A1: LSeg (|[0,1]|,|[1,1]|) meets LSeg (|[1,0]|,|[1,1]|) by TOPREAL1:18;
A2: LSeg (|[0,0]|,|[1,0]|) meets LSeg (|[1,0]|,|[1,1]|) by TOPREAL1:16;
A3: LSeg (|[0,0]|,|[0,1]|) meets LSeg (|[0,1]|,|[1,1]|) by TOPREAL1:15;
R^2-unit_square = (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[0,0]|)) by TOPREAL1:def 2, XBOOLE_1:4;
hence R^2-unit_square is connected by A3, A2, A1, JORDAN1:5; :: thesis: verum