let P be Subset of (TOP-REAL 2); :: thesis: ( P is being_S-P_arc implies P is compact )
A1: I[01] is compact by HEINE:11, TOPMETR:27;
assume P is being_S-P_arc ; :: thesis: P is compact
then reconsider P = 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 TOPREAL1:36;
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_1:23;
hence P is compact by COMPTS_1:12; :: thesis: verum