let f be FinSequence of the carrier of (TOP-REAL 2); :: thesis: { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } is Subset-Family of (TOP-REAL 2)
set F = { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } ;
{ (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } c= bool (REAL 2)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } or x in bool (REAL 2) )
assume x in { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } ; :: thesis: x in bool (REAL 2)
then consider i being Element of NAT such that
A1: LSeg f,i = x and
( 1 <= i & i + 1 <= len f ) ;
x is Subset of (REAL 2) by A1, EUCLID:25;
hence x in bool (REAL 2) ; :: thesis: verum

end;

hence { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } is Subset-Family of (TOP-REAL 2) by EUCLID:25; :: thesis: verum