
theorem Thm23:
  for a,b be Real holds circle(a,b,0) = { |[a,b]| }
  proof
    let a,b be Real;
    now
      hereby
        let t be object;
        assume t in {p where p is Point of TOP-REAL 2: |.p-|[a,b]|.|=0};
        then consider p0 be Point of TOP-REAL 2 such that
A1:     t=p0 and
A2:     |.p0-|[a,b]|.|=0;
        p0=|[a,b]| by A2,EUCLID_6:42;
        hence t in { |[a,b]| } by A1,TARSKI:def 1;
      end;
      let t be object;
      assume
A3:   t in { |[a,b]| }; then
A4:   t = |[a,b]| by TARSKI:def 1;
      reconsider p0=t as Point of TOP-REAL 2 by A3,TARSKI:def 1;
      |.p0-|[a,b]|.|=0 by A4,EUCLID_6:42;
      hence t in {p where p is Point of TOP-REAL 2: |.p-|[a,b]|.|=0};
    end;
    then {p where p is Point of TOP-REAL 2: |.p-|[a,b]|.|=0} c= { |[a,b]| } &
    { |[a,b]| } c= {p where p is Point of TOP-REAL 2: |.p-|[a,b]|.|=0};
    hence thesis by JGRAPH_6:def 5;
  end;
