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

theorem Th6:
  p2<>p1 implies |.p3-p2.| * sin angle(p3,p2,p1) = |.p3-p1.| * sin
  angle(p2,p1,p3)
proof
  the_area_of_polygon3(p1,p2,p3) = the_area_of_polygon3(p3,p1,p2);
  then
  |.p1-p2.|*|.p3-p2.|*sin angle(p3,p2,p1) / 2 = the_area_of_polygon3(p3,p1
  ,p2) by Th5;
  then |.p1-p2.|*|.p3-p2.|*sin angle(p3,p2,p1) / 2 = |.p3-p1.|*|.p2-p1.|*sin
  angle(p2,p1,p3) / 2 by Th5;
  then
  |.p1-p2.|*|.p3-p2.|*sin angle(p3,p2,p1) = |.p3-p1.|*|.p1-p2.|*sin angle(
  p2,p1,p3) by Lm2;
  then
A1: |.p3-p2.|*sin angle(p3,p2,p1)*|.p1-p2.| = |.p3-p1.|*sin angle(p2,p1,p3)*
  |.p1-p2.|;
  assume p2<>p1;
  then |.p1-p2.|<>0 by Lm1;
  hence thesis by A1,XCMPLX_1:5;
end;
