reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for addGroup-like non empty addMagma;
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 addGroup;
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 Th147:
  for G being finite addGroup, H being Subgroup of G holds card G =
  card H * index H
proof
  let G be finite addGroup, H be Subgroup of G;
  reconsider C = Left_Cosets H as finite set;
  now
    let X be set;
    assume
A1: X in C;
    then reconsider x = X as Subset of G;
    x is finite;
    then reconsider B = X as finite set;
    take B;
    thus B = X;
    consider a being Element of G such that
A2: x = a + H by A1,Def15;
    ex B,C being finite set st B = a + H & C = H + a & card H = card B &
    card H = card C by Th133;
    hence card B = card H by A2;
    let Y;
    assume that
A3: Y in C and
A4: X <> Y;
    reconsider y = Y as Subset of G by A3;
A5: ex b being Element of G st y = b + H by A3,Def15;
    hence X,Y are_equipotent by A2,Th128;
    thus X misses Y by A2,A4,A5,Th115;
  end;
  then
A6: ex D being finite set st D = union C & card D = card H * card C
    by GROUP_2:156;
  union Left_Cosets H = the carrier of G by Th137;
  hence thesis by A6,Def18;
end;
