 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 H being strict Subgroup of G holds Left_Cosets H is finite &
    Index H = card G implies G is finite & H = (1).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;
