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 Th7:
  for i being Nat holds 1 <= i implies LSeg(f^g,len f + i) = LSeg(g ,i)
proof
  let i be Nat;
  assume
A1: 1 <= i;
  per cases;
  suppose
A2: i+1 <= len g;
    len f +(i+1) = len f+i+1;
    then len f +i+1 <= len f + len g by A2,XREAL_1:6;
    then
A3: len f+i+1 <= len(f^g) by FINSEQ_1:22;
    i <= len f+i by NAT_1:11;
    then
A4: 1 <= len f+i by A1,XXREAL_0:2;
A5: i+1 in dom g by A1,A2,SEQ_4:134;
A6: i in dom g by A1,A2,SEQ_4:134;
    thus LSeg(g,i) = LSeg(g/.i,g/.(i+1)) by A1,A2,TOPREAL1:def 3
      .= LSeg((f^g)/.(len f+i),g/.(i+1)) by A6,FINSEQ_4:69
      .= LSeg((f^g)/.(len f+i), (f^g)/.(len f + (i+1))) by A5,FINSEQ_4:69
      .= LSeg(f^g,len f + i) by A4,A3,TOPREAL1:def 3;
  end;
  suppose
A7: i+1 > len g;
    then len f + (i + 1) > len f + len g by XREAL_1:6;
    then len f + i + 1 > len(f^g) by FINSEQ_1:22;
    hence LSeg(f^g,len f + i) = {} by TOPREAL1:def 3
      .= LSeg(g,i) by A7,TOPREAL1:def 3;
  end;
end;
