reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th55:
  r1<>r2 & r19<>r29 implies [.r1,r2,r19,r29.] is special_polygonal
proof
  assume that
A1: r1<>r2 and
A2: r19<>r29;
  set p1 = |[r1,r19]|, p2 = |[r1,r29]| , p3 = |[r2,r29]|, p4 = |[r2,r19]|;
A3: p3`1 = r2 by EUCLID:52;
  take p1,p3;
  thus p1 <> p3 by A1,FINSEQ_1:77;
A4: p4`1 = r2 by EUCLID:52;
A5: p3`2 = r29 by EUCLID:52;
A6: p4`2 = r19 by EUCLID:52;
  set f1 = <* p1,p2,p3 *>, f2 = <* p1,p4,p3 *>;
A7: p1`1 = r1 by EUCLID:52;
A8: len f2 = 3 by FINSEQ_1:45;
A9: len f1 = 3 by FINSEQ_1:45;
A10: p1`2 = r19 by EUCLID:52;
  then reconsider f1,f2 as S-Sequence_in_R2 by A1,A2,A7,A3,A5,TOPREAL3:34,35;
  take f1,f2;
  thus p1 = f1/.1 & p1 = f2/.1 & p3 = f1/.len f1 & p3 = f2/.len f2 by A9,A8,
FINSEQ_4:18;
A11: L~f2 = LSeg(p3,p4) \/ LSeg(p4,p1) by TOPREAL3:16;
  L~f1 = LSeg(p1,p2) \/ LSeg(p2,p3) by TOPREAL3:16;
  hence
  L~f1 /\ L~f2 = ((LSeg(p1,p2) \/ LSeg(p2,p3)) /\ LSeg(p3,p4)) \/ ((LSeg(
  p1,p2) \/ LSeg(p2,p3)) /\ LSeg(p4,p1)) by A11,XBOOLE_1:23
    .= ((LSeg(p2,p1) \/ LSeg(p3,p2)) /\ LSeg(p4,p3)) \/ {p1} by A2,A7,A10,A5,A6
,TOPREAL3:27
    .= {p3} \/ {p1} by A1,A7,A3,A5,A4,TOPREAL3:28
    .= {p1,p3} by ENUMSET1:1;
  thus thesis by A11,TOPREAL3:16;
end;
