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;
reserve x,y,y1,y2 for set;
reserve G for addGroup;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;
reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty addMagma,
   P, Q, P1, Q1 for Subset of H,
   h for Element of H;
 reserve a for Element of G;

theorem Th18:
  (+a)/" = +(-a)
proof
  set f = +a, g = +(-a);
A1: now
    reconsider h = f as Function;
    let y be object;
    assume y in the carrier of G;
    then reconsider y1 = y as Element of G;
    rng f = the carrier of G by FUNCT_2:def 3;
    then
A2: y1 in rng f;
    dom f = the carrier of G by FUNCT_2:def 1;
    then
A3: g.y1 in dom f & f/".y1 in dom f;
    f.(g.y) = (g.y1)+a by Def2
      .= y1+(-a)+a by Def2
      .= y1+(-a+a) by RLVECT_1:def 3
      .= y1+(0_G) by Def5
      .= y by Def4
      .= h.(h".y) by A2,FUNCT_1:35
      .= f.(f/".y) by TOPS_2:def 4;
    hence f/".y = g.y by A3,FUNCT_1:def 4;
  end;
  thus thesis by A1;
end;
