reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,c9,x,y,z,p,q,q9 for POINT of S;

theorem Prelim10:
  (Middle a,b,c or Middle c,b,a) & (a <> b or b <> c) implies
  (Line(b,a) = Line(b,c) & Line(a,b) = Line(b,c) &
  Line(a,b) = Line(c,b) & Line(b,a) = Line(c,b))
  proof
    assume that
A1: (Middle a,b,c or Middle c,b,a) and
A2: a <> b or b <> c;
A3: Middle a,b,c by A1,GTARSKI3:96;
    then between a,b,c by GTARSKI3:def 12;
    then a4: Collinear a,b,c by GTARSKI1:def 17;
    per cases by A2;
    suppose
A5:   a <> b;
      then b <> c by A3,GTARSKI1:def 7,GTARSKI3:def 12;
      hence thesis by a4,A5,GTARSKI3:82,LemmaA1;
    end;
    suppose
A6:   b <> c;
      then a <> b by A1,Prelim09;
      hence thesis by a4,A6,GTARSKI3:82,LemmaA1;
    end;
  end;
