
theorem Thm27:
  for A,B,C be Point of TOP-REAL 2 holds
  the_area_of_polygon3(A,B,C)=the_area_of_polygon3(B,C,A) &
  the_area_of_polygon3(A,B,C)=the_area_of_polygon3(C,A,B)
  proof
    let A,B,C be Point of TOP-REAL 2;
A1: the_area_of_polygon3(A,B,C)= ((A`1*B`2-B`1*
    A`2)+(B`1*C`2-C`1*B`2)+(C`1*A`2-A`1*C`2))/2 by EUCLID_6:def 1;
    the_area_of_polygon3(B,C,A)= ((B`1*C`2-C`1*
    B`2)+(C`1*A`2-A`1*C`2)+(A`1*B`2-B`1*A`2))/2 by EUCLID_6:def 1;
    hence the_area_of_polygon3(A,B,C)=the_area_of_polygon3(B,C,A) by A1;
    the_area_of_polygon3(C,A,B)= ((C`1*A`2-A`1*
    C`2)+(A`1*B`2-B`1*A`2)+(B`1*C`2-C`1*B`2))/2 by EUCLID_6:def 1;
    hence thesis by A1;
  end;
