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;
reserve S         for non empty satisfying_Lower_Dimension_Axiom
                                satisfying_Tarski-model
                                TarskiGeometryStruct,
        a,b,c,d,m,p,q,r,s,x for POINT of S,
        A,A9,E              for Subset of S;

theorem Th19:
  A out a,b & A out b,c implies A out a,c
  proof
    assume that
A1: A out a,b and
A2: A out b,c;
    consider x be POINT of S such that
A3: between a,A,x and
A4: between b,A,x by A1;
    consider y be POINT of S such that
A5: between b,A,y and
A6: between c,A,y by A2;
    between x,A,b & between y,A,b by A4,A5,GTARSKI3:14;
    then between y,A,c & A out y,x by A6,GTARSKI3:14;
    then between x,A,c by Th14;
    then between c,A,x by GTARSKI3:14;
    hence thesis by A3;
  end;
