reserve k,n,i,j for Nat;

theorem Th10:
  for a,b being Element of Group_of_Perm 2 st (ex p being Element
  of Permutations 2 st p=a & p is being_transposition)& (ex q being Element of
  Permutations 2 st q=b & q is being_transposition) holds a*b = <*1,2*>
proof
  let a,b be Element of Group_of_Perm 2;
  assume that
A1: ex p being Element of Permutations 2 st p=a & p is being_transposition and
A2: ex q being Element of Permutations 2 st q=b & q is being_transposition;
  consider p being Element of Permutations 2 such that
A3: p=a and
A4: p is being_transposition by A1;
  the carrier of Group_of_Perm 2 =Permutations 2 by MATRIX_1:def 13;
  then
A5: a*b = <*1,2*> or a*b = <*2,1*> by Th3,TARSKI:def 2;
  reconsider p2=p as FinSequence by A3,A4,Th8;
A6: a= <*2,1*> by A1,Th8;
  then len p2=2 by A3,FINSEQ_1:44;
  then 1 in Seg len p2;
  then
A7: 1 in dom p2 by FINSEQ_1:def 3;
  consider q being Element of Permutations 2 such that
A8: q=b and
A9: q is being_transposition by A2;
  reconsider q2=q as FinSequence by A8,A9,Th8;
  b= <*2,1*> by A2,Th8;
  then
A10: q2.2=1 by A8;
A11: a*b=q*p by A3,A8,MATRIX_1:def 13;
  p2.1=2 by A6,A3;
  then (q*p).1=1 by A10,A7,FUNCT_1:13;
  hence thesis by A5,A11;
end;
