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;

theorem ThMak1:
  for S being non empty TarskiGeometryStruct holds
  S is satisfying_Continuity_Axiom iff S is (Co)
  proof
    let S be non empty TarskiGeometryStruct;
    hereby
      assume
A1:   S is satisfying_Continuity_Axiom;
      now
        let X,Y be set;
        reconsider X9 = X /\ the carrier of S,
        Y9 = Y /\ the carrier of S as Subset of S by XBOOLE_1:17;
        assume ex a being POINT of S st
          (for x,y being POINT of S st x in X & y in Y holds between a,x,y);
        then consider a be POINT of S such that
A3:     (for x,y being POINT of S st x in X & y in Y holds between a,x,y);
        for x,y be POINT of S st x in X9 & y in Y9 holds between a,x,y
        proof
          let x,y be POINT of S;
          assume x in X9 & y in Y9;
          then x in X & y in Y by XBOOLE_0:def 4;
          hence thesis by A3;
        end;
        then consider b be POINT of S such that
A4:     for x,y be POINT of S st x in X9 & y in Y9 holds between x,b,y by A1;
        for x,y be POINT of S st x in X & y in Y holds between x,b,y
        proof
          let x,y be POINT of S;
          assume x in X & y in Y;
          then x in X9 & y in Y9 by XBOOLE_0:def 4;
          hence thesis by A4;
        end;
        hence ex b being POINT of S st
          (for x,y being POINT of S st x in X & y in Y holds between x,b,y);
      end;
      hence S is (Co);
    end;
    assume S is (Co);
    hence S is satisfying_Continuity_Axiom;
  end;
