let n be Element of NAT ; :: thesis: for P being Subset of (TOP-REAL n)
for p1, p2 being Point of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
ex p3 being Point of (TOP-REAL n) st
( p3 in P & p3 <> p1 & p3 <> p2 )
let P be Subset of (TOP-REAL n); :: thesis: for p1, p2 being Point of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
ex p3 being Point of (TOP-REAL n) st
( p3 in P & p3 <> p1 & p3 <> p2 )
let p1, p2 be Point of (TOP-REAL n); :: thesis: ( P is_an_arc_of p1,p2 implies ex p3 being Point of (TOP-REAL n) st
( p3 in P & p3 <> p1 & p3 <> p2 ) )
assume P is_an_arc_of p1,p2 ; :: thesis: ex p3 being Point of (TOP-REAL n) st
( p3 in P & p3 <> p1 & p3 <> p2 )
then consider f being Function of I[01] ,((TOP-REAL n) | P) such that
A1: ( f is being_homeomorphism & f . 0 = p1 & f . 1 = p2 ) by TOPREAL1:def 2;
1 / 2 in [#] I[01] by BORSUK_1:83, XXREAL_1:1;
then A2: 1 / 2 in dom f by A1, TOPS_2:def 5;
then f . (1 / 2) in rng f by FUNCT_1:def 5;
then f . (1 / 2) in the carrier of ((TOP-REAL n) | P) ;
then A3: f . (1 / 2) in P by PRE_TOPC:29;
then reconsider p = f . (1 / 2) as Point of (TOP-REAL n) ;
A4: f is one-to-one by A1, TOPS_2:def 5;
0 in [#] I[01] by BORSUK_1:83, XXREAL_1:1;
then 0 in dom f by A1, TOPS_2:def 5;
then A5: p1 <> p by A1, A2, A4, FUNCT_1:def 8;
1 in [#] I[01] by BORSUK_1:83, XXREAL_1:1;
then 1 in dom f by A1, TOPS_2:def 5;
then f . 1 <> f . (1 / 2) by A2, A4, FUNCT_1:def 8;
hence ex p3 being Point of (TOP-REAL n) st
( p3 in P & p3 <> p1 & p3 <> p2 ) by A1, A3, A5; :: thesis: verum