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 Th146:
  Left_Cosets H is finite implies (ex B being finite set st B =
Left_Cosets H & index H = card B) & ex C being finite set st C = Right_Cosets H
  & index H = card C
proof
  assume Left_Cosets H is finite;
  then reconsider B = Left_Cosets H as finite set;
  hereby
    take B;
    thus B = Left_Cosets H & index H = card B by Def18;
  end;
  then reconsider C = Right_Cosets H as finite set by Th136,CARD_1:38;
  take C;
  index H = card B & B, C are_equipotent by Def18,Th136;
  hence thesis by CARD_1:5;
end;
