reserve p,p1,p2,q for Point of TOP-REAL 2,
  f,f1,f2,g,g1,g2 for FinSequence of TOP-REAL 2,
  r,s for Real,

  n,m,i,j,k for Nat,
  G for Go-board,
  x for set;

theorem
  len f1 < n & n+1 <= len (f1^f2) & m+len f1 = n implies LSeg(f1^f2,n) =
  LSeg(f2,m)
proof
  set f = f1^f2;
  assume that
A1: len f1 < n and
A2: n+1 <= len f and
A3: m+len f1 = n;
A4: 1<=m by A1,A3,NAT_1:19;
  reconsider p=f/.n, q=f/.(n+1) as Point of TOP-REAL 2;
A5: n+1 = m+1+len f1 by A3;
  len f=len f1+len f2 by FINSEQ_1:22;
  then
A6: m+1<=len f2 by A2,A5,XREAL_1:6;
  then
A7: f/.(n+1)=f2/.(m+1) by A5,NAT_1:11,SEQ_4:136;
  m <= m + 1 by NAT_1:11;
  then m <= len f2 by A6,XXREAL_0:2;
  then
A8: f/.n=f2/.m by A3,A4,SEQ_4:136;
  0+1<=n by A1,NAT_1:13;
  hence LSeg(f,n)=LSeg(p,q) by A2,TOPREAL1:def 3
    .=LSeg(f2,m) by A4,A6,A8,A7,TOPREAL1:def 3;
end;
