let A, B be Point of (TOP-REAL 2); ( A <> B implies the_perpendicular_bisector (A,B) is being_line )
assume
A <> B
; the_perpendicular_bisector (A,B) is being_line
then consider L1, L2 being Element of line_of_REAL 2 such that
A1:
the_perpendicular_bisector (A,B) = L2
and
L1 = Line (A,B)
and
A2:
L1 _|_ L2
and
L1 /\ L2 = {(the_midpoint_of_the_segment (A,B))}
by Def2;
thus
the_perpendicular_bisector (A,B) is being_line
by A1, A2, EUCLIDLP:67; verum