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

theorem Th21:
  p1,p2,p3 is_a_triangle & p4,p5,p6 is_a_triangle & angle(p1,p2,p3
) = angle(p4,p5,p6) & angle(p3,p1,p2) = angle(p6,p4,p5) implies |.p3-p2.|*|.p4-
p6.| = |.p1-p3.|*|.p6-p5.| & |.p3-p2.|*|.p5-p4.| = |.p2-p1.|*|.p6-p5.| & |.p1-
  p3.|*|.p5-p4.| = |.p2-p1.|*|.p4-p6.|
proof
  assume p1,p2,p3 is_a_triangle;
  then
A1: p1,p2,p3 are_mutually_distinct by Th20;
  then
A2: p3<>p2 by ZFMISC_1:def 5;
A3: p2<>p1 by A1,ZFMISC_1:def 5;
  then
A4: euc2cpx(p2)<> euc2cpx(p1) by EUCLID_3:4;
A5: p3<>p1 by A1,ZFMISC_1:def 5;
  then
A6: euc2cpx(p3)<> euc2cpx(p1) by EUCLID_3:4;
  assume
A7: p4,p5,p6 is_a_triangle;
  then
A8: angle(p4,p5,p6)<>PI & angle(p5,p6,p4)<>PI by Th20;
A9: p4,p5,p6 are_mutually_distinct by A7,Th20;
  then
A10: p5<>p6 by ZFMISC_1:def 5;
  then
A11: euc2cpx(p5)<> euc2cpx(p6) by EUCLID_3:4;
A12: p6<>p4 by A9,ZFMISC_1:def 5;
  then
A13: euc2cpx(p6)<> euc2cpx(p4) by EUCLID_3:4;
A14: p5<>p4 by A9,ZFMISC_1:def 5;
  then
A15: euc2cpx(p5)<> euc2cpx(p4) by EUCLID_3:4;
  assume
A16: angle(p1,p2,p3) = angle(p4,p5,p6) & angle(p3,p1,p2) = angle(p6,p4, p5);
A17: euc2cpx(p3)<> euc2cpx(p2) by A2,EUCLID_3:4;
A18: angle(p2,p3,p1) = angle(p5,p6,p4)
  proof
    per cases by A17,A6,A4,A11,A15,A13,COMPLEX2:88;
    suppose
      angle(p3,p1,p2)+angle(p1,p2,p3)+angle(p2,p3,p1) = PI & angle(p6
      ,p4,p5)+angle(p4,p5,p6)+angle(p5,p6,p4) = PI;
      hence thesis by A16;
    end;
    suppose
      angle(p3,p1,p2)+angle(p1,p2,p3)+angle(p2,p3,p1) = 5*PI & angle(
      p6,p4,p5)+angle(p4,p5,p6)+angle(p5,p6,p4) = 5*PI;
      hence thesis by A16;
    end;
    suppose
A19:  angle(p3,p1,p2)+angle(p1,p2,p3)+angle(p2,p3,p1) = PI & angle(p6
      ,p4,p5)+angle(p4,p5,p6)+angle(p5,p6,p4) = 5*PI;
      angle(p2,p3,p1)>=0 & -angle(p5,p6,p4) > -2*PI by COMPLEX2:70,XREAL_1:24;
      then
A20:  angle(p2,p3,p1)+(-angle(p5,p6,p4)) > 0+(-2*PI) by XREAL_1:8;
      angle(p2,p3,p1)-angle(p5,p6,p4) = -4*PI by A16,A19;
      then 4*PI < 2*PI by A20,XREAL_1:24;
      then 4*PI/PI < 2*PI/PI by XREAL_1:74;
      then 4 < 2*PI/PI by XCMPLX_1:89;
      then 4 < 2 by XCMPLX_1:89;
      hence thesis;
    end;
    suppose
A21:  angle(p3,p1,p2)+angle(p1,p2,p3)+angle(p2,p3,p1)=5*PI & angle(p6
      ,p4,p5)+angle(p4,p5,p6)+angle(p5,p6,p4)=PI;
      angle(p2,p3,p1)<2*PI & angle(p5,p6,p4)>=0 by COMPLEX2:70;
      then angle(p2,p3,p1)+(-angle(p5,p6,p4)) < 2*PI+(-0) by XREAL_1:8;
      then 4*PI/PI < 2*PI/PI by A16,A21,XREAL_1:74;
      then 4 < 2*PI/PI by XCMPLX_1:89;
      then 4 < 2 by XCMPLX_1:89;
      hence thesis;
    end;
  end;
A22: angle(p6,p4,p5)<>PI by A7,Th20;
  hence |.p3-p2.|*|.p4-p6.| = |.p1-p3.|*|.p6-p5.| by A2,A5,A3,A8,A10,A14,A12
,A16,A18,Lm18;
  thus |.p3-p2.|*|.p5-p4.| = |.p2-p1.|*|.p6-p5.| by A2,A5,A3,A8,A22,A10,A14,A12
,A16,A18,Lm18;
  thus thesis by A2,A5,A3,A8,A22,A10,A14,A12,A16,A18,Lm18;
end;
