theorem
  for G being strict addGroup, H being strict Subgroup of G holds
  Left_Cosets H is finite & index H = 1 implies H = G
proof
  let G be strict addGroup, H be strict Subgroup of G;
  assume that
A1: Left_Cosets H is finite and
A2: index H = 1;
  reconsider B = Left_Cosets H as finite set by A1;
  card B = 1 by A2,Def18;
  then consider x being object such that
A3: Left_Cosets H = {x} by CARD_2:42;
  union {x} = x & union Left_Cosets H = the carrier of G by Th137,ZFMISC_1:25;
  hence thesis by A3,Th143;
end;
