
theorem LmEMDetX52:
  for L, E being Z_Module, I being Subset of L, J be Subset of E
  st the ModuleStr of L = the ModuleStr of E & I = J
  holds Lin I = Lin J
  proof
    let L, E be Z_Module, I be Subset of L, J be Subset of E;
    assume that
    A1: the ModuleStr of L = the ModuleStr of E and
    A2: I = J;
    L is Submodule of E by A1,LmEMDetX53;
    hence Lin I = Lin J by A2,ZMODUL03:20;
  end;
