
theorem Th31:
  for I being non empty set for J being Poset-yielding non-Empty
ManySortedSet of I for X being Subset of product J holds ex_inf_of X, product J
  iff for i being Element of I holds ex_inf_of pi(X,i), J.i
proof
  let I be non empty set;
  let J be Poset-yielding non-Empty ManySortedSet of I;
  let X be Subset of product J;
  hereby
    set f = inf X;
    assume
A1: ex_inf_of X, product J;
    let i be Element of I;
A2: f is_<=_than X by A1,YELLOW_0:31;
A3: now
      let x be Element of J.i;
      set g = f+*(i,x);
A4:   dom g = dom f by FUNCT_7:30;
      dom f = I by WAYBEL_3:27;
      then
A5:   g.i = x by FUNCT_7:31;
      now
        let j be Element of I;
        g.j = f.j or g.j = x & j = i by A5,FUNCT_7:32;
        hence g.j is Element of J.j;
      end;
      then reconsider g as Element of product J by A4,WAYBEL_3:27;
      assume
A6:   pi(X,i) is_>=_than x;
      X is_>=_than g
      proof
        let h be Element of product J;
        assume
A7:     h in X;
        then
A8:     h.i in pi(X, i) by CARD_3:def 6;
A9:     h >= f by A2,A7;
        now
          let j be Element of I;
          g.j = f.j or g.j = x & j = i by A5,FUNCT_7:32;
          hence h.j >= g.j by A6,A9,A8,WAYBEL_3:28;
        end;
        hence thesis by WAYBEL_3:28;
      end;
      then f >= g by A1,YELLOW_0:31;
      hence f.i >= x by A5,WAYBEL_3:28;
    end;
    f.i is_<=_than pi(X, i) by A2,Th29;
    hence ex_inf_of pi(X, i), J.i by A3,YELLOW_0:31;
  end;
  assume for i being Element of I holds ex_inf_of pi(X,i), J.i;
  then ex f being Element of product J st ( for i being Element of I holds f.i
  = inf pi(X,i))& f is_<=_than X & for g being Element of product J st X
  is_>=_than g holds f >= g by Lm2;
  hence thesis by YELLOW_0:31;
end;
