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
  r <> s implies circle(a,b,r) misses circle(a,b,s)
  proof
    assume
A1: r <> s;
    per cases;
    suppose not r is positive or not s is positive;
      then per cases;
      suppose
A2:     r = 0;
        then
A3:     circle(a,b,r) = {|[a,b]|} by EUCLID10:36;
        assume circle(a,b,r) meets circle(a,b,s);
        then consider p be object such that
A4:     p in circle(a,b,r) and
A5:     p in circle(a,b,s) by XBOOLE_0:3;
        p in {p where p is Point of TOP-REAL 2: |.p - |[a,b]|.| = s}
            by A5,JGRAPH_6:def 5;
        then consider p1 be Point of TOP-REAL 2 such that
A6:     p = p1 and
A7:     |.p1 - |[a,b]|.| = s;
        s = |.|[a,b]| - |[a,b]|.| by A6,A7,A4,A3,TARSKI:def 1
         .= 0 by EUCLID_6:42;
        hence contradiction by A2,A1;
      end;
      suppose r < 0 or s < 0;
        then circle(a,b,r) is empty or circle(a,b,s) is empty;
        hence thesis;
      end;
      suppose
A8:     s = 0;
        then
A9:     circle(a,b,s) = {|[a,b]|} by EUCLID10:36;
        assume circle(a,b,r) meets circle(a,b,s);
        then consider p be object such that
A10:    p in circle(a,b,s) and
A11:    p in circle(a,b,r) by XBOOLE_0:3;
        p in {p where p is Point of TOP-REAL 2: |.p - |[a,b]|.| = r}
              by A11,JGRAPH_6:def 5;
        then consider p1 be Point of TOP-REAL 2 such that
A12:    p = p1 and
A13:    |.p1 - |[a,b]|.| = r;
        r = |.|[a,b]| - |[a,b]|.| by A12,A13,A10,A9,TARSKI:def 1
         .= 0 by EUCLID_6:42;
        hence contradiction by A8,A1;
      end;
    end;
    suppose
      r is positive & s is positive;
      assume circle(a,b,r) meets circle(a,b,s);
      then consider p be object such that
A15:  p in circle(a,b,r) and
A16:  p in circle(a,b,s) by XBOOLE_0:3;
      p in {p where p is Point of TOP-REAL 2: |. p - |[a,b]|.| = r}
             by A15,JGRAPH_6:def 5;
      then consider p1 be Point of TOP-REAL 2 such that
A17:  p = p1 and
A18:  |.p1 - |[a,b]|.| = r;
      p in {p where p is Point of TOP-REAL 2: |. p - |[a,b]|.| = s}
             by A16,JGRAPH_6:def 5;
      then consider p2 be Point of TOP-REAL 2 such that
A19:  p = p2 and
A20:  |.p2 - |[a,b]|.| = s;
      thus contradiction by A17,A19,A18,A1,A20;
    end;
  end;
