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;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
        a,b,m for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              TarskiGeometryStruct,
        a,b,m for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_SAS
              TarskiGeometryStruct,
        a for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_SAS
              TarskiGeometryStruct,
  a,p,p9 for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_SAS
              satisfying_Pasch
              TarskiGeometryStruct,
  a,p,p9 for POINT of S;
reserve S for satisfying_CongruenceIdentity
                satisfying_CongruenceSymmetry
                satisfying_CongruenceEquivalenceRelation
                satisfying_SegmentConstruction
                satisfying_BetweennessIdentity
                satisfying_SAS
                TarskiGeometryStruct,
        a,p for POINT of S;
reserve              S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d,m,p,p9,q,r,s for POINT of S;
reserve         S for non empty satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d,p for POINT of S;
reserve a1,a2,b1,b2,m1,m2 for POINT of S;
reserve S           for non empty
                        satisfying_Lower_Dimension_Axiom
                        satisfying_Tarski-model
                        TarskiGeometryStruct,
        a,b,c,p,q,r for POINT of S;
reserve S for TarskiGeometryStruct;
reserve S for TarskiGeometryStruct;

theorem ThMak3:
  for S being TarskiGeometryStruct st
    S is (RE) & S is (TE) holds
      (S is (FS) iff S is (FS'))
  proof
    let S be TarskiGeometryStruct;
    assume that
A1: S is (RE) and
A2: S is (TE);
    hereby
      assume
A3:   S is (FS);
      thus S is (FS')
      proof
        let a,b,c,d,a9,b9,c9,d9 be POINT of S;
        assume a <> b & between a,b,c & between a9,b9,c9 &
          a,b equiv a9,b9 & b,c equiv b9,c9 & a,d equiv a9,d9 &
          b,d equiv b9,d9;
        then c,d equiv c9,d9 & c,d equiv d,c by A1,A3;
        hence d,c equiv c9,d9 by A2;
      end;
    end;
    assume
A4: S is (FS');
    let a,b,c,d,a9,b9,c9,d9 be POINT of S;
    assume a <> b & between a,b,c & between a9,b9,c9 &
      a,b equiv a9,b9 & b,c equiv b9,c9 & a,d equiv a9,d9 & b,d equiv b9,d9;
    then d,c equiv c9,d9 & d,c equiv c,d by A1,A4;
    hence c,d equiv c9,d9 by A2;
  end;
