reserve X,x for set;
reserve k,m,n for Element of NAT,
  p,q,r,s,r9,s9 for Element of HP-WFF,
  T1,T2 for Tree;
reserve T1,T2 for DecoratedTree;
reserve t,t1 for FinSequence;

theorem Th18:
  p^q = r^s implies p = r & q = s
proof
  assume
A1: p^q = r^s;
  per cases;
  suppose
    len p <= len r;
    then ex t st p^t = r by A1,FINSEQ_1:47;
    hence p = r by Th17;
    hence thesis by A1,FINSEQ_1:33;
  end;
  suppose
    len p >= len r;
    then ex t st r^t = p by A1,FINSEQ_1:47;
    hence p = r by Th17;
    hence thesis by A1,FINSEQ_1:33;
  end;
end;
