reserve V for non empty RLSStruct;
reserve x,y,y1 for set;
reserve v for VECTOR of V;
reserve a,b for Real;

theorem Th10:
  for V being add-associative right_zeroed right_complementable
  scalar-distributive scalar-unital vector-distributive non empty RLSStruct,
  v being Element of V holds
  a = 0 or v = 0.V implies a * v = 0.V
proof
  let V be add-associative right_zeroed right_complementable
  scalar-distributive scalar-unital vector-distributive non empty RLSStruct,
  v be Element of V;
  assume a = 0 or v = 0.V;
  then per cases;
    suppose
A1:   a = 0;
      v + 0 * v = 1 * v + 0 * v by Def8
        .= (1 + 0) * v by Def6
        .= v + 0.V by Def8;
      hence thesis by A1,Th8;
    end;
    suppose
A2:   v = 0.V;
      a * 0.V + a * 0.V = a * (0.V + 0.V) by Def5
        .= a * 0.V + 0.V;
      hence thesis by A2,Th8;
    end;
end;
