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 Th143:
  for G being strict addGroup, H being strict Subgroup of G holds
  Left_Cosets H = {the carrier of G} implies H = G
proof
  let G be strict addGroup, H be strict Subgroup of G;
  assume Left_Cosets H = {the carrier of G};
  then
A1: the carrier of G in Left_Cosets H by TARSKI:def 1;
  then reconsider T = the carrier of G as Subset of G;
  consider a being Element of G such that
A2: T = a + H by A1,Def15;
  now
    let g be Element of G;
    ex b being Element of G st a + g = a + b & b in H by A2,Th103;
    hence g in H by Th6;
  end;
  hence thesis by Th62;
end;
