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

theorem
for X being RealUnitarySpace,
    M be Subset of X
holds Ort_CompSet(M) is linearly-closed
proof
let X be RealUnitarySpace,
    M be Subset of X;
A1:for v, u being VECTOR of X st
   v in Ort_CompSet(M)
 & u in Ort_CompSet(M) holds
          v + u in Ort_CompSet(M)
proof
  let v, u be VECTOR of X;
  assume A2: v in Ort_CompSet(M)
 & u in Ort_CompSet(M);
now let x be Point of X;
 assume x in M; then
 A3: x.|. v = 0 & x.|. u = 0 by A2,Def1;
 thus x.|. (u+v) = x.|. u + x.|. v by BHSP_1:2
            .=0 by A3;
end;
hence v + u in Ort_CompSet(M) by Def1;
end;

for a being Real
         for v being VECTOR of X st v in Ort_CompSet(M)
       holds a * v in Ort_CompSet(M)
proof
let a be Real, u be VECTOR of X;
  assume A4: u in Ort_CompSet(M);
now let x be Point of X;
 assume A5:x in M;
 thus x.|. (a*u) = a*x.|. u by BHSP_1:3
            .= a*0 by A5,A4,Def1
            .=0;
end;
hence a*u in Ort_CompSet(M) by Def1;
end;
hence thesis by A1,RLSUB_1:def 1;
end;
