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 Th7:
  [x,y] in FreeAtoms(M0) iff x in dom M0 & y in (Carrier M0).x
proof
  hereby
    assume [x,y] in FreeAtoms(M0);
    then x in dom Carrier M0 & y in (Carrier M0).x by WAYBEL26:38;
    hence x in dom M0 & y in (Carrier M0).x by PRALG_1:def 14;
  end;
  assume A1: x in dom M0 & y in (Carrier M0).x;
  then x in dom Carrier M0 by PRALG_1:def 14;
  hence thesis by A1, WAYBEL26:38;
end;
