let n be 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

p1 <> 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

p1 <> p2

let p1, p2 be Point of (TOP-REAL n); :: thesis: ( P is_an_arc_of p1,p2 implies p1 <> p2 )

assume P is_an_arc_of p1,p2 ; :: thesis: p1 <> p2

then consider f being Function of I[01],((TOP-REAL n) | P) such that

A1: f is being_homeomorphism and

A2: f . 0 = p1 and

A3: f . 1 = p2 by TOPREAL1:def 1;

1 in [#] I[01] by BORSUK_1:40, XXREAL_1:1;

then A4: 1 in dom f by A1, TOPS_2:def 5;

A5: f is one-to-one by A1, TOPS_2:def 5;

0 in [#] I[01] by BORSUK_1:40, XXREAL_1:1;

then 0 in dom f by A1, TOPS_2:def 5;

hence p1 <> p2 by A2, A3, A4, A5, FUNCT_1:def 4; :: thesis: verum

for p1, p2 being Point of (TOP-REAL n) st P is_an_arc_of p1,p2 holds

p1 <> 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

p1 <> p2

let p1, p2 be Point of (TOP-REAL n); :: thesis: ( P is_an_arc_of p1,p2 implies p1 <> p2 )

assume P is_an_arc_of p1,p2 ; :: thesis: p1 <> p2

then consider f being Function of I[01],((TOP-REAL n) | P) such that

A1: f is being_homeomorphism and

A2: f . 0 = p1 and

A3: f . 1 = p2 by TOPREAL1:def 1;

1 in [#] I[01] by BORSUK_1:40, XXREAL_1:1;

then A4: 1 in dom f by A1, TOPS_2:def 5;

A5: f is one-to-one by A1, TOPS_2:def 5;

0 in [#] I[01] by BORSUK_1:40, XXREAL_1:1;

then 0 in dom f by A1, TOPS_2:def 5;

hence p1 <> p2 by A2, A3, A4, A5, FUNCT_1:def 4; :: thesis: verum