reserve p,p1,p2,p3,p11,p22,q,q1,q2 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r,r1,r2,s,s1,s2 for Real,
  u,u1,u2,u5 for Point of Euclid 2,
  n,m,i,j,k for Nat,
  N for Nat,
  x,y,z for set;
reserve lambda for Real;

theorem
  L~(f|i) c= L~f
proof
  set h = f|i, Mh = {LSeg(h,n) : 1<=n & n+1<=len h}, Mf = {LSeg(f,m) : 1<=m &
  m+1<=len f};
  let x be object;
A1: h = f|Seg i by FINSEQ_1:def 16;
A2: dom f = Seg len f by FINSEQ_1:def 3;
  assume
A3: x in L~h;
  then consider X be set such that
A4: x in X and
A5: X in Mh by TARSKI:def 4;
  consider k such that
A6: X = LSeg(h,k) and
A7: 1<=k and
A8: k+1<=len h by A5;
  per cases;
  suppose
A9: i in dom f;
A10: dom h=dom f /\ Seg i by A1,RELAT_1:61;
A11: i in NAT by ORDINAL1:def 12;
A12: i<=len f by A9,FINSEQ_3:25;
    then Seg i c= dom f by A2,FINSEQ_1:5;
    then dom h = Seg i by A10,XBOOLE_1:28;
    then len h <= len f by A12,A11,FINSEQ_1:def 3;
    then
A13: k+1<=len f by A8,XXREAL_0:2;
    k <= k+1 by NAT_1:12;
    then k <= len h by A8,XXREAL_0:2;
    then
A14: k in dom h by A7,FINSEQ_3:25;
    1<=k+1 by NAT_1:12;
    then k+1 in dom h by A8,FINSEQ_3:25;
    then X = LSeg(f,k) by A6,A14,Th17;
    then X in Mf by A7,A13;
    hence thesis by A4,TARSKI:def 4;
  end;
  suppose
A15: not i in dom f;
    now
      per cases by A15,FINSEQ_3:25;
      case
        i<1;
        then i<0+1;
        then i<=0 by NAT_1:13;
        then Seg i = {};
        then dom h = dom f /\ {} by A1,RELAT_1:61;
        then dom h = {};
        then Seg len h = {} by FINSEQ_1:def 3;
        then len h = 0;
        hence contradiction by A3,TOPREAL1:22;
      end;
      case
        len f<i;
        then Seg len f c= Seg i by FINSEQ_1:5;
        then dom f c= Seg i by FINSEQ_1:def 3;
        then
A16:    dom f = dom f /\ (Seg i) by XBOOLE_1:28;
        for x being object st x in dom h holds h.x = f.x by A1,FUNCT_1:47;
        then h = f by A1,A16,RELAT_1:61;
        hence thesis by A4,A5,TARSKI:def 4;
      end;
    end;
    hence thesis;
  end;
end;
