let n be Nat; :: thesis: for P being Subset of ()
for p1, p2 being Point of () st P is_an_arc_of p1,p2 holds
ex p3 being Point of () st
( p3 in P & p3 <> p1 & p3 <> p2 )

let P be Subset of (); :: thesis: for p1, p2 being Point of () st P is_an_arc_of p1,p2 holds
ex p3 being Point of () st
( p3 in P & p3 <> p1 & p3 <> p2 )

let p1, p2 be Point of (); :: thesis: ( P is_an_arc_of p1,p2 implies ex p3 being Point of () st
( p3 in P & p3 <> p1 & p3 <> p2 ) )

assume P is_an_arc_of p1,p2 ; :: thesis: ex p3 being Point of () st
( p3 in P & p3 <> p1 & p3 <> p2 )

then consider f being Function of I[01],(() | P) such that
A1: f is being_homeomorphism and
A2: f . 0 = p1 and
A3: f . 1 = p2 by TOPREAL1:def 1;
1 / 2 in [#] I[01] by ;
then A4: 1 / 2 in dom f by ;
then f . (1 / 2) in rng f by FUNCT_1:def 3;
then f . (1 / 2) in the carrier of (() | P) ;
then A5: f . (1 / 2) in P by PRE_TOPC:8;
then reconsider p = f . (1 / 2) as Point of () ;
A6: f is one-to-one by ;
0 in [#] I[01] by ;
then 0 in dom f by ;
then A7: p1 <> p by ;
1 in [#] I[01] by ;
then 1 in dom f by ;
then f . 1 <> f . (1 / 2) by ;
hence ex p3 being Point of () st
( p3 in P & p3 <> p1 & p3 <> p2 ) by A3, A5, A7; :: thesis: verum