 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;
reserve G for non empty multMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;

theorem Th18:
  {g} * {h} = {g * h}
proof
  thus {g} * {h} c= {g * h}
  proof
    let x be object;
    assume x in {g} * {h};
    then consider g1,g2 such that
A1: x = g1 * g2 and
A2: g1 in {g} & g2 in {h};
    g1 = g & g2 = h by A2,TARSKI:def 1;
    hence thesis by A1,TARSKI:def 1;
  end;
  let x be object;
  assume x in {g * h};
  then
A3: x = g * h by TARSKI:def 1;
  g in {g} & h in {h} by TARSKI:def 1;
  hence thesis by A3;
end;
