reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  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;
