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

theorem Th23:
  p1,p2,p3 are_mutually_distinct & angle(p1,p2,p3)<=PI implies
  angle(p2,p3,p1)<=PI & angle(p3,p1,p2)<=PI
proof
A1: angle(p1,p2,p3)>=0 by COMPLEX2:70;
  assume
A2: p1,p2,p3 are_mutually_distinct;
  then p1<>p3 by ZFMISC_1:def 5;
  then
A3: euc2cpx(p1)<> euc2cpx(p3) by EUCLID_3:4;
  p2<>p3 by A2,ZFMISC_1:def 5;
  then
A4: euc2cpx(p2)<> euc2cpx(p3) by EUCLID_3:4;
  p1<>p2 by A2,ZFMISC_1:def 5;
  then euc2cpx(p1)<> euc2cpx(p2) by EUCLID_3:4;
  then
A5: angle(p1,p2,p3)+angle(p2,p3,p1)+angle(p3,p1,p2)=PI or angle(p1,p2,p3)+
  angle(p2,p3,p1)+angle(p3,p1,p2)=5*PI by A3,A4,COMPLEX2:88;
  angle(p2,p3,p1)< 2*PI & angle(p3,p1,p2)<2*PI by COMPLEX2:70;
  then
A6: angle(p2,p3,p1)+angle(p3,p1,p2)<2*PI+2*PI by XREAL_1:8;
  assume angle(p1,p2,p3)<=PI;
  then
A7: angle(p1,p2,p3)+(angle(p2,p3,p1)+angle(p3,p1,p2))<PI+4*PI by A6,XREAL_1:8;
A8: angle(p3,p1,p2)>=0 by COMPLEX2:70;
  thus angle(p2,p3,p1)<=PI
  proof
    assume angle(p2,p3,p1)>PI;
    then angle(p1,p2,p3)+angle(p2,p3,p1)>0+PI by A1,XREAL_1:8;
    hence contradiction by A5,A7,A8,XREAL_1:8;
  end;
A9: angle(p2,p3,p1)>= 0 by COMPLEX2:70;
  thus angle(p3,p1,p2)<=PI
  proof
    assume angle(p3,p1,p2)>PI;
    then angle(p2,p3,p1)+angle(p3,p1,p2)>0+PI by A9,XREAL_1:8;
    hence contradiction by A5,A7,A1,XREAL_1:8;
  end;
end;
