let f, g be FinSequence of (TOP-REAL 2); :: thesis: ( f,g are_in_general_position implies (L~ f) /\ (L~ g) is finite )
assume A1: f,g are_in_general_position ; :: thesis: (L~ f) /\ (L~ g) is finite
set BL = { (LSeg g,j) where j is Element of NAT : ( 1 <= j & j + 1 <= len g ) } ;
set AL = { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } ;

A2: now

let Z be set ; :: thesis: ( Z in INTERSECTION { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } ,{ (LSeg g,j) where j is Element of NAT : ( 1 <= j & j + 1 <= len g ) } implies Z is finite )
assume Z in INTERSECTION { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } ,{ (LSeg g,j) where j is Element of NAT : ( 1 <= j & j + 1 <= len g ) } ; :: thesis: Z is finite
then consider X, Y being set such that
A3: ( X in { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } & Y in { (LSeg g,j) where j is Element of NAT : ( 1 <= j & j + 1 <= len g ) } ) and
A4: Z = X /\ Y by SETFAM_1:def 5;
( ex i being Element of NAT st
( X = LSeg f,i & 1 <= i & i + 1 <= len f ) & ex j being Element of NAT st
( Y = LSeg g,j & 1 <= j & j + 1 <= len g ) ) by A3;
hence Z is finite by A1, A4, Th10; :: thesis: verum

end;

( (L~ f) /\ (L~ g) = union (INTERSECTION { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } ,{ (LSeg g,j) where j is Element of NAT : ( 1 <= j & j + 1 <= len g ) } ) & INTERSECTION { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } ,{ (LSeg g,j) where j is Element of NAT : ( 1 <= j & j + 1 <= len g ) } is finite ) by Th11, SETFAM_1:39;
hence (L~ f) /\ (L~ g) is finite by A2, FINSET_1:25; :: thesis: verum