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
P is connected
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
P is connected
let p1, p2 be Point of (TOP-REAL n); :: thesis: ( P is_an_arc_of p1,p2 implies P is connected )
assume A1: P is_an_arc_of p1,p2 ; :: thesis: P is connected
then consider f being Function of I[01] ,((TOP-REAL n) | P) such that
A2: f is being_homeomorphism and
f . 0 = p1 and
f . 1 = p2 by TOPREAL1:def 2;
reconsider P9 = P as non empty Subset of (TOP-REAL n) by A1, TOPREAL1:4;
A3: f is continuous Function of I[01] ,((TOP-REAL n) | P9) by A2, TOPS_2:def 5;
A4: [#] I[01] is connected by CONNSP_1:28, TREAL_1:22;
A5: rng f = [#] ((TOP-REAL n) | P) by A2, TOPS_2:def 5;
A6: [#] ((TOP-REAL n) | P) = P by PRE_TOPC:def 10;
dom f = [#] I[01] by A2, TOPS_2:def 5;
then A7: P = f .: ([#] I[01] ) by A5, A6, RELAT_1:146;
f .: ([#] I[01] ) is connected by A3, A4, TOPS_2:75;
hence P is connected by A7, CONNSP_1:24; :: thesis: verum