
theorem
  for a,b being Real,r being positive Real, C be Subset of Euclid 2 st
  C = circle(a,b,r) holds diameter C = 2 * r
  proof
    let a,b be Real,r be positive Real, C be Subset of Euclid 2 such that
A1: C=circle(a,b,r);
A2: circle(a,b,r)={p where p is Point of TOP-REAL 2: |.p-|[a,b]|.|=r}
    by JGRAPH_6:def 5;
A3: for x,y being Point of Euclid 2 st x in C & y in C holds dist(x,y) <= 2 * r
    proof
      let x,y be Point of Euclid 2 such that
A4:   x in C and
A5:   y in C;
    consider JA be Point of TOP-REAL 2 such that
A6: x=JA and
A7: |.JA-|[a,b]|.|=r by A1,A4,A2;
    consider JB be Point of TOP-REAL 2 such that
A8: y=JB and
A9: |.JB-|[a,b]|.|=r by A1,A5,A2;
    reconsider KA=JA,KB=JB as Element of REAL 2 by EUCLID:22;
    reconsider O=|[a,b]| as Element of REAL 2 by EUCLID:22;
    reconsider O2=|[a,b]| as Point of Euclid 2 by EUCLID:67;
A10: dist(x,y)<=dist(x,O2)+dist(y,O2) by METRIC_1:4;
    dist(y,O2)=|.KB-O.| by A8,SPPOL_1:5;
    then
    dist(x,y) <= |.KA-O.| + |.KB-O.| by A10,A6,SPPOL_1:5;
    hence thesis by A7,A9;
  end;
A12: C is bounded by A1,JORDAN2C:11;
  for s be Real st (for x,y being Point of Euclid 2 st x in C & y in C holds
  dist(x,y) <= s) holds 2 * r <= s
  proof
    let s be Real;
    assume
A13: for x,y being Point of Euclid 2 st x in C & y in C holds
     dist(x,y) <= s;
    assume
A14: s < 2 * r;
    set A1=|[a+r,b]|;
    set B1=|[a-r,b]|;
A15: A1-|[a,b]|=|[a+r-a,b-b]| by EUCLID:62
    .=|[r,0]|;
A16: B1-|[a,b]|=|[a-r-a,b-b]| by EUCLID:62
    .=|[-r,0]|;
A17: r^2 + 0^2 =r^2 + 0 * 0 by SQUARE_1:def 1
    .=r^2;
    Re (r + 0 * <i>)=r & Im (r + 0 * <i>)=0 by COMPLEX1:12;
    then |.cpx2euc(r + 0 * <i>).| = sqrt (r^2) by A17,EUCLID_3:24
    .=r by SQUARE_1:22; then
A18: |.A1-|[a,b]|.|=r by A15,EUCLID_3:5; then
A19: A1 in circle(a,b,r) by A2;
A20: A1 in C by A18,A1,A2;
A21: (-r)^2 + 0^2 =(-r)^2 + 0 * 0 by SQUARE_1:def 1
    .=r^2 by SQUARE_1:3;
    Re (-r + 0 * <i>)=-r & Im (-r + 0 * <i>)=0 by COMPLEX1:12;
    then |.cpx2euc(-r + 0 * <i>).| = sqrt (r^2) by A21,EUCLID_3:24
    .=r by SQUARE_1:22; then
A22: |.|[-r,0]|.|=r by EUCLID_3:5; then
A23: B1 in circle(a,b,r) by A2,A16;
A24: B1 in C by A1,A2,A22,A16;
A25: |[a,b]| in LSeg(A1,B1)
    proof
      |[a+r,b]|+|[a-r,b]|=|[a+r+(a-r),b+b]| by EUCLID:56;
      then 1/2*(A1+B1)=|[1/2*(2*a),1/2*(2*b)]| by EUCLID:58
        .=|[a,b]|;
      hence thesis by RLTOPSP1:69;
    end;
A26:  A1<>B1
      proof
        assume A1=B1;
        then a+r=a-r & b=b by SPPOL_2:1;
        then r=0;
        hence contradiction;
      end;
A27:  A1<>|[a,b]|
      proof
        assume A1=|[a,b]|;
        then a+r=a & b=b by SPPOL_2:1;
        then r=0;
        hence contradiction;
      end;
    A1,B1,|[a,b]| are_mutually_distinct
    proof
      B1<>|[a,b]|
      proof
        assume B1=|[a,b]|;
        then a-r=a & b=b by SPPOL_2:1;
        then -r=0;
        hence contradiction;
      end;
      hence thesis by A26,A27;
    end; then
A28: |.A1-B1.|=2 * r by A19,A23,A25,Thm38;
    reconsider a1=A1,b1=B1 as Point of Euclid 2 by EUCLID:67;
    reconsider A2=A1,B2=B1 as Element of REAL 2 by EUCLID:22;
    Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7;
    then dist(a1,b1)=(Pitag_dist 2).(A1,B1) by METRIC_1:def 1
    .=|. A2 - B2 .| by EUCLID:def 6
    .= 2 * r by A28;
    hence contradiction by A14,A20,A24,A13;
  end;
  hence thesis by A3,A12,A1,TBSP_1:def 8;
end;
