theorem Th8:
  for P be Permutation of Seg n st P is being_transposition for i,j
st i < j holds P.i = j iff i in dom P & j in dom P & P.i=j & P.j=i & for k st k
  <> i & k <> j & k in dom P holds P.k = k
proof
  let P;
  assume P is being_transposition;
  then consider i9,j9 be Nat such that
  i9 in dom P and
  j9 in dom P and
  i9<>j9 and
A1: P.i9=j9 and
A2: P.j9=i9 and
A3: for k st k <>i9 & k<>j9 & k in dom P holds P.k=k;
  let i,j such that
A4: i < j;
  thus P.i=j implies (i in dom P & j in dom P & P.i=j & P.j=i & for k st k<>i
  & k<>j & k in dom P holds P.k = k)
  proof
A5: dom P=Seg n by FUNCT_2:52;
A6: rng P=Seg n by FUNCT_2:def 3;
    assume
A7: P.i=j;
    then
A8: i in dom P by A4,FUNCT_1:def 2;
    then i=j9 or i=i9 by A4,A3,A7;
    hence thesis by A1,A2,A3,A7,A8,A6,A5,FUNCT_1:def 3;
  end;
  thus thesis;
end;
