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

theorem Th22:
  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(p5,p6,p4) implies |.p2-p3.| * |.
p4-p6.| = |.p3-p1.| * |.p5-p4.| & |.p2-p3.| * |.p6-p5.| = |.p1-p2.| * |.p5-p4.|
  & |.p3-p1.| * |.p6-p5.| = |.p1-p2.| * |.p4-p6.|
proof
  assume
A1: p1,p2,p3 is_a_triangle;
  then
A2: p1,p2,p3 are_mutually_distinct by Th20;
  then
A3: p3<>p2 by ZFMISC_1:def 5;
A4: angle(p3,p1,p2)<>PI by A1,Th20;
A5: p3<>p1 by A2,ZFMISC_1:def 5;
  then
A6: euc2cpx(p3)<> euc2cpx(p1) by EUCLID_3:4;
A7: p2<>p1 by A2,ZFMISC_1:def 5;
  then
A8: euc2cpx(p2)<> euc2cpx(p1) by EUCLID_3:4;
  assume
A9: p4,p5,p6 is_a_triangle;
  then
A10: p4,p5,p6 are_mutually_distinct by Th20;
  then
A11: p4<>p5 by ZFMISC_1:def 5;
  then
A12: euc2cpx(p4)<> euc2cpx(p5) by EUCLID_3:4;
A13: p5<>p6 by A10,ZFMISC_1:def 5;
  then
A14: euc2cpx(p5)<> euc2cpx(p6) by EUCLID_3:4;
A15: angle(p6,p4,p5)<>PI by A9,Th20;
A16: p4<>p6 by A10,ZFMISC_1:def 5;
  then
A17: euc2cpx(p4)<> euc2cpx(p6) by EUCLID_3:4;
  assume
A18: angle(p1,p2,p3) = angle(p4,p5,p6) & angle(p3,p1,p2) = angle(p5,p6, p4);
A19: euc2cpx(p3)<> euc2cpx(p2) by A3,EUCLID_3:4;
A20: angle(p2,p3,p1) = angle(p6,p4,p5)
  proof
    per cases by A19,A6,A8,A12,A17,A14,COMPLEX2:88;
    suppose
      angle(p3,p1,p2)+angle(p1,p2,p3)+angle(p2,p3,p1) = PI & angle(p5
      ,p6,p4)+angle(p6,p4,p5)+angle(p4,p5,p6) = PI;
      hence thesis by A18;
    end;
    suppose
      angle(p3,p1,p2)+angle(p1,p2,p3)+angle(p2,p3,p1) = 5*PI & angle(
      p5,p6,p4)+angle(p6,p4,p5)+angle(p4,p5,p6) = 5*PI;
      hence thesis by A18;
    end;
    suppose
A21:  angle(p3,p1,p2)+angle(p1,p2,p3)+angle(p2,p3,p1) = PI & angle(p5
      ,p6,p4)+angle(p6,p4,p5)+angle(p4,p5,p6) = 5*PI;
      angle(p2,p3,p1)>=0 & -angle(p6,p4,p5) > -2*PI by COMPLEX2:70,XREAL_1:24;
      then
A22:  angle(p2,p3,p1)+(-angle(p6,p4,p5)) > 0+(-2*PI) by XREAL_1:8;
      angle(p2,p3,p1)-angle(p6,p4,p5) = -4*PI by A18,A21;
      then 4*PI < 2*PI by A22,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
A23:  angle(p3,p1,p2)+angle(p1,p2,p3)+angle(p2,p3,p1)=5*PI & angle(p5
      ,p6,p4)+angle(p6,p4,p5)+angle(p4,p5,p6)=PI;
      angle(p2,p3,p1)<2*PI & angle(p6,p4,p5)>=0 by COMPLEX2:70;
      then angle(p2,p3,p1)+(-angle(p6,p4,p5)) < 2*PI+(-0) by XREAL_1:8;
      then 4*PI/PI < 2*PI/PI by A18,A23,XREAL_1:74;
      then 4 < 2*PI/PI by XCMPLX_1:89;
      then 4 < 2 by XCMPLX_1:89;
      hence thesis;
    end;
  end;
  angle(p1,p2,p3)<>PI & angle(p2,p3,p1)<>PI by A1,Th20;
  hence |.p2-p3.|*|.p4-p6.| = |.p3-p1.|*|.p5-p4.| by A4,A3,A5,A7,A15,A11,A16
,A13,A18,Lm19;
A24: angle(p4,p5,p6)<>PI & angle(p5,p6,p4)<>PI by A9,Th20;
  hence |.p2-p3.|*|.p6-p5.| = |.p1-p2.|*|.p5-p4.| by A3,A5,A7,A15,A11,A16,A13
,A18,A20,Lm19;
  thus thesis by A3,A5,A7,A24,A15,A11,A16,A13,A18,A20,Lm19;
end;
