reserve n,i,k,m for Nat;
reserve r,r1,r2,s,s1,s2 for Real;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL 2,
  f,f1,f2 for FinSequence of the carrier of TOP-REAL 2,
  p,p1,p2,p3,q,q3 for Point of TOP-REAL 2;

theorem Th20:
  f is one-to-one & 1 <= i & i+1 <= len f implies LSeg(f,i) is non trivial
proof
  assume
A1: f is one-to-one;
A2: i <> i+1;
  assume
A3: 1 <= i & i+1 <= len f;
  then i in dom f & i+1 in dom f by SEQ_4:134;
  then
A4: f/.i<>f/.(i+1) by A1,A2,PARTFUN2:10;
A5: f/.i in LSeg(f/.i,f/.(i+1)) & f/.(i+1) in LSeg(f/.i,f/.(i+1)) by
RLTOPSP1:68;
  LSeg(f/.i,f/.(i+1)) = LSeg(f,i) by A3,TOPREAL1:def 3;
  hence thesis by A4,A5,ZFMISC_1:def 10;
end;
