reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;

theorem Th22:
  v in W implies - v in W
proof
  assume v in W;
  then (- jj) * v in W by Th21;
  hence thesis by RLVECT_1:16;
end;
