 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;

theorem Th14:
  G is uniquely-decomposable iff the multF of G is having_a_unity &
  for a,b being Element of G st a*b = the_unity_wrt the multF of G holds a = b
  & b = the_unity_wrt the multF of G
proof
  thus G is uniquely-decomposable implies op(G) is having_a_unity & for a,b
being Element of G st a*b = the_unity_wrt op(G) holds a = b & b = the_unity_wrt
  the multF of G
  proof
    assume that
A1: op(G) is having_a_unity and
A2: for a,b being Element of G st op(G).(a,b) = the_unity_wrt op(G)
    holds a = b & b = the_unity_wrt op(G);
    thus op(G) is having_a_unity by A1;
    let a,b be Element of G;
    thus thesis by A2;
  end;
  assume that
A3: op(G) is having_a_unity and
A4: for a,b being Element of G st a*b = the_unity_wrt op(G) holds a = b
  & b = the_unity_wrt op(G);
  thus op(G) is having_a_unity by A3;
  let a,b be Element of G;
  a*b = op(G).(a,b);
  hence thesis by A4;
end;
