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;
