 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;
reserve G for non empty multMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for Group-like non empty multMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for Group;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;

theorem
  for G being strict Group, H being strict Subgroup of G holds
  Left_Cosets H is finite & index H = 1 implies H = G
proof
  let G be strict Group, 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;
