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 Th5:
  between a,b,c & a <> b & A is_line & a in A & b in A implies c in A
  proof
    assume that
A1: between a,b,c and
A2: a <> b and
A3: A is_line and
A4: a in A and
A5: b in A;
    Collinear a,b,c by A1;
    then c in Line(a,b);
    hence thesis by A3,A4,A5,A2,GTARSKI3:87;
  end;
