reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,c9,x,y,z,p,q,q9 for POINT of S;
reserve              S for satisfying_Tarski-model TarskiGeometryStruct,
        a,a9,b,b9,c,c9 for POINT of S;
reserve S                 for non empty satisfying_Tarski-model
                                    TarskiGeometryStruct,
        A,A9              for Subset of S,
        x,y,z,a,b,c,c9,d,u,p,q,q9 for POINT of S;
reserve S                 for non empty
                            satisfying_Lower_Dimension_Axiom
                            satisfying_Tarski-model
                            TarskiGeometryStruct,
        a,b,c,p,q,x,y,z,t for POINT of S;

theorem Satz8p22:
  for S being non empty satisfying_Lower_Dimension_Axiom
  satisfying_Tarski-model TarskiGeometryStruct
  for a,b being POINT of S holds
  ex x being POINT of S st Middle a,x,b
  proof
    let S be non empty satisfying_Lower_Dimension_Axiom
      satisfying_Tarski-model TarskiGeometryStruct;
    let a,b be POINT of S;
    set c = the POINT of S;
    per cases;
    suppose a = b;
      hence thesis by GTARSKI3:97;
    end;
    suppose
A1:   a <> b;
      then consider q,t be POINT of S such that
A2:   are_orthogonal b,a,q,b and
      Collinear b,a,t and
      between c,t,q by Satz8p21;
A3:   are_orthogonal a,b,q,b by A2;
      consider p,t9 be POINT of S such that
A4:   are_orthogonal a,b,p,a and
A5:   Collinear a,b,t9 and
A6:   between q,t9,p by A1,Satz8p21;
      per cases by GTARSKI3:64;
      suppose a,p <= b,q;
        hence thesis by Satz8p22lemma,A3,A4,A5,A6;
      end;
      suppose
A7:     b,q <= a,p;
        are_orthogonal b,a,p,a & are_orthogonal b,a,q,b & Collinear b,a,t9 &
          between p,t9,q by A6,A4,A2,A5,GTARSKI3:14,45;
        then ex x be POINT of S st Middle b,x,a by Satz8p22lemma,A7;
        hence thesis by GTARSKI3:96;
      end;
    end;
  end;
