reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th57:
  P is being_S-P_arc implies P is compact
proof
A1: I[01] is compact by HEINE:4,TOPMETR:20;
  assume P is being_S-P_arc;
  then reconsider P as being_S-P_arc Subset of TOP-REAL 2;
  consider f being Function of I[01], (TOP-REAL 2)|P such that
A2: f is being_homeomorphism by TOPREAL1:29;
A3: rng f = [#]((TOP-REAL 2)|P) by A2,TOPS_2:def 5;
  f is continuous by A2,TOPS_2:def 5;
  then (TOP-REAL 2)|P is compact by A1,A3,COMPTS_1:14;
  hence thesis by COMPTS_1:3;
end;
