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 Def18;
A3: Index H = Index(H * a) by Th71;
  then Left_Cosets(H * a) is finite by A1;
  hence thesis by A2,A3,Def18;
end;
