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 Th4:
  between a,b,c & A is_line & a in A & c in A implies b in A
  proof
    assume that
A1: between a,b,c and
A2: A is_line and
A3: a in A and
A4: c in A;
    Collinear a,b,c by A1;
    then Collinear a,c,b by GTARSKI3:45;
    then
A5: b in Line(a,c);
    a<>c or a=c;
    hence thesis by A2,A4,A5,GTARSKI3:87,A1,A3,GTARSKI1:def 10;
  end;
