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;
