reserve            S for satisfying_CongruenceSymmetry
                         satisfying_CongruenceEquivalenceRelation
                         TarskiGeometryStruct,
         a,b,c,d,e,f for POINT of S;
reserve S for satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_SAS
              TarskiGeometryStruct,
        q,a,b,c,a9,b9,c9,x1,x2 for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
        a,b,c,d for POINT of S;
reserve       S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d for POINT of S;
reserve         S for satisfying_CongruenceIdentity
                      satisfying_SegmentConstruction
                      satisfying_BetweennessIdentity
                      satisfying_Pasch
                      TarskiGeometryStruct,
        a,b,c,d,e for POINT of S;
reserve       S for satisfying_Tarski-model
                    TarskiGeometryStruct,
      a,b,c,d,p for POINT of S;
reserve                   S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d,a9,b9,c9,d9 for POINT of S;
reserve S for satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,a9,b9,c9,d9,p,q for POINT of S;
reserve                       S for satisfying_Tarski-model
                                    TarskiGeometryStruct,
        a,b,c,d,e,f,a9,b9,c9,d9 for POINT of S;
reserve p for POINT of S;
reserve r for POINT of S;
reserve x,y for POINT of S;
reserve S for non empty satisfying_Tarski-model TarskiGeometryStruct;
reserve p,q,r,s for POINT of S;
reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
  a,b,p,q for POINT of S;
reserve S for non empty satisfying_Tarski-model TarskiGeometryStruct,
                  A,B for Subset of S,
        a,b,c,p,q,r,s for POINT of S;

theorem
  for S being satisfying_Tarski-model TarskiGeometryStruct
  for a,b,c,a1,b1,c1 being Element of S st
  b out a,c & b1 out a1,c1 & b,a equiv b1,a1 & b,c equiv b1,c1 holds
  a,c equiv a1,c1
  proof
    let S be satisfying_Tarski-model TarskiGeometryStruct;
    let a,b,c,a1,b1,c1 be Element of S;
    assume that
A1: b out a,c and
A2: b1 out a1,c1 and
A3: b,a equiv b1,a1 and
A4: b,c equiv b1,c1 and
A5: not a,c equiv a1,c1;
    now
      thus b out c,a by A1;
      thus b1 out c1,a1 by A2;
      b,a equiv a1,b1 & b,c equiv c1,b1 & not a,c equiv c1,a1
        by Satz2p5,A3,A4,A5;
      hence not c,a equiv c1,a1 & a,b equiv a1,b1 & c,b equiv c1,b1 by Satz2p4;
      hence not (between c,a,b & between c1,a1,b1) &
            not (between a,c,b & between a1,c1,b1) by Satz4p3,A5;
      thus (b,a <= b,c or b1,c1 <= b1,a1) & (b,c <= b,a or b1,a1 <= b1,c1)
        by A3,Satz6p28Lem02,A4;
    end;
    hence contradiction by Satz6p28Lem01,A1,A2,Satz3p2;
end;
