let p be Point of (Tunit_circle 2); :: thesis: Topen_unit_circle p, I(01) are_homeomorphic
set P = Topen_unit_circle p;
set D = Sphere (0. (TOP-REAL 2)),p1;
reconsider p2 = p as Point of (TOP-REAL 2) by PRE_TOPC:55;
(Sphere (0. (TOP-REAL 2)),p1) \ {p} c= Sphere (0. (TOP-REAL 2)),p1 by XBOOLE_1:36;
then reconsider A = (Sphere (0. (TOP-REAL 2)),p1) \ {p} as Subset of (Tcircle (0. (TOP-REAL 2)),1) by Th9;
Topen_unit_circle p = (Tcircle (0. (TOP-REAL 2)),1) | A by Lm14, Th23, EUCLID:58
.= (TOP-REAL 2) | ((Sphere (0. (TOP-REAL 2)),p1) \ {p2}) by GOBOARD9:4 ;
hence Topen_unit_circle p, I(01) are_homeomorphic by Lm14, BORSUK_4:77, EUCLID:58; :: thesis: verum