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;
reserve B,C for Coset of W;

theorem Th61:
  0.V in v + W iff v in W
proof
  thus 0.V in v + W implies v in W
  proof
    assume 0.V in v + W;
    then consider u such that
A1: 0.V = v + u and
A2: u in W;
    v = - u by A1,RLVECT_1:def 10;
    hence thesis by A2,Th41;
  end;
  assume v in W;
  then
A3: - v in W by Th41;
  0.V = v - v by RLVECT_1:15
    .= v + (- v);
  hence thesis by A3;
end;
