A1: I[01] is compact by HEINE:4, TOPMETR:20;
consider P1, P2 being non empty Subset of (TOP-REAL 2) such that
A2: P1 is being_S-P_arc and
A3: P2 is being_S-P_arc and
A4: R^2-unit_square = P1 \/ P2 by TOPREAL1:27;
consider f being Function of I[01],((TOP-REAL 2) | P1) such that
A5: f is being_homeomorphism by A2, TOPREAL1:29;
A6: rng f = [#] ((TOP-REAL 2) | P1) by A5;
consider f0 being Function of I[01],((TOP-REAL 2) | P2) such that
A7: f0 is being_homeomorphism by A3, TOPREAL1:29;
A8: rng f0 = [#] ((TOP-REAL 2) | P2) by A7;
reconsider P2 = P2 as non empty Subset of (TOP-REAL 2) ;
f0 is continuous by A7;
then (TOP-REAL 2) | P2 is compact by A1, A8, COMPTS_1:14;
then A9: P2 is compact by COMPTS_1:3;
reconsider P1 = P1 as non empty Subset of (TOP-REAL 2) ;
f is continuous by A5;
then (TOP-REAL 2) | P1 is compact by A1, A6, COMPTS_1:14;
then P1 is compact by COMPTS_1:3;
hence R^2-unit_square is compact by A4, A9, COMPTS_1:10; :: thesis: verum