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 Th50:
  p1,p2 split P implies p2,p1 split P
proof
  assume
A1: p1 <> p2;
  given f1,f2 being S-Sequence_in_R2 such that
A2: p1 = f1/.1 and
A3: p1 = f2/.1 and
A4: p2 = f1/.len f1 and
A5: p2 = f2/.len f2 and
A6: L~f1 /\ L~f2 = {p1,p2} and
A7: P = L~f1 \/ L~f2;
  thus p2 <> p1 by A1;
  reconsider g1 = Rev f1, g2 = Rev f2 as S-Sequence_in_R2;
  take g1, g2;
A8: len g2 = len f2 by FINSEQ_5:def 3;
  len g1 = len f1 by FINSEQ_5:def 3;
  hence
  p2 = g1/.1 & p2 = g2/.1 & p1 = g1/.len g1 & p1 = g2/.len g2 by A2,A3,A4,A5,A8
,FINSEQ_5:65;
  L~g1 = L~f1 by Th22;
  hence thesis by A6,A7,Th22;
end;
