r out a,b & between r,b,a holds between b,A,c)
  proof
    assume that
A1: between a,A,c and
A2: m in A and
A3: Middle a,m,c and
A4: r in A;
A5: between c,m,a by A3,GTARSKI3:14;
    let b;
    assume that
A6: r out a,b and
A7: between r,b,a;
    consider t be POINT of S such that
A8: between b,t,c and
A9: between m,t,r by A7,A5,GTARSKI1:def 11;
    Collinear r,b,a by A7;
    then a in Line(r,b);
    hence between b,A,c by A8,A9,A2,Th4,A1,A4,A6,GTARSKI3:87;
  end;
