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 Th8:
  f is non empty & g is non empty implies LSeg(f^g,len f) = LSeg(f
  /.len f,g/.1)
proof
  assume that
A1: f is non empty and
A2: g is non empty;
A3: 1 in dom g by A2,FINSEQ_5:6;
  then 1 <= len g by FINSEQ_3:25;
  then len f + 1 <= len f + len g by XREAL_1:6;
  then
A4: len f + 1 <= len(f^g) by FINSEQ_1:22;
A5: len f in dom f by A1,FINSEQ_5:6;
  then 1 <= len f by FINSEQ_3:25;
  hence LSeg(f^g,len f) = LSeg((f^g)/.(len f),(f^g)/.(len f+1)) by A4,
TOPREAL1:def 3
    .= LSeg(f/.len f,(f^g)/.(len f+1)) by A5,FINSEQ_4:68
    .= LSeg(f/.len f,g/.1) by A3,FINSEQ_4:69;
end;
