reserve p1,p2,p3,p4,p5,p6,p,pc for Point of TOP-REAL 2;
reserve a,b,c,r,s for Real;

theorem Th32:
  p1 in circle(a,b,r) & r>0 implies ex p2 st p1<>p2 & p2 in circle
  (a,b,r) & |[a,b]| in LSeg(p1,p2)
proof
  set pc = |[a,b]|;
  set p2 = 2*pc - p1;
  assume
A1: p1 in circle(a,b,r);
  then
A2: |. p1 - pc .| = r by TOPREAL9:43;
  assume
A3: r>0;
  take p2;
  thus p1<>p2
  proof
    assume p1=p2;
    then 1*p1 + p1 = 2*pc -p1 + p1 by RLVECT_1:def 8;
    then 1*p1 + 1*p1 = 2*pc -p1 + p1 by RLVECT_1:def 8;
    then (1+1)*p1 = 2*pc -p1 +p1 by RLVECT_1:def 6;
    then 2*p1 = 2*pc -(p1-p1) by RLVECT_1:29;
    then 2*p1 = 2*pc -0.TOP-REAL 2 by RLVECT_1:5;
    then 2*p1 = 2*pc + 0.TOP-REAL 2 by RLVECT_1:12;
    then 2*p1 = 2*pc by RLVECT_1:4;
    then p1 = pc by RLVECT_1:36;
    then |. pc - pc .| = r by A1,TOPREAL9:43;
    hence contradiction by A3,Lm1;
  end;
  |. p2 - pc .| = |. 2*pc - p1 - pc .| .= |. 2*pc + -p1 - pc .|
    .= |. 2*pc +(-pc) +(-p1) .| by RLVECT_1:def 3
    .= |. 2*pc +((-1)*pc) +(-p1) .|
    .= |. (2+(-1))*pc +(-p1) .| by RLVECT_1:def 6
    .= |. pc - p1 .| by RLVECT_1:def 8
    .= r by A2,Lm2;
  hence p2 in circle(a,b,r) by TOPREAL9:43;
  (1-1/2)*p1+(1/2)*p2 = (1/2)*(p1+p2) by RLVECT_1:def 5
    .= (1/2)*(p1+(-p1)+2*pc) by RLVECT_1:def 3
    .= (1/2)*(0.TOP-REAL 2+2*pc) by RLVECT_1:5
    .= (1/2)*(2*pc) by RLVECT_1:4
    .= ((1/2)*2)*pc by RLVECT_1:def 7
    .= pc by RLVECT_1:def 8;
  hence thesis;
end;
