 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
  ex H1 being strict Subgroup of G st the carrier of H1 = a * H2 * a"
proof
  set A = a * H2 * a";
  set x = the Element of a * H2;
A1: a * H2 <> {} by Th108;
  then reconsider x as Element of G by Lm1;
A2: now
    let g;
    assume g in A;
    then consider g1 such that
A3: g = g1 * a" and
A4: g1 in a * H2 by Th28;
    consider g2 such that
A5: g1 = a * g2 and
A6: g2 in H2 by A4,Th103;
    g2" in H2 by A6,Th51;
    then
A7: g2" * a" in H2 * a" by Th104;
    g" = a"" * (a * g2)" by A3,A5,GROUP_1:17
      .= a * (g2" * a") by GROUP_1:17;
    then g" in a * (H2 * a") by A7,Th27;
    hence g" in A by Th10;
  end;
A8: now
    let g1,g2;
    assume that
A9: g1 in A and
A10: g2 in A;
    consider g such that
A11: g1 = g * a" and
A12: g in a * H2 by A9,Th28;
    consider h being Element of G such that
A13: g = a * h and
A14: h in H2 by A12,Th103;
    A = a * (H2 * a") by Th10;
    then consider b such that
A15: g2 = a * b and
A16: b in H2 * a" by A10,Th27;
    consider c being Element of G such that
A17: b = c * a" and
A18: c in H2 by A16,Th104;
    h * c in H2 by A14,A18,Th50;
    then
A19: a * (h * c) in a * H2 by Th103;
    g1 * g2 = (a * h) * (a" * (a * (c * a"))) by A11,A15,A13,A17,GROUP_1:def 3
      .= (a * h) * (a" * a * (c * a")) by GROUP_1:def 3
      .= (a * h) * (1_G * (c * a")) by GROUP_1:def 5
      .= (a * h) * (c * a") by GROUP_1:def 4
      .= a * h * c * a" by GROUP_1:def 3
      .= a * (h * c) * a" by GROUP_1:def 3;
    hence g1 * g2 in A by A19,Th28;
  end;
  x * a" in A by A1,Th28;
  hence thesis by A8,A2,Th52;
end;
