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;
reserve G for addGroup-like non empty addMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for addGroup;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve x,y,y1,y2 for set;
reserve G for addGroup;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem
  A <> {} implies (0_G) * A = {0_G}
proof
  set y = the Element of A;
  assume
A1: A <> {};
  then reconsider y as Element of G by TARSKI:def 3;
  thus (0_G) * A c= {0_G}
  proof
    let x be object;
    assume x in (0_G) * A;
    then ex a st x = (0_G) * a & a in A by ThB42;
    then x = 0_G by Th17;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {0_G};
  then x = 0_G by TARSKI:def 1;
  then (0_G) * y = x by Th17;
  hence thesis by A1,ThB42;
end;
