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;

theorem
  for H being strict Subgroup of G holds Left_Cosets H = the set of all {a}
 implies H = (0).G
proof
  let H be strict Subgroup of G;
  assume
A1: Left_Cosets H = the set of all {a} ;
A2: the carrier of H c= {0_G}
  proof
    set a = the Element of G;
    let x be object;
    assume x in the carrier of H;
    then reconsider h = x as Element of H;
A3: h in H;
    reconsider h as Element of G by Th41,STRUCT_0:def 5;
    a + H in Left_Cosets H by Def15;
    then consider b such that
A4: a + H = {b} by A1;
    a + h in a + H by A3,Th103;
    then
A5: a + h = b by A4,TARSKI:def 1;
    a + 0_G in a + H by Th46,Th103;
    then a + 0_G = b by A4,TARSKI:def 1;
    then h = 0_G by A5,Th6;
    hence thesis by TARSKI:def 1;
  end;
  0_G in H by Th46;
  then {0_G} = the carrier of H by A2,ZFMISC_1:31;
  hence thesis by Def7;
end;
