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 Th142:
  Left_Cosets (Omega).G = {the carrier of G} & Right_Cosets
  (Omega).G = {the carrier of G}
proof
  set a = the Element of G;
A1: Left_Cosets (Omega).G c= {the carrier of G}
  proof
    let x be object;
    assume
A2: x in Left_Cosets (Omega).G;
    then reconsider X = x as Subset of G;
    consider a such that
A3: X = a + (Omega).G by A2,Def15;
    a + (Omega).G = the carrier of G by Th111;
    hence thesis by A3,TARSKI:def 1;
  end;
A4: Right_Cosets (Omega).G c= {the carrier of G}
  proof
    let x be object;
    assume
A5: x in Right_Cosets (Omega).G;
    then reconsider X = x as Subset of G;
    consider a such that
A6: X = (Omega).G + a by A5,Def16;
    (Omega).G + a = the carrier of G by Th111;
    hence thesis by A6,TARSKI:def 1;
  end;
  (Omega). G + a = the carrier of G by Th111;
  then
A7: the carrier of G in Right_Cosets(Omega).G by Def16;
  a + (Omega).G = the carrier of G by Th111;
  then the carrier of G in Left_Cosets (Omega).G by Def15;
  hence thesis by A7,ZFMISC_1:31,A1,A4;
end;
