 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
  inverse_op(H) = inverse_op(G) | the carrier of H
proof
  the carrier of H c= the carrier of G by Def5;
  then
A1: (the carrier of G) /\ (the carrier of H) = the carrier of H by XBOOLE_1:28;
A2: now
    let x be object;
    assume x in dom(inverse_op(H));
    then reconsider a = x as Element of H;
    reconsider b = a as Element of G by Th42;
    thus inverse_op(H).x = a" by GROUP_1:def 6
      .= b" by Th48
      .= inverse_op(G).x by GROUP_1:def 6;
  end;
  dom(inverse_op(H)) = the carrier of H & dom(inverse_op(G)) = the carrier
  of G by FUNCT_2:def 1;
  hence thesis by A1,A2,FUNCT_1:46;
end;
