reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem Th17:
  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) = a*(g.y1) by Def1
      .= a*(a"*y1) by Def1
      .= a*a"*y1 by GROUP_1:def 3
      .= 1_G*y1 by GROUP_1:def 5
      .= y by GROUP_1:def 4
      .= 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;
