 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;

theorem LmTriv:
  for M being strict multMagma
  st (ex x being object st the carrier of M = {x})
  ex G being strict trivial Group st M = G
proof
  let M be strict multMagma;
  given x being object such that
  A1: the carrier of M = {x};
  reconsider M as non empty multMagma by A1;
  reconsider x as Element of M by A1, TARSKI:def 1;
  A2: for a,b,c being Element of M holds (a * b) * c = a * (b * c)
  proof
    let a,b,c be Element of M;
    (a * b) * c = x & a * (b * c) = x by A1, TARSKI:def 1;
    hence (a * b) * c = a * (b * c);
  end;
  ex e being Element of M st
  for h being Element of M holds
  (h * e = h & e * h = h & ex g being Element of M st (h * g = e & g * h = e))
  proof
    take e = x;
    let h be Element of M;
    h = x by A1, TARSKI:def 1;
    hence h * e = h & e * h = h by A1, TARSKI:def 1;
    take g = x;
    thus thesis by A1, TARSKI:def 1;
  end;
  then reconsider G=M as strict trivial Group by A1, A2, GROUP_1:def 2,def 3;
  take G;
  thus thesis;
end;
