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 Th18:
  len f >= 2 implies rng f c= L~f
proof
  assume
A1: len f >= 2;
  let x be object;
  assume x in rng f;
  then consider i being Element of NAT such that
A2: i in dom f and
A3: f/.i = x by PARTFUN2:2;
A4: 1 <= i by A2,FINSEQ_3:25;
A5: i <= len f by A2,FINSEQ_3:25;
A6: f/.i in LSeg(f/.i,f/.(i+1)) by RLTOPSP1:68;
  per cases;
  suppose
A7: i = len f;
    then consider j be Nat such that
A8: j+1 = i by A1,NAT_1:6;
    reconsider j as Element of NAT by ORDINAL1:def 12;
    1+1 <= j+1 by A1,A7,A8;
    then
A9: 1 <= j by XREAL_1:6;
    f/.(j+1) in LSeg(f/.j,f/.(j+1)) by RLTOPSP1:68;
    hence thesis by A3,A7,A8,A9,Th15;
  end;
  suppose
    i <> len f;
    then i < len f by A5,XXREAL_0:1;
    then i+1 <= len f by NAT_1:13;
    hence thesis by A3,A4,A6,Th15;
  end;
end;
