reserve x,y,z for object, X for set, I for non empty set, i,j for Element of I,
    M0 for multMagma-yielding Function,
    M for non empty multMagma-yielding Function,
    M1, M2, M3 for non empty multMagma,
    G for Group-like multMagma-Family of I,
    H for Group-like associative multMagma-Family of I;

theorem Th10:
  M0 = {} iff FreeAtoms(M0) = {}
proof
  A1: dom M0 = dom Carrier M0 by PRALG_1:def 14;
  hereby
    assume M0 = {};
    then Carrier M0 = {} by A1;
    then rng disjoin Carrier M0 = {} by CARD_3:3;
    then Union disjoin Carrier M0 = {} by ZFMISC_1:2, CARD_3:def 4;
    hence FreeAtoms(M0) = {};
  end;
  assume A2: FreeAtoms(M0) = {};
  assume A3: M0 <> {};
  then consider i being object such that
    A4: i in dom M0 by XBOOLE_0:def 1;
  reconsider i as Element of dom M0 by A4;
  M0.i in rng M0 by A4, FUNCT_1:3;
  then reconsider R = M0.i as non empty multMagma by GROUP_7:def 1;
  set x = the Element of R;
  x in the carrier of R;
  hence contradiction by A2, A3, Th8;
end;
