reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem Th10:
  for P being Subset of TOP-REAL n, p1,p2 being Point of TOP-REAL n st
  P is_an_arc_of p1,p2 holds P is connected
proof
  let P be Subset of TOP-REAL n, p1,p2 be Point of TOP-REAL n;
  assume
A1: P is_an_arc_of p1,p2;
  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 1;
  reconsider P9 = P as non empty Subset of TOP-REAL n by A1,TOPREAL1:1;
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_1:27,TREAL_1:19;
A5: rng f=[#]((TOP-REAL n)|P) by A2,TOPS_2:def 5;
A6: [#]((TOP-REAL n)|P)=P by PRE_TOPC:def 5;
  dom f=[#]I[01] by A2,TOPS_2:def 5;
  then
A7: P=f.:([#]I[01]) by A5,A6,RELAT_1:113;
  f.:([#]I[01]) is connected by A3,A4,TOPS_2:61;
  hence thesis by A7,CONNSP_1:23;
end;
