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 Th27:
  x in g + A iff ex h st x = g + h & h in A
proof
  thus x in g + A implies ex h st x = g + h & h in A
  proof
    assume x in g + A;
    then consider g1,g2 such that
A1: x = g1 + g2 and
A2: g1 in {g} and
A3: g2 in A;
    g1 = g by A2,TARSKI:def 1;
    hence thesis by A1,A3;
  end;
  given h such that
A4: x = g + h & h in A;
  g in {g} by TARSKI:def 1;
  hence thesis by A4;
end;
