reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:COMPLEX,D:],D;

theorem Th55:
  (0).W = (0).V
proof
  the carrier of (0).W = {0.W} & the carrier of (0).V = {0.V} by Def9;
  then
A1: the carrier of (0).W = the carrier of (0).V by Def8;
  (0).W is Subspace of V by Th46;
  hence thesis by A1,Th49;
end;
