let f be FinSequence of the carrier of (TOP-REAL 2); :: thesis: for k being Nat holds { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } is finite
let k be Nat; :: thesis: { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } is finite
set F = { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } ;
set F1 = { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f ) } ;
{ (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } c= { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f ) }

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } or x in { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f ) } )
assume x in { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } ; :: thesis: x in { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f ) }
then ex i being Nat st
( LSeg (f,i) = x & 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) ;
hence x in { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f ) } ; :: thesis: verum

end;

hence { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } is finite by Th23, FINSET_1:1; :: thesis: verum