theorem
  Left_Cosets H is finite implies index H = index(H |^ a)
proof
  assume
A1: Left_Cosets H is finite;
  then
A2: ex B being finite set st B = Left_Cosets H & index H = card B by
GROUP_2:def 18;
A3: Index H = Index(H |^ a) by Th71;
  then Left_Cosets H,Left_Cosets(H |^ a) are_equipotent by CARD_1:5;
  then Left_Cosets(H |^ a) is finite by A1,CARD_1:38;
  hence thesis by A2,A3,GROUP_2:def 18;
end;
