 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem
for X being RealUnitarySpace,
    M be Subspace of X,
    x,m1,m2 be Point of X
st
m1 in M & m2 in M
&
( for m be Point of X
       st m in M holds ||.x-m1.|| <= ||.x-m.|| )
 &
( for m be Point of X
       st m in M holds ||.x-m2.|| <= ||.x-m.|| )
holds m1=m2
proof
let X be RealUnitarySpace,
    M be Subspace of X,
    x,m1,m2 be Point of X;
assume that
A1: m1 in M and
A2: m2 in M and
A3:( for m be Point of X
       st m in M holds ||.x-m1.|| <= ||.x-m.|| )
 and
A4:( for m be Point of X
       st m in M holds ||.x-m2.|| <= ||.x-m.|| );
  m1-m2 in M by A1,A2,RUSUB_1:17; then
A5:(x-m1) .|. (m1-m2) = 0 by Lm2,A3,A1;
A6:(x-m1) + (m1-m2) =x-m1 + m1-m2 by RLVECT_1:28
                 .=x-m2 by RLVECT_4:1;
  assume m1 <> m2; then
  m1-m2 <> 0.X by RLVECT_1:21; then
  ||.m1-m2.|| <> 0 by BHSP_1:26; then
  ||.x-m1.|| ^2 + 0 < ||.x-m1.|| ^2 + ||.m1-m2.|| ^2
    by XREAL_1:8,SQUARE_1:12; then
A7: ||.x-m1.|| ^2 < ||.x-m2.|| ^2 by A6,BHSP_5:6,BHSP_1:def 3,A5;
  0 <= ||.x-m2.|| by BHSP_1:28;
  hence contradiction by A7,SQUARE_1:15,A1,A4;
end;
