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

theorem Th30:
  p1 in circle(a,b,r) & p3 in circle(a,b,r) & p4 in circle(a,b,r)
  & p in LSeg(p1,p3) & p in LSeg(p1,p4) & p3<>p4 implies p=p1
proof
  assume
A1: p1 in circle(a,b,r);
  assume
A2: p3 in circle(a,b,r);
  assume
A3: p4 in circle(a,b,r);
  assume
A4: p in LSeg(p1,p3);
  assume
A5: p in LSeg(p1,p4);
  assume
A6: p3<>p4;
  per cases;
  suppose
A7: p1=p3 or p1=p4;
    per cases by A7;
    suppose
      p1=p3;
      then p in {p1} by A4,RLTOPSP1:70;
      hence thesis by TARSKI:def 1;
    end;
    suppose
      p1=p4;
      then p in {p1} by A5,RLTOPSP1:70;
      hence thesis by TARSKI:def 1;
    end;
  end;
  suppose
A8: p1<>p3 & p1<>p4;
    per cases;
    suppose
A9:   p<>p1;
A10:  inside_of_circle(a,b,r) misses circle(a,b,r) by TOPREAL9:54;
      per cases by A4,A5,A6,A9,Th12;
      suppose
A11:    p3 in LSeg(p1,p4);
        not p3 in {p1,p4} by A6,A8,TARSKI:def 2;
        then
A12:    p3 in LSeg(p1,p4) \ {p1,p4} by A11,XBOOLE_0:def 5;
        LSeg(p1,p4) \ {p1,p4} c= inside_of_circle(a,b,r) by A1,A3,TOPREAL9:60;
        then p3 in inside_of_circle(a,b,r) /\ circle(a,b,r) by A2,A12,
XBOOLE_0:def 4;
        hence thesis by A10,XBOOLE_0:def 7;
      end;
      suppose
A13:    p4 in LSeg(p1,p3);
        not p4 in {p1,p3} by A6,A8,TARSKI:def 2;
        then
A14:    p4 in LSeg(p1,p3) \ {p1,p3} by A13,XBOOLE_0:def 5;
        LSeg(p1,p3) \ {p1,p3} c= inside_of_circle(a,b,r) by A1,A2,TOPREAL9:60;
        then p4 in inside_of_circle(a,b,r) /\ circle(a,b,r) by A3,A14,
XBOOLE_0:def 4;
        hence thesis by A10,XBOOLE_0:def 7;
      end;
    end;
    suppose
      p=p1;
      hence thesis;
    end;
  end;
end;
