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;

theorem Th1:
  z = 0 or v = 0.V implies z * v = 0.V
proof
  assume
A1: z = 0 or v = 0.V;
  now
    per cases by A1;
    suppose
A2:   z = 0c;
      v + 0c * v = 1r * v + 0c * v by Def5
        .= (1r + 0c) * v by Def3
        .= v by Def5
        .= v + 0.V by RLVECT_1:4;
      hence thesis by A2,RLVECT_1:8;
    end;
    suppose
A3:   v = 0.V;
      z * 0.V + z * 0.V = z * (0.V + 0.V) by Def2
        .= z * 0.V by RLVECT_1:4
        .= z * 0.V + 0.V by RLVECT_1:4;
      hence thesis by A3,RLVECT_1:8;
    end;
  end;
  hence thesis;
end;
