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;

theorem Th20:
  V1 <> {} & V1 is linearly-closed implies 0.V in V1
proof
  assume that
A1: V1 <> {} and
A2: V1 is linearly-closed;
  set x = the Element of V1;
  reconsider x as Element of V by A1,TARSKI:def 3;
  0c * x in V1 by A1,A2;
  hence thesis by Th1;
end;
