reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th36:
  for f,g being ext-real-valued FinSequence holds
  f^g is increasing implies f is increasing & g is increasing
  proof
    let f,g be ext-real-valued FinSequence;
    assume
A1: f^g is increasing;
A2: dom f c= dom(f^g) by FINSEQ_1:26;
    thus f is increasing
    proof
      let e1,e2 be ExtReal;
      assume e1 in dom f;
      then
A3:   e1 in dom (f^g) & f.e1 = (f^g).e1 by A2,FINSEQ_1:def 7;
      assume e2 in dom f;
      then e2 in dom (f^g) & f.e2 = (f^g).e2 by A2,FINSEQ_1:def 7;
      hence thesis by A1,A3,VALUED_0:def 13;
    end;
    thus g is increasing
    proof
      let e1,e2 be ExtReal;
      assume
A4:   e1 in dom g;
      then consider k1 being Nat such that
A5:   e1 = k1 and
A6:   len f + k1 in dom(f^g) by FINSEQ_1:27;
A7:   g.e1 = (f^g).(len f + k1) by A4,A5,FINSEQ_1:def 7;
      assume
A8:   e2 in dom g;
      then consider k2 being Nat such that
A9:   e2 = k2 and
A10:  len f + k2 in dom(f^g) by FINSEQ_1:27;
A11:  g.e2 = (f^g).(len f + k2) by A8,A9,FINSEQ_1:def 7;
      assume e1 < e2;
      then len f + k1 < len f + k2 by A5,A9,XREAL_1:8;
      hence thesis by A1,A6,A7,A10,A11,VALUED_0:def 13;
    end;
  end;
