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
  v <> 0.V & z1 * v = z2 * v implies z1 = z2
proof
  assume that
A1: v <> 0.V and
A2: z1 * v = z2 * v;
  0.V = z1 * v - z2 * v by A2,RLVECT_1:15
    .= (z1 - z2) * v by Th10;
  then (- z2) + z1 = 0 by A1,Th2;
  hence thesis;
end;
