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 Th9:
  for N being multMagma-Family of I
  holds [i,x] in FreeAtoms(N) iff x in the carrier of (N.i)
proof
  let N be multMagma-Family of I;
  consider R being 1-sorted such that
    A1: R = N.i & (Carrier N).i = the carrier of R by PRALG_1:def 15;
  thus [i,x] in FreeAtoms(N) implies x in the carrier of (N.i) by A1, Th7;
  assume A2: x in the carrier of (N.i);
  dom N = I by PARTFUN1:def 2;
  hence thesis by A1, A2, Th7;
end;
