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 Th11:
  FreeAtoms(M) = union the set of all [:{i},the carrier of M.i :]
    where i is Element of dom M
proof
  set S=the set of all [:{i},the carrier of M.i :] where i is Element of dom M;
  now
    let z be object;
    hereby
      assume A1: z in FreeAtoms(M);
      then consider i,g being object such that
        A2: z = [i,g] by RELAT_1:def 1;
      reconsider i as Element of dom M by A1, A2, Th7;
      A3: g in the carrier of (M.i) by A1, A2, Th8;
      i in {i} by TARSKI:def 1;
      then A4: z in [:{i},the carrier of M.i :] by A2, A3, ZFMISC_1:def 2;
      [:{i},the carrier of M.i :] in S;
      hence z in union S by A4, TARSKI:def 4;
    end;
    assume z in union S;
    then consider Z being set such that
      A5: z in Z & Z in S by TARSKI:def 4;
    consider i being Element of dom M such that
      A6: Z = [:{i},the carrier of M.i :] by A5;
    consider j,g being object such that
      A7: j in {i} & g in the carrier of M.i & z = [j,g]
      by A5, A6, ZFMISC_1:def 2;
    z = [i,g] by A7, TARSKI:def 1;
    hence z in FreeAtoms(M) by A7, Th8;
  end;
  hence thesis by TARSKI:2;
end;
