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

theorem Th24:
for X being RealUnitarySpace,
    S,T be non empty Subset of X
 st S c= T
holds
Ort_Comp (T) is Subspace of Ort_Comp(S)
proof
let X be RealUnitarySpace,
    S,T be non empty Subset of X;
assume A1:S c= T;
A2:the carrier of Ort_Comp(S) = Ort_CompSet(S)
 & the carrier of Ort_Comp(T) = Ort_CompSet(T) by Lm5;
Ort_CompSet(T) c= Ort_CompSet(S)
proof let x0 be object;
  assume A3:x0 in Ort_CompSet(T); then
   reconsider x=x0 as Point of X;
  for y be Point of X st y in S
  holds y .|. x = 0 by A1,A3,Def1;
  hence x0 in Ort_CompSet(S) by Def1;
end;
hence thesis by RUSUB_1:22,A2;
end;
