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 Th37:
  for f,g being ext-real-valued FinSequence holds
  f^g is positive-yielding implies
  f is positive-yielding & g is positive-yielding
  proof
    let f,g be ext-real-valued FinSequence;
    assume
A1: f^g is positive-yielding;
    thus f is positive-yielding
    proof
      let r;
      assume r in rng f;
      then consider x being object such that
A2:   x in dom f and
A3:   f.x = r by FUNCT_1:def 3;
A4:   f.x in rng f by A2,FUNCT_1:def 3;
      rng f c= rng(f^g) by FINSEQ_1:29;
      hence thesis by A1,A3,A4,PARTFUN3:def 1;
    end;
    thus g is positive-yielding
    proof
      let r;
      assume r in rng g;
      then consider x being object such that
A5:   x in dom g and
A6:   g.x = r by FUNCT_1:def 3;
A7:   g.x in rng g by A5,FUNCT_1:def 3;
      rng g c= rng(f^g) by FINSEQ_1:30;
      hence thesis by A1,A6,A7,PARTFUN3:def 1;
    end;
  end;
