 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 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 the carrier of G in Right_Cosets(Omega).G by Def16;
  then
A7: {the carrier of G} c= Right_Cosets(Omega).G by ZFMISC_1:31;
  a * (Omega).G = the carrier of G by Th111;
  then the carrier of G in Left_Cosets (Omega).G by Def15;
  then {the carrier of G} c= Left_Cosets (Omega).G by ZFMISC_1:31;
  hence thesis by A7,A1,A4;
end;
