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 Th8:
  for i being Element of dom M
  holds [i,x] in FreeAtoms(M) iff x in the carrier of (M.i)
proof
  let i be Element of dom M;
  consider R being 1-sorted such that
    A1: R = M.i & (Carrier M).i = the carrier of R by PRALG_1:def 14;
  thus [i,x] in FreeAtoms(M) implies x in the carrier of (M.i) by A1, Th7;
  assume x in the carrier of (M.i);
  hence thesis by A1, Th7;
end;
