theorem
  for H being strict Subgroup of G holds Left_Cosets H is finite & Index
  H = card G implies G is finite & H = (0).G
proof
  let H be strict Subgroup of G;
  assume that
A1: Left_Cosets H is finite and
A2: Index H = card G;
  thus
A3: G is finite by A1,A2;
  ex B being finite set st B = Left_Cosets H & index H = card B by A1,Def18;
  hence thesis by A2,A3,Th154;
end;
