reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

theorem
  the carrier of Tunit_circle(2) = circle(0,0,1)
  proof
A1: the carrier of Tunit_circle(2) c= circle(0,0,1)
    proof
      let u be object;
      assume u in the carrier of Tunit_circle(2);
      then u in the carrier of Tcircle(0.TOP-REAL 2,1) by TOPREALB:def 7;
      then u in Sphere(0.TOP-REAL 2,1) by TOPREALB:9;
      hence thesis by EUCLID:54,TOPREAL9:52;
    end;
    circle(0,0,1) c= the carrier of Tunit_circle(2)
    proof
      let u be object;
      assume u in circle(0,0,1);
      then u in Sphere(0.TOP-REAL 2,1) by TOPREAL9:52,EUCLID:54;
      then u in the carrier of Tcircle(0.TOP-REAL 2,1) by TOPREALB:9;
      hence thesis by TOPREALB:def 7;
    end;
    hence thesis by A1;
  end;
