theorem Th24:
  w is_atlas_of S,G implies for a,b,c holds @(w).(a,b) = c iff w.(
  a,c) = w.(c,b)
proof
  assume
A1: w is_atlas_of S,G;
  let a,b,c;
  thus @(w).(a,b) = c implies w.(a,c) = w.(c,b) by A1,Def9;
  thus w.(a,c) = w.(c,b) implies @(w).(a,b) = c
  proof
    defpred P[Element of S,Element of S,Element of S] means w.($1,$3) = w.($3,
    $2);
    assume
A2: w.(a,c) = w.(c,b);
A3: for a,b,c,c9 st P[a,b,c] & P[a,b,c9] holds c = c9
    proof
      let a,b,c,c9 such that
A4:   ( P[a,b,c])& P[a,b,c9];
      w.(c,c9) = w.(c,a) + w.(a,c9) by A1
        .= w.(c9,b) + w.(b,c) by A1,A4,Th5
        .= w.(c9,c) by A1
        .= -w.(c,c9) by A1,Th4;
      then w.(c,c9) = 0.G by Th16;
      hence thesis by A1,Th3;
    end;
    set c9 = @(w).(a,b);
    P[a,b,c9] by A1,Def9;
    hence thesis by A2,A3;
  end;
end;
