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 Th127:
  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;
A7: -g2 + -a in H2 + -a by A6,Th51,Th104;
    -g = -(g1 + -a) by A3
      .= --a + -(a + g2) by A5,Th16
      .= a + (-g2 + -a) by Th16;
    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;
A19: a + (h + c) in a + H2 by A14,A18,Th50,Th103;
    g1 + g2 = (a + h) + (-a + (a + (c + -a))) by A11,A15,A13,A17,RLVECT_1:def 3
      .= (a + h) + (-a + a + (c + -a)) by RLVECT_1:def 3
      .= (a + h) + (0_G + (c + -a)) by Def5
      .= (a + h) + (c + -a) by Def4
      .= a + h + c + -a by RLVECT_1:def 3
      .= a + (h + c) + -a by RLVECT_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;
