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

theorem Th10:
  p in LSeg(p2,p3) & p<>p2 implies angle(p1,p2,p3) = angle(p1,p2,p )
proof
  assume
A1: p in LSeg(p2,p3);
  assume
A2: p<>p2;
  then
A3: angle(p3,p2,p1) = angle(p,p2,p1) by A1,Th9;
  per cases;
  suppose
A4: angle(p1,p2,p3) = 0;
    then angle(p3,p2,p1) = 0 by EUCLID_3:36;
    then
A5: angle(p,p2,p1) = 0 by A1,A2,Th9;
    thus angle(p1,p2,p3) = angle(p3,p2,p1) by A4,EUCLID_3:36
      .= angle(p,p2,p1) by A1,A2,Th9
      .= angle(p1,p2,p) by A5,EUCLID_3:36;
  end;
  suppose
A6: angle(p1,p2,p3) <> 0;
    then
A7: angle(p,p2,p1) <> 0 by A3,EUCLID_3:36;
    thus angle(p1,p2,p3) = 2*PI - angle(p3,p2,p1) by A6,EUCLID_3:38
      .= 2*PI - angle(p,p2,p1) by A1,A2,Th9
      .= angle(p1,p2,p) by A7,EUCLID_3:37;
  end;
end;
