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

theorem Th20:
  p1,p2,p3 is_a_triangle iff p1,p2,p3 are_mutually_distinct &
  angle(p1,p2,p3)<>PI & angle(p2,p3,p1)<>PI & angle(p3,p1,p2)<>PI
proof
  hereby
    assume
A1: p1,p2,p3 is_a_triangle;
    then
A2: not p2 in LSeg(p3,p1) by TOPREAL9:67;
    then
A3: p2<>p3 by RLTOPSP1:68;
A4: not p1 in LSeg(p2,p3) by A1,TOPREAL9:67;
    then p1<>p2 & p1<>p3 by RLTOPSP1:68;
    hence p1,p2,p3 are_mutually_distinct by A3,ZFMISC_1:def 5;
    not p3 in LSeg(p1,p2) by A1,TOPREAL9:67;
    hence
    angle(p1,p2,p3)<>PI & angle(p2,p3,p1)<>PI & angle(p3,p1,p2)<>PI by A4,A2
,Th11;
  end;
  assume
A5: p1,p2,p3 are_mutually_distinct;
  then
A6: p1<>p2 by ZFMISC_1:def 5;
  assume
A7: angle(p1,p2,p3)<>PI;
A8: p1<>p3 by A5,ZFMISC_1:def 5;
A9: p2<>p3 by A5,ZFMISC_1:def 5;
  assume angle(p2,p3,p1)<>PI;
  then
A10: not p3 in LSeg(p2,p1) by A8,A9,Th8;
  assume angle(p3,p1,p2)<>PI;
  then
A11: not p1 in LSeg(p3,p2) by A6,A8,Th8;
  not p2 in LSeg(p1,p3) by A6,A9,A7,Th8;
  hence thesis by A10,A11,TOPREAL9:67;
end;
