reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;

theorem Th26:
  p^r = q^r or r^p = r^q implies p = q
proof
A1: now
    assume
A2: p^r = q^r;
    then len p + len r = len(q^r) by Def3;
    then
A3: len p + len r = len q + len r by Def3;
    for k st k in dom p holds p.k=q.k
    proof
      let k;
      assume
A4:   k in dom p;
      hence p.k=(q^r).k by A2,Def3
        .=q.k by A3,A4,Def3;
    end;
    hence thesis by A3;
  end;
A5: now
    assume
A6: r^p=r^q;
    then
A7: len r + len p = len(r^q) by Def3
      .=len r + len q by Def3;
    for k st k in dom p holds p.k=q.k
    proof
      let k;
      assume
A8:   k in dom p;
      hence p.k = (r^q).(len r + k) by A6,Def3
        .= q.k by A7,A8,Def3;
    end;
    hence thesis by A7;
  end;
  assume p^r = q^r or r^p = r^q;
  hence thesis by A1,A5;
end;
