reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem
  for f being FinSequence of TOP-REAL n st 2<=len f holds f.1 in L~f & f
  /.1 in L~f & f.len f in L~f & f/.len f in L~f
proof
  let f be FinSequence of TOP-REAL n;
  assume
A1: 2<=len f;
  then
A2: 1+1<=len f;
  then
A3: LSeg(f,1) in {LSeg(f,i):1<=i & i+1<=len f};
  f/.1 in LSeg(f/.1,f/.(1+1)) by RLTOPSP1:68;
  then f/.1 in LSeg(f,1) by A1,TOPREAL1:def 3;
  then f/.1 in union{LSeg(f,i):1<=i & i+1<=len f} by A3,TARSKI:def 4;
  then
A4: f/.1 in L~f by TOPREAL1:def 4;
A5: len f-'1+1=len f by A2,NAT_D:46,XREAL_1:235;
A6: 1<=len f-'1 by A2,NAT_D:49;
  then
A7: LSeg(f,len f-'1) in {LSeg(f,i):1<=i & i+1<=len f} by A5;
  f/.len f in LSeg(f/.(len f-'1),f/.(len f-'1+1)) by A5,RLTOPSP1:68;
  then f/.len f in LSeg(f,len f-'1) by A6,A5,TOPREAL1:def 3;
  then f/.len f in union{LSeg(f,i):1<=i & i+1<=len f} by A7,TARSKI:def 4;
  then
A8: f/.len f in L~f by TOPREAL1:def 4;
  1<=len f by A2,NAT_D:46;
  hence thesis by A4,A8,FINSEQ_4:15;
end;
