reserve G for strict Group,
  a,b,x,y,z for Element of G,
  H,K for strict Subgroup of G,
  p for Element of NAT,
  A for Subset of G;
reserve G for Group;
reserve H, B, A for Subgroup of G,
  D for Subgroup of A;

theorem Th19:
  D = A /\ B & G is finite implies index(G,B) >= index(A,D)
proof
  assume that
A1: D = A /\ B and
A2: G is finite;
  reconsider LCB = Left_Cosets B as finite non empty set by A2;
  reconsider LCD = Left_Cosets D as finite non empty set by A2;
A3: now
    let x,y be Element of G;
    let x9,y9 be Element of A such that
A4: x9 = x and
A5: y9 = y;
A6: y9" = y" by A5,GROUP_2:48;
A7: y9"*x9 in A;
    x*B = y*B iff y"*x in B by GROUP_2:114;
    then x*B = y*B iff y9"*x9 in B by A4,A6,GROUP_2:43;
    then x*B = y*B iff y9"*x9 in D by A1,A7,GROUP_2:82;
    hence x*B = y*B iff x9*D = y9*D by GROUP_2:114;
  end;
  defpred P[set, set] means ex a being Element of G, a9 being Element of A st
  a = a9 & $2 = a*B & $1 = a9*D;
A8: for x being Element of LCD ex y being Element of LCB st P[x,y]
  proof
    let x be Element of LCD;
    x in LCD;
    then consider a9 being Element of A such that
A9: x = a9*D by GROUP_2:def 15;
    reconsider a = a9 as Element of G by GROUP_2:42;
    reconsider y = a*B as Element of LCB by GROUP_2:def 15;
    take y,a,a9;
    thus thesis by A9;
  end;
  consider F being Function of LCD, LCB such that
A10: for a being Element of LCD holds P[a,F.a] from FUNCT_2:sch 3(A8);
A11: dom F = LCD by FUNCT_2:def 1;
A12: rng F c= LCB by RELAT_1:def 19;
A13: index D = card LCD by GROUP_2:def 18;
A14: index B = card LCB by GROUP_2:def 18;
  F is one-to-one
  proof
    let x1,x2 be object;
    assume that
A15: x1 in dom F and
A16: x2 in dom F;
    reconsider z1 = x1, z2 = x2 as Element of LCD by A15,A16,FUNCT_2:def 1;
A17: ex a being Element of G, a9 being Element of A st ( a = a9)&
    ( F.z1 = a*B)&( z1 = a9*D) by A10;
    ex b being Element of G, b9 being Element of A st ( b = b9)&
    ( F.z2 = b*B)&( z2 = b9*D) by A10;
    hence thesis by A3,A17;
  end;
  then Segm index D c= Segm index B by A11,A12,A13,A14,CARD_1:10;
  hence thesis by NAT_1:39;
end;
