 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;
reserve G for non empty multMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for Group-like non empty multMagma;
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 Group;
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 Group, H being strict Subgroup of G holds
  Left_Cosets H = {the carrier of G} implies H = G
proof
  let G be strict Group, 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 GROUP_1:6;
  end;
  hence thesis by Th62;
end;
