reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve x,y,z,y1,y2 for Element of REAL 2;
reserve L,L1,L2,L3,L4 for Element of line_of_REAL 2;
reserve D,E,F for Point of TOP-REAL 2;
reserve b,c,d,r,s for Real;

theorem
  A,B,C is_a_triangle &
  A in circle(a,b,r) & B in circle(a,b,r) & C in circle(a,b,r) &
  A in circle(c,d,s) & B in circle(c,d,s) & C in circle(c,d,s)
  implies a=c & b=d & r=s
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: A in circle(a,b,r) and
A3: B in circle(a,b,r) and
A4: C in circle(a,b,r) and
A5: A in circle(c,d,s) and
A6: B in circle(c,d,s) and
A7: C in circle(c,d,s);
A8: |[a,b]| = the_circumcenter(A,B,C) & r = |.the_circumcenter(A,B,C)-A.| &
    |[c,d]| = the_circumcenter(A,B,C) & s = |.the_circumcenter(A,B,C)-A.|
    by A1,A2,A3,A4,A5,A6,A7,Th49;
    |[a,b]|`1 = a & |[a,b]|`2 = b & |[c,d]|`1 = c & |[c,d]|`2 = d by EUCLID:52;
    hence thesis by A8;
  end;
